\documentstyle[12pt,a4]{report}
\begin{document}
 
\begin{titlepage}
   \begin{center}
	\Large
	{\bf Analysis of the Motion\\ 
	of a Bipolar Motor }

	\vspace*{6cm}

	\Large
	Neville R. Jones

	\vspace*{4cm}
	
	\normalsize
	A thesis submitted in partial fulfillment of the requirements \\
	for the degree of Bachelor of Science with Honours.

	\vspace*{3cm}
	\large
	The University of Tasmania\\
	Department of Physics

	\vspace*{2cm}
	November 16, 1992
   \end{center}
\end{titlepage} 

\begin{abstract}
Chaos theory is reshaping ideas about the behaviour of seemingly simple
systems. An experimental device of a dipole free to rotate in a sinusoidally
varying magnetic field has been utilized to study this field. The time
evolution of the system is governed by a second-order nonlinear differential
equation. Despite the ease with which the equation is derived,
full analytical solutions are impossible. 

The experimental system exhibits a rich variety of behaviour including
periodic, period-doubled and chaotic. Support equipment permits
detailed study of the dynamics from which numerous results have been derived.
The transition points between the types of motion have been accurately
determined, and a uniform band structure has been recognised where periodic
solutions are separated by bands of chaotic behaviour. Universal features of
chaotic systems have also been observed. The experimental results are confirmed
by numerical simulation of the system.

The diversity of interesting behaviour the apparatus displays and its ability 
to produce replicable results make it ideal for use in the undergraduate
experimental laboratory.

\end{abstract}
\begin{center}
   \Large
	\subsubsection{Acknowledgments}
\end{center}
\normalsize
I would like to take the opportunity to thank the following people:
   \begin{itemize}
	\item   Professor R. Delbourgo, for offering the project which was
		ideally suited for my need to compromise between theoretical
		and experimental physics. His encouragement, assistance and
		advice throughout the year was much appreciated.
	\item   Mr K. Bolton \& Mr B. Wilson, whose technical and mechanical
		expertise were constantly depended upon. Their knowledge,
		patience and generosity enabled me to overcome many problems
		associated with the device.
	\item   Miss P. Coleman \& Mrs H. Stock, for their help with numerous
		matters, particularly with using \LaTeX . 
	\item   Mr K. Parker \& Mr D. Dainton, for supplying much of the
		necessary equipment. 
	\item   Dr B. Watson, for loan of a precise signal generator belonging
		to the optical astronomy group.
	\item   Dr R. Warner, Dr M. Yung \& Tony Waites, for helping with various
		aspects of computing.
	\item   Mr F. Koolhof, for photographically producing several incremental
		encoder wheels.
	\item   My brothers, for the good times. 
	\item   My parents, who have always offered support free from pressure 
		to do well.
	\item   Suellen, for her understanding. 
	\item   Matthew Roberts, for company and friendship throughout the 
		year, and for making the Physics Honours group of 1992 greater 
		than one. 
   \end{itemize}
All of these people and more played an important part in enabling me to
complete the Honours year. 

\tableofcontents




\chapter{Introduction}

The study of dynamical systems, that is systems which change in space and time,
has greatly increased in the last few decades. This is singularly attributable
to the development of {\em chaos theory}.

The growth in popularity is due to the numerous limitations of Newtonian
mechanics. Those who are only familiar with this mechanics believe that
dynamical systems run along a predictable and regular course, ending up in some
periodic and steady state. Unfortunately this popular misconception leads one
to believe that the behaviour of a dynamical system is entirely predictable
provided the initial conditions and the equations which govern its evolution
are known. 

At the turn of the century Poincar\'{e} challenged this notion by showing the
time evolution of the classical three-body problem of celestial mechanics was
highly sensitive to initial conditions. In times since many other simple
systems have been found which have very complicated motions. Their behaviour is
characterised by extreme sensitivity to the precise starting conditions and the
inability to settle into any reasonable final state with a recognised fixed
pattern. This phenomenon goes under the name of chaos, and perfect
predictability can only be achieved if the initial conditions are known and
specified with infinite precision. Any errors grow exponentially, resulting in
irregular and random long-time behaviour of the system.

Extensive theoretical work into chaos has only been carried out over the past
few decades.  Before the development of chaos theory the irregular behaviour of
many physical systems was inexplicable by conventional means. Numerous cases of
it had been encountered by scientists and engineers over the last century only
to be dismissed as noise. It is such a wide spread phenomenon that it has been
reported in almost every scientific discipline: astronomy, biology, chemistry,
engineering, mathematics, medicine, meteorology, the social sciences and physics
(\cite{bib-Gle,bib-Hol}).

Systems in which chaotic behaviour occurs are described by nonlinear
equations. The mathematical tools which have been developed over the last few
centuries are very limited to the type of equations for which they can be used.
Analytical solutions of nonlinear equations are difficult to obtain, whereas
linear systems are most often studied because solutions are easily derived.
Subsequently linear systems show no surprises, the solution is known so that
unexpected behaviour does not occur.  With nonlinearity a necessary condition
for chaotic behaviour, the real world, dominated by nonlinearities, is expected
to show many ramifications of chaos. It is only our basic education which makes
us believe the physical world is easily described in terms of predictable
linear equations. 

To date the most significant achievement of chaos theory is in the
improved understanding of turbulent fluid flow. Despite the ease with which
turbulent fluid flow can be obtained, and thus experimentally investigated, it
stands as one of the oldest problems in classical physics. Only in the last
twenty years, with chaos theory as a guiding light, has a more comprehensive
account been derived.

Since chaos theory goes a long way in explaining many other problems which have
plagued science for centuries, it has generated an interest matched only by the
revolutionary theories of relativity and quantum mechanics. 
Interestingly, the main analytical advances in the study of
nonlinear systems were made by workers familiar with techniques of quantum
mechanics. Their mathematical tools are better equipped to deal with difficult
equations, with further study into chaotic systems refining these methods to an
even greater extent. Similarly, a whole new geometry has been created with
chaos, in which objects have an intermediate dimension, no longer specified by
an integer number. These mathematical objects, called {\em fractals}, are the
most complicated strutures ever encountered. Surprisingly, fractals can be
generated by simple relations, in much the same way that chaos can occur in
comparatively short, simple looking equations.

The most celebrated discovery of chaos is that unrelated systems can behave in
very similar manners. The transition from orderly, periodic behaviour to
chaotic motion usually occurs as some {\em control} parameter of the system is
changed. Importantly, this transition follows a robust {\em route} at a
specific rate, independent of the governing equations. Establishing such {\em
universal} properties in many classes of systems has been the goal of many
workers. Since theoretical foundations for the occurrence of universal aspects
in chaotic systems are just over ten years old, there are many systems yet to
be studied.

\subsubsection{Importance}
A system which has received some degree of attention is the {\em bipolar
motor}, consisting of a permanent magnet free to rotate in a harmonically
varying magnetic field (\cite{bib-Bal,bib-Cro,bib-Mei,bib-Tis}). This simple
mechanical apparatus has many advantages over other devices which demonstrate
chaos. Its mechanical nature makes it attractive for use in the undergraduate
laboratory where an introduction to nonlinear dynamics at this level is sorely
overdue. It gives the user a greater understanding and intuitive feel of key
aspects of chaotic behaviour than that gained by the study of the more popular
electrical circuits. 

The apparatus, originally constructed by Tisch (1989), has been extensively
tested and improved. It is now ready for use as a third year experiment into
chaos, and can also be used as a lecture demonstration, highlighting the
diversity of applications of the device.

Although a reasonable undergraduate experiment involving the apparatus has been
formulated, this is not to say all the dynamical features of the system have
been investigated. The device shows such a variety of behaviour that further
advanced study into the dynamics will hopefully be undertaken. Interestingly,
the equation of motion governing the behaviour of the magnet, although easily
derived, is particularly hard to investigate analytically. Despite many
advances which have been made in obtaining solutions of nonlinear equations,
apparently no technique exists which is capable of unravelling the mysteries of
the bipolar motor.

\subsubsection{Thesis Structure}
Since the apparatus is intended for use in the undergraduate laboratory a
comprehensive background of chaos theory has been given in
Chapter~\ref{chpt-genthy}.  Chapter~\ref{chpt-specthy} goes on to look at the
theoretical aspects which are specific to the experimental system. The exact
form and operational characteristics of the setup are detailed in
Chapter~\ref{chpt-expset}, followed by description of a well determined
experimental method in Chapter~\ref{chpt-expmeth}. This is invaluable to future
users of the device, enabling collection of excellent data in the minimum of
time. The ability of the device to produce quality data is shown in
Chapter~\ref{chpt-expres}  where the experimental results obtained by the
author are given.  Chapter~\ref{chpt-numres}  describes the numerical modelling
of the system which has been performed, showing excellent agreement with the
experimental results. Both numerical and experimental results are discussed
and compared with those of other workers in Chapter~\ref{chpt-Discussion},
while Chapter~\ref{chpt-conclus} is reserved for concluding remarks. Several
appendices containing experimental data are included at the end.



\chapter{General Theory}
\label{chpt-genthy}

Chaotic phenomena can occur in a variety of systems. These may be in the form
of iterative mappings, where a function is recursively used to evaluate newer
values. Such mappings are very common in mathematics and practical situations.
Mathematicians often use recursive relations to numerically find solutions 
which are analytically difficult.  For example numerous recursion algorithms 
exist which are useful for finding the zeros of a complex function, such as 
the Newton-Raphson Process and the Secant Method to name a few.
Another simple example occurs in the modelling of population dynamics.
Clearly the size of the population at a given time is an important factor in
determining the future population size. So any estimates of population size use
the previous value in the evaluation. Under certain conditions such models are 
known to exhibit chaotic behaviour.

For the rest of the thesis, nonlinear differential equations shall be
considered, as the occurrence of chaotic phenomena in such equations is
relevant to the experimental situation that has been investigated.

\section{Differential Equations}
\label{sec-diffeqs}
The time evolution of many dynamical systems is governed by a single, or set of
$d$-dimensional differential equations.  These equations are typically of the
form
\begin{equation}
 \frac{dx_{i}(t)}{dt} \stackrel{\rm def}{=} \dot{x}_{i}(t) = 
		       f_{i,\mu}(x_{1}(t) , \ldots , x_{d}(t)) \label{eq:1ode} 
\end{equation}
where \( i=1, \ldots ,d \) and $ \mu $ refers to a {\em control parameter} of
the system (for example the intensity of the force used in a driven pendulum).
Equation~\ref{eq:1ode} is a coupled set of first order differential equations. 
Because differential equations of any order can be re-expressed in terms of 
coupled first order differential equations, this is the general 
form for the differential evolution equation of a dynamical system.

   A solution of Equation~\ref{eq:1ode} is called a trajectory of the system. 
Clearly this solution depends on the initial values of the \( x_{i} \)'s.  
Such dependence is known as the initial value problem of differential 
equations. With known values of the \( x_{i} \) at a certain time, \( t=0 \) 
say, then a unique solution does exist. Such  a solution is written as
\begin{equation}
   x_{i}(t) = \phi_{i, \mu}(x_{1}(0), \ldots ,x_{d}(0),t)  \label{eq:sol}
\end{equation}  
where 
\begin{equation}
   \frac{d}{dt}  \phi_{i,\mu} = f_{i,\mu}                   \label{eq:flow}
\end{equation}
  To simplify the equations, and to enable some physical intuition into what
the trajectory is, one can express the \( x_{i}, f_{i} \) and \( \phi_{i} \) as
vectors. Thereby one allocates
\begin{equation}
   {\bf x}           = \left( \begin{array}{c}
				 x_{1} \\  \vdots  \\  x_{d}
			      \end{array}                      \right)
   , \: {\bf f}_{\mu}     = \left( \begin{array}{c}
				 f_{1,\mu} \\  \vdots  \\  f_{d,\mu}
			      \end{array}                      \right)
   , \: \mbox{\boldmath $\phi$}_{\mu} = \left( \begin{array}{c}
				 \phi_{1,\mu}  \\  \vdots  \\  \phi_{d,\mu}
			      \end{array}                      \right)
\end{equation} 

In this vector notation~\ref{eq:1ode} becomes
\[ \dot{{\bf x}} = {\bf f}_{\mu}({\bf x}(t))   \]
Now if one takes the $ d $-dimensional vector $ {\bf x} $ at a time $ t $, it
can be thought of as a point in a $ d $-dimensional space called {\em phase
space}. Equation~\ref{eq:sol} describes the time dependence of the position of the
point $ {\bf x} $ in the phase space. In vector notation~\ref{eq:sol} is
equivalent to 
\begin{equation}
   {\bf x} (t) = \mbox{\boldmath $\phi$}_{\mu} ( {\bf x} (0) )
\end{equation}
where $ \mbox{\boldmath $\phi$}_{\mu} $ is called the {\em time evolution 
operator} or {\em flow} of the system. Thus under the action of this operator, 
the initial point ${\bf x}(0)$ moves through phase space, tracing out a 
curve as time goes by. This curve is precisely the trajectory of the system, 
its name now becoming obvious. 

Since the trajectory is a (smooth) curve in phase space, tangents can be taken
at any point $ {\bf x} $ in time. The tangent at the point $ {\bf x} (t) $ is
given by 
\begin{equation}
   \dot{{\bf x}} (t) = \left(  \begin{array}{c}
				  \dot{x_{1}}  \\  \vdots  \\ \dot{x}_{d}
			       \end{array}                           \right)
\end{equation}
However, from~\ref{eq:1ode} 
\begin{equation}
   \left(    \begin{array}{c}
		\dot{x}_{1}  \\ \vdots \\ \dot{x}_{d}
	     \end{array}                           \right)
  = \left(   \begin{array}{c}
		f_{1, \mu}  \\  \vdots \\ f_{d, \mu}
	     \end{array}                          \right)
 \equiv {\bf f}_{\mu}
\end{equation}
Thus $ {\bf f}_{\mu} $ is the {\em tangent} or {\em vector field}.
From Equation~\ref{eq:flow}, we see the tangent field $ {\bf f}_{\mu} $ is related to
the flow field $ \mbox{\boldmath $\phi$}_{\mu} $ through the equation:
\begin{equation}
   \frac {d}{dt} \mbox{\boldmath $\phi$}_{\mu} = {\bf f}_{\mu}
\end{equation}

It is suitable now to introduce another definition: Systems of the form
\ref{eq:1ode} are termed {\em autonomous}, as the vector field $ {\bf f}_{\mu}
$ does not contain time explicitly. The value of $x_{i}$ are time dependent, 
but this only means that the position of the representative point, ${\bf x}$, 
in phase space changes. Thus an autonomous vector/tangent field is static in 
time, and the observed flow of the representative  point is manifested by this 
vector field.  

A {\em nonautonomous} differential equation is one in which the vector field
has an explicit time dependence. Such equations differ from~\ref{eq:1ode} in
that time appears as an explicit parameter of ${\bf f}_{\mu}$, so that
\begin{equation}
   \dot{{\bf x}}(t) = {\bf f}_{\mu}({\bf x}(t),t)         \label{eq:nonauto}
\end{equation}
Equations of the form~\ref{eq:nonauto} are more difficult to solve as time
translations of the solution are not necessarily themselves solutions of the
equation (as is the case for autonomous differential equations).  However, all
is not lost, as nonautonomous equations can be expressed as autonomous
equations by redefining time as a new {\em dependent} variable. Such a
convenient simplification results in enlargement of phase space, time becoming
an extra coordinate, but this drawback is negligible, especially as shall be
seen in the case of nonautonomous equations with periodic time dependence.
%example of change of variables

In summary, a dynamical system can be thought of as a flow on a differentiable
space (termed the {\em manifold}) arising from the influence of a vector field.

\section{Volume Elements}
\label{sec-vol_els}

In the introduction, it was emphasized how the behaviour of a dynamical system
may be inherently unpredictable. In such systems, accurate long term
predictions could only be made if all conditions are known to infinite
precision. To see how such unpredictable motion can arise, one considers a
small, finite volume element 
\[ \delta  V = \delta  x_{1}  \ldots  \delta x_{d} \]
in phase space. 

The $ \delta x_{i} $ can be thought of as specifying either the `width' of the
$ i^{th} $ coordinate in phase space, or in the uncertainty associated with 
the measurement of $x_{i}$ . (Such uncertainty arises from the Uncertainty 
Principle of Heisenberg at the quantum level, or in practice from the limited
precision of measurement devices.) 
Thus at time $ t $ we have, in short-hand notation:
\begin{equation}
   \delta V(t) = \prod_{i=1}^{d}  \delta x_{i}(t)    \label{eq:vol}
\end{equation}
From \ref{eq:1ode} and use of the chain rule, the temporal evolution of the
uncertainties, $ x_{i} $, can be ascertained:
\begin{equation}
 \frac {d}{dt}  \delta  x_{i}(t) = \sum_{j=1}^{d} \frac{ \partial
				     f_{i,\mu} ( {\bf x} (t) ) }{ \partial
				       x_{j}}   \delta x_{j}(t)  
\end{equation}
the coefficients $ \frac{ \partial f_{i,\mu} ( {\bf x} (t)) }{ \partial x_{j} }
$ form a matrix $ L_{ij} ( {\bf x} (t) ) $. In the abbreviated notation:
\begin{equation}  
   L ( {\bf x} (t) ) = \frac { \partial {\bf f}_{\mu} ( {\bf x} (t)) }{
			\partial {\bf x} }  \label{eq:jacL}  
\end{equation}                        
 
With this, the volume element at time $ t $ can be related to the volume
element at some earlier time:
\begin{equation}
   \delta V(t) =  \det J({\bf x}(t)) \: \delta V(0)          \label{eq:dVdt}
\end{equation}
where $\det J$ is called the Jacobian. $ J({\bf x}(t)) $ is the {\em Jacobi
matrix} of $ {\bf x} $, with elements defined by: 
\begin{equation}
   J_{ij} ( {\bf x} (t) ) = \frac { \partial x_{i} (t) }{ \partial x_{j} (0) }
						       \label{eq:jacobi} 
\end{equation}
It follows that the matrix $ L({\bf x}(t)) $ is in fact the Jacobian matrix of
the vector field $ {\bf f}_{\mu} $ at $ {\bf x} $ at time $ t $. 

To quantify the rate of change of the volume element with time,  $ \delta
\dot{V} (t) $, one must first investigate the rate of change of the Jacobian
with time. Using definition~\ref{eq:jacobi} McCauley (1988) shows
\begin{equation}
   \det \dot{J} ( {\bf x} (t)) = \sum_{i=1}^{d} \frac{ \partial
				  \dot{x}_{i}(t)}{ \partial x_{i} (t) }
				  \det J( {\bf x} (t) )
						      \label{eq:djacdt}  
\end{equation}
But from~\ref{eq:1ode}
\begin{equation}
   \sum_{i=1}^{d} \frac{ \partial \dot{x}_{i} (t) }{ \partial x_{i} (t) }
   =    \sum_{i=1}^{d} \frac{ \partial f_{i,\mu} ({\bf x} (t)) }{ \partial
	x_{i} (t) }
   \equiv \nabla \cdot {\bf f}_{\mu}( {\bf x} (t) )
\end{equation}
where $ \nabla \cdot {\bf f} $ is the divergence of the vector field $ {\bf f}
$ .Thus
   \begin{equation}
	 \det \dot{J} ( {\bf x} (t)) = \nabla \cdot {\bf f}_{\mu}( {\bf x} (t))
					\det J ({\bf x} (t))
						\label{eq:dotJ,divf}
   \end{equation} 

Hence, combining these results \ref{eq:dVdt} becomes
\begin{eqnarray}
   \delta \dot{V} (t) & = & \det \dot{J}({\bf x}(t)) \: \delta V(0) \nonumber \\
		      & = & \nabla \cdot {\bf f}_{\mu}({\bf x}(t))   \:
			      \det J({\bf x}(t))  \: \delta V(0)     \nonumber \\
		      & = & \nabla \cdot {\bf f}_{\mu}({\bf x}(t)) \: \delta V(t)
			     \label{eq:divf} 
\end{eqnarray}
\ref{eq:divf} signifies that the rate of change in the volume element with
time is related to the outward flux of the vector field locally passing through
the volume element at that time.  Thus the number and location of sources and
sinks of $ {\bf f}_{\mu} $ determines the rate of change of the volume element.

Such an understanding enables one to differentiate between conservative and
dissipative behaviour in dynamical systems.

\subsection{Conservative Systems}
\label{subsec-cons}

Conservative systems are such that there is no energy loss in the system.
Subsequently it is not necessary to provide energy to the system to maintain
its evolution. In practice this means there are no sources or sinks of the
vector field, and thus the divergence of the vector field is zero. This is
easily confirmed in the Hamiltonian equations of motion as shown below. 

\subsubsection{Hamiltonian Systems}
\label{subsec-Hameqs}
	Hamiltonians are regarded a special type of conservative system
because they are governed by the following equations of motion
   \begin{eqnarray}
	\dot{x}_{i} & = & \frac{\partial H}{\partial p_{i}} \nonumber \\
	\dot{p}_{i} & = & - \frac{\partial H}{\partial x_{i}}         
						\label{eq:HamEqs}
   \end{eqnarray}
where $i = 1 \ldots N$, with $N$ specifying the number of degrees of freedom.
The dimension of the system of equations is obviously $n = 2N$. $H$ is called
the Hamiltonian of the system and is a function of both position $x$ and
momentum $p$ in phase space so that
	\[ H = H(x_{1}, \ldots, x_{N}, p_{1}, \ldots, p_{N})    \]
For Hamiltonian systems it is easily established that $\nabla \cdot {\bf
f}_{\mu} = 0$. From Equations~\ref{eq:HamEqs}
   \[   \nabla \cdot {\bf f}_{\mu} = \sum_{i=1}^{N} \left( \frac{\partial
					\dot{x}_{i}}{\partial x_{i}} +
					\frac{\partial \dot{p}_{i}}{\partial
					p_{i}}          \right)         
				   = \sum_{i=1}^{N} \left( \frac{
					\partial^{2} H}{\partial x_{i} \partial
					p_{i}} - \frac{\partial^{2} H}{\partial
					p_{i} \partial x_{i}}  \right)  
				   = 0
   \]
\\
Extending this to conservative systems, where this result is assumed, shows
\begin{equation}
   \nabla \cdot {\bf f}_{\mu}({\bf x}(t)) = 0 \;\; \mbox{  for all  }\;\; 
					{\bf x} \label{eq:divf=0}
\end{equation}
and so by~\ref{eq:divf}  
\begin{equation}
   \delta \dot{V} (t) = 0      
\end{equation}
signifying that in conservative systems, phase space volumes do not
change (in magnitude) with time. However, this does not mean the shape
of the volume element cannot change, as will be displayed in
Section~\ref{sec-LyapExps}. 

From~\ref{eq:djacdt} the condition \ref{eq:divf=0} gives 
\[  \det \dot{J} ({\bf x}(t)) = 0 \]
so 
\[ \det J({\bf x}(t)) =  \mbox{constant of integration} \]
This constant can be obtained from Equation~\ref{eq:jacobi}. Since the
Jacobi is a constant, evaluation of it at time $ t=0 $ is possible. Thus
from the definition,
\[  J_{ij} = \frac{ \partial x_{i} (0) }{ \partial x_{j}(0) } = \delta_{ij} \]
where $ \delta_{ij} $ is the Kr\"{o}necker delta.  Thus the Jacobi is a unit
matrix, and trivially
\begin{equation} 
   \det J = 1 
\end{equation}

\subsection{Dissipative Systems}
\label{subsec-diss}

Dissipative systems, as their name implies, lose energy. Such behaviour is
associated with a sink somewhere in the vector field, so that the divergence of
the field is non-zero, or more specifically negative. Therefore
\[ \nabla \cdot {\bf f}_{\mu}({\bf x}(t)) < 0 \] 
With this condition, integration of~\ref{eq:divf} yields
\begin{equation} 
  \det J({\bf x}(t)) < 1
\end{equation}
and this shows, via~\ref{eq:dVdt}, that dissipative systems are
characterized by the shrinkage of phase volume elements with time; 
\begin{equation}
  \delta V(t) < \delta V(0)                     \label{eq:volshrink}
\end{equation}

\section{Fixed Points and Attractors}
\label{sec-attractors} 

A solution, \( \bar{{\bf x}} \), of the differential equation~\ref{eq:1ode},
where the vector field vanishes,  i.e. \({\bf f}_{\mu}(\bar{{\bf x}}) = 0\), is
called a {\em fixed point}. Many fixed points have the property that a
solution, \({\bf x}(t)\), that is close to \( \bar{{\bf x}} \) will remain near
for all time resulting in the fixed point being termed {\em
stable}.  This definition can be extended if the solution moves toward the fixed
point, that is,
\[ {\bf x}(t) \rightarrow \bar{{\bf x}} \; \; \mbox{  as  } \; \;  t
						\rightarrow \infty \] 
Then the fixed point is {\em asymptotically stable}.  Asymptotically stable
fixed points are called {\em sinks}, a term met previously, but now it has been
given a more solid definition.\footnote{As is the case for all definitions in
this thesis, it is only an operational one.  Strict mathematical definitions
exist, but they do not readily give an intuitive feel of the phenomena under
consideration.}

It is reasonable to ask how the fixed points affect trajectories in phase space.
To this end, recall the behaviour of phase space volume elements in dissipative
systems. From Subsection~\ref{subsec-diss}, the shrinkage of volume in phase
space means that an observed motion that is both stable and persistent must be
sustained on a structure that has dimension less than that of the phase space.
Such a structure is called an {\em attractor}.

The name attractor is synonymous with its meaning.  Almost any motion that
starts in some volume of phase space nearby is destined to end up being
confined to the attractor, simply due to phase volume shrinkage in dissipative
systems (see Equation~\ref{eq:volshrink}).  The volume for which all motion eventually
ends up on the attractor is called the {\em basin of attraction}.  If motion in
phase space begins distantly from the attractor, but still within the basin of
attraction, the trajectories will follow complicated {\em transient} paths
. After sufficient time (as $t \! \rightarrow \! \infty$), they will approach
the attractor. When the trajectory is on the attractor it cannot leave; it is 
confined to the attractor and all transient motion has ceased.

The three types of attractors that receive much attention in texts are the
following. The first two are presented as simple examples of attractors, while
the third reveals some of the bizarre behaviour that can be encountered in the 
chaotic regime. 

\subsection{Fixed Point Attractors}

	These type of attractors are the simplest type.  They are named because
all trajectories in the basin of attraction end up at the fixed point.  Thus
the fixed point is a sink.  In the case of a one-dimensional flow, this is the
only type of attractor possible.  Figure~\ref{fig:fixed} shows an example of
this type of attractor.
   \begin{figure}
	\vspace{5cm}
	\caption{Trajectories leading to a (a) fixed point attractor, and 
		(b) a limit cycle attractor. (a) $\dot{x} = - y, \; \dot{y} = x - y $ 
		;(b) the Van der Pol oscillator $\dot{x} = -y, \; \dot{y} 
		= x + y \mu (1-x^{2})$ [from Holden (1986)].}
	\label{fig:fixed}
   \end{figure}

\subsection{Limit Cycle Attractors}

	Such attractors commonly occur in two-dimensional flows.  The
trajectories on a limit cycle attractor represent a periodic solution to the
equation of motion~\ref{eq:1ode}. Thus the attractor is a simple closed curve
in phase space.

\subsection{Strange Attractors}

	In three-dimensional systems, both of the above attractors can occur
under varying conditions (namely the value of $\mu$). However, there also
exists the potential to find a much more exotic attractor, the {\em strange
attractor}.

At a given $\mu$, the dissipative system will possess a single attractor.
Thus, after transients have died down, the trajectory will be `locked' onto the
attractor. Intuitively, this suggests the initial conditions are not of great
importance. It appears sufficient that they should lie in the basin of
attraction, and provided one waits long enough, all trajectories will end up on
the attractor. However, in strange attractors, there is a peculiar dependence
on the initial conditions. Such a dependency does not seem possible, after all
Equation~\ref{eq:volshrink} suggests that trajectories passing through a large volume
of phase space will end up passing through a smaller volume later.
But the separation between neighbouring trajectories was not considered in the
argument, and so \ref{eq:volshrink} does not imply this is also contracting.
Strange attractors have the property that closely neighbouring points give
trajectories that, although confined to the attractor, rapidly separate. Such
behaviour does show there is extreme sensitivity to the initial conditions.
Figure~\ref{fig-LorenzAttr} is an illustration of the first strange attractor 
discovered. 
Quanitative measure and study of strange attractors seems an impossibility, but
statistical analysis of their properties is undertaken by {\em ergodic} theory.
   \begin{figure}
	\vspace{5cm}
	\caption{The famous Lorenz ``butterfly'' attractor [from Berg\'{e} 
	{\em et al.} (1984)]. }
	\label{fig-LorenzAttr}
   \end{figure}

\section{Lyapunov Exponents} 
\label{sec-LyapExps}
	Section \ref{sec-attractors} shows that the behaviour of trajectories
on attractors can vary greatly depending on the type of attractor present.  It
has been seen that despite phase space volume shrinkage associated with
dissipative systems, the behaviour of neighbouring trajectories can be very
different.

In the chaotic regime, two nearby trajectories that correspond to close initial
conditions exponentially diverge apart (on average). This is because they
are on the strange attractor. Thus two initial conditions which appear
identical due to the limited resolving power of our measurement devices will
rapidly be seen to behave in quite different manners. The system is exhibiting
chaotic behaviour. 

Compare this with the behaviour of nearby trajectories in a non-chaotic regime.
In this situation the neighbouring trajectories either converge toward one
another exponentially fast, or at worst, exhibit a slower than exponential
divergence. In such cases long term prediction should be possible.

A measure of how neighbouring trajectories will behave under various
circumstances is thus required. Such a measure is provided by the so-called
{\em Lyapunov exponents}. They give information about the exponential rate of
the divergence or convergence of nearby trajectories, and are thereby important
tools in the study of chaos. Positive Lyapunov exponents correspond to a
separation of near trajectories, implying chaos, while negative exponents are
associated with the convergence of trajectories, and thus regular behaviour. 

To obtain a mathematical definition of the Lyapunov exponents, recall the work
of Section~\ref{sec-vol_els}, where a small volume element was specified by the
product of small `widths' $ \delta x_{i} $ in phase space.  This concept is
utilized in this section by specifying a point ${\bf x}$ in phase space, and a
close point ${\bf x + \delta x}$ where
\begin{equation}
   {\bf \delta x} = \left( \begin{array}{c}
			      \delta x_{1} \\ \vdots \\ \delta x_{d}  
			   \end{array}
		    \right)
\end{equation}
Now compare how the solutions will evolve under the vector field.
From Equation~\ref{eq:1ode}
\[ \frac{d}{dt}({\bf x + \delta x}) = {\bf f}_{\mu}( {\bf x + \delta x})  \]
Decomposing this form using a Taylor series expansion gives:
\[ \dot{{\bf x}} +  \delta \dot{{\bf x}} = {\bf f}_{\mu}({\bf x}) +
   \frac{\partial {\bf f}_{\mu}({\bf x})}{\partial {\bf x}} \delta {\bf x} +
					 \ldots \mbox{(higher order terms)}
\]
Ignoring the higher order terms, this reduces to give the rate of change of the
$\delta x_{i}$, a result previously stated in Section~\ref{sec-vol_els} but
unsubstantiated 
\begin{equation}
   \delta  \dot{x}_{i}(t) = \sum_{j=1}^{d} \frac{ \partial
				     f_{i,\mu} ( {\bf x} (t) ) }{ \partial
				       x_{j}}   \delta x_{j}(t)
						  \label{eq:delxi}
\end{equation}
Recall the  coefficients $ \frac{ \partial f_{i,\mu} ( {\bf x} (t)) }{ \partial x_{j} }
$ form a matrix such that
\[   L_{ij} ( {\bf x} (t) ) =  \frac{ \partial f_{i,\mu} ( {\bf x} (t)) }
					{ \partial x_{j} }      \]           
From Equation~\ref{eq:delxi}, a common time dependence is sought by
supposing the solution is of the form
\begin{equation}
	\delta x_{i} (t) = a_{i} e^{\lambda t}    \label{eq:gendelxi_sol}
\end{equation}
where $a_{i}$ is some amplitude term and an exponential growth or decay with
time is assumed. Substituting this into Equation \ref{eq:delxi} one
obtains 
	\[ \lambda a_{i} = \sum_{j=1}^{d} L_{ij} a_{j}   \]
and this is true for $i=1, \ldots ,d$ so it can be written in vector form as
\begin{equation}
	L \, {\bf a} = \lambda \, {\bf a}
\end{equation}
which is precisely the general eigenvalue problem of matrix algebra. Thus there
are $d$ eigenvalues $\lambda_{i}$ which are called the Lyapunov exponents of
the system.  Now from Equation~\ref{eq:gendelxi_sol} the uncertainty in each
dimension of phase space can be related to an earlier uncertainty through the
relation
\begin{equation}
   \delta x_{i} (t) = \delta x_{i} (t_{\rm o}) e^{\lambda_{i}(t-t_{\rm o})} 
						\label{eq:delxisol}
\end{equation}
where $t_{\rm o}$ is some initial time. Here one can see how the Lyapunov exponent
provides a quantitative measure of the way in which nearby trajectories
diverge/converge in each direction, depending on the sign (and magnitude) of
the exponent.

The diversity of applications of the Lyapunov exponents shall now be
highlighted. They are useful in both the chaotic and regular regimes and can
readily enable one to discern many properties of the dynamic system.

\subsection{Conservative versus Dissipative Systems}
Returning to the previous work on volume elements, Lyapunov exponents can also
provide a characterization between conservative and dissipative systems.
Recall \( \delta V(t) = \prod_{i=1}^{d} \delta x_{i} \), using the
solution~\ref{eq:delxisol}, one obtains
\begin{equation}
   \delta V(t) = \delta V(t_{\rm o}) e^{\Sigma_{i=1}^{d} \lambda_{i}(t-t_{\rm o})}
\end{equation}
Thus the criterion for conservative systems, whereby phase volume is conserved
(see Subsection~\ref{subsec-cons}) implies \( \delta V(t) = \delta V(t_{\rm o}) \)
which is only realised if
\begin{equation}
   \sum_{i=1}^{d} \lambda_{i} = 0
\end{equation}
Similarly, dissipative systems (see Subsection~\ref{subsec-diss}) are
characterized by the shrinkage of phase space volumes so that
\begin{equation}
   \sum_{i=1}^{d} \lambda_{i} < 0
\end{equation}
It must be stressed that these conditions only refer to the sum of the Lyapunov
exponents, and do not restrict the sign or size of individual components.

\subsection{Deformation of Phase Space Volumes}
The individual components determine how the volume evolves in each direction of
phase space. A positive exponent $\lambda_{i}$ in the co-ordinate $x_{i}$ means
that the uncertainty in this component of the phase space element $\delta
x_{i}$ rapidly grows, resulting in unpredictable behaviour due to the
divergence of neighbouring trajectories. However, the nature of these
deformations of the phase space volume is very different for conservative and
dissipative systems.

\subsubsection{Chaos in Conservative Systems}
The fact that phase space volumes remain constant in conservative systems as
they evolve has very important ramifications in regard to chaos in such
systems. Chaotic motion, characterised by at least one positive Lyapunov
exponent means an ensemble of initial conditions, called a {\em phase drop},
will rapidly lose its shape with time. However, since the sum of Lyapunov
exponents is zero (volume preserved), other regions may grow or contract,
leading to a pronounced deformation of the drop. Its boundary becomes highly
irregular as shown in Figure~\ref{fig-consChaos}. Chaotic motion in a
Hamiltonian system is specifically referred to as {\em stochastic} motion.
   \begin{figure}
	\vspace{5cm}
	\caption{Chaotic motion in a conservative system.}
	\label{fig-consChaos}
   \end{figure}

\subsubsection{Chaos in Dissipative Systems}
The phase space volume contraction property causes a distinctly different
phenomenon in dissipative systems.

Several positive Lyapunov exponents cause the expansion of a phase drop in some
directions so that trajectories diverge in these directions. However, the net
sum of the exponents must be less than zero, so that on average the volume
contracts. The phase drop will be stretched into a line with time as shown in
Figure~\ref{fig-dissChaos}. 
   \begin{figure}
	\vspace{5cm}
	\caption{Chaotic motion in a dissipative system.}
	\label{fig-dissChaos}
   \end{figure}

\subsection{Lyapunov Spectrum}
\label{subsec-LyapSpec}
As can be seen by the definition of the Lyapunov exponents, the number of 
exponents corresponds to the dimension of the system.  The set of Lyapunov
exponents constitutes the Lyapunov spectrum. The exponents may be ordered such
that 
\[ \lambda_{1} \geq \ldots \geq \lambda_{d}  \]

Examination of the signs of the spectrum enables one to discern what sort of
attractor the system possesses at a given state (or $\mu$). This approach is
summarised for a three-dimensional dissipative system such as the experimental
set-up in Table~\ref{tab-LyapSpec}.
 \begin{table}
	\begin{center}
		\begin{tabular}{|c|l|} \hline \hline
\multicolumn{1}{c}{Spectral Type} & \multicolumn{1}{|c}{Attractor} \\ \hline
		(-- , -- , -- ) & point-like attractor  \\
		(0  , -- , -- ) & limit cycle attractor \\
		(0  ,  0 , -- ) & torus attractor       \\
		(+  ,  0 , -- ) & strange attractor     \\   \hline  \hline
		\end{tabular}
	\end{center}
	\caption{Lyapunov spectrum and corresponding attractor}
	\label{tab-LyapSpec}
 \end{table}
The sequence of signs under the spectral type heading corresponds to the
sign of \( \lambda_{1}, \lambda_{2}, \lambda_{3} \).

\section{Poincar\'{e} Section}
\label{sec-PoinSec1}
The Poincar\'{e} section is a powerful geometric tool used in the study 
differential equations. Poincar\'{e} first developed the method in 1899 when 
he was investigating the three body problem in celestial mechanics. In essence
, the Poincar\'{e} section reduces the study of continuous time systems (flows) 
to the study of an associated discrete time system (map). This technique offers 
several advantages, three of which are:
\begin{itemize}
   \item {\em Dimensional Reduction.} In constructing a Poincar\'{e} section
	 {\em at least} one of the variables of the problem is eliminated, so
	 the problem is reduced to a lower dimension.
   \item {\em Global Dynamics.} Poincar\'{e} sections of low
	 dimension problems can give a remarkable insight into the global dynamics
	 of the system.
   \item {\em Conceptual Clarity.} Abstract concepts that are difficult to state 
	 for ordinary differential equations are usually much simpler and clearer
	 to state for the associated Poincar\'{e} section.
\end{itemize}         

A Poincar\'{e} section is essentially a slice through the flow.  In doing this
one chooses a cross-section through which the flow is everywhere transverse to
it.  Thus the section is not necessarily a plane.  Mathematically, one takes a
$d-1$ dimensional {\em hypersurface} in the $d$-dimensional phase space.  This
surface will be penetrated by any trajectory of the system since it is
transverse to the flow.  Thus the dynamics has been simplified from $d$
dimensions to $d-1$. An example of the method is shown below where a
3-dimensional flow is sliced by a 2-dimensional plane, the Poincar\'{e}
section.
   \begin{figure}[h]
	\vspace*{6cm}
	\caption{Poincar\'{e} plane intersecting a 3-dimensional flow}
   \end{figure}
Despite the usefulness of the Poincar\'{e} section, it is difficult 
to describe general methods for their construction, however one can identify
certain requirements from the nature of the problem.

In the experimental set-up, the differential equation is nonlinear and
nonautonomous. However, the nonautonomy is in the form of periodic forcing,
and a Poincar\'{e} section can be readily constructed, as will be seen in
Section~\ref{sec-PoinSec2}.

\section{Bifurcations}
\label{sec-bifs(a)}
The past sections have highlighted the diversity of behaviour that can be
encountered in physical systems.  This diversity is manifested by the control
parameter $\mu$. As the parameter varies, the structure of the solution may
greatly change. The points at which the solution qualitatively changes
are called {\em bifurcations}. The phrase was originally coined by Poincar\'{e}
to describe the `splitting' of equilibrium solutions in a family of
differential equations.  Such bifurcation of equilibria produce changes in the
type of flow.  A value $\mu_{0}$ of equation~\ref{eq:1ode} for which the flow
is not structurally stable is a {\em bifurcation value} of $\mu$.

There are many means by which a structural instability can occur in the flow
and there are different types of bifurcation to explain each. There are two
main classes of bifurcation, local and global. Local bifurcations are studied
by examining the nature of the vector field in a small region about the
bifurcating point. Global bifurcations are the opposite, dealing with
qualitative changes in the flow structure for extended regions of phase space.
At present local bifurcations are understood better as they are the simplest
to study. The types particular to the experimental setup are described in
Section~\ref{sec-bifs(b)}. 

\section{Period Doubling}
An important bifurcation type, common to many deterministic systems is the
period-doubling bifurcation. In this instance, as the control parameter is
increased beyond the bifurcation point the solution type goes from a single
stable value to two stable values, with the solution oscillating between them.
Further increase in the control parameter may result in another period-doubling
bifurcation so that the system oscillates between four stable solutions. At
each bifurcation the frequency of oscillation halves so that the period
doubles, the reason for its name now becoming obvious. 

If this succession of period-doubling bifurcations continues as the control
parameter is increased then the period-doubling route to chaos is observed. At
each bifurcation the period of oscillation doubles so that after $n$
bifurcations the period of oscillation is $2^{n}$. Thus the solution repeats
itself after $2^{n}$ periods of the fundamental. As $n$ gets very large, after
many bifurcations, it takes much longer for a solution to get repeated. As $n$
continues to increase the period of oscillation becomes infinite (in the limit)
and irregular behaviour ensues. This period-doubling route to chaos occurs in
many unrelated systems. The fact that different systems have solutions which
evolve in a similar way is one of the most significant aspects of chaos theory.

Associated with this is the discovery that certain numbers are common to
unrelated systems, another triumph in the study of deterministic chaos. This
property and the Universal Constants are the topic of the next section.

\section{Universality}
The remarkable property that unrelated dynamical systems have features in
common is the most substantial discovery of chaos theory. Thus there are
underlying {\em universal} properties in these dynamical
systems not necessarily determined by the equations of motion. This implies one
no longer needs know the specific details of the equations to be able
to predict features inherent to systems capable of chaotic behaviour. This is
in stark contrast to past beliefs where emphasis was placed on knowing the
equations of motion exactly. Without an understanding of the equations of
motion it was assumed impossible to quantify or predict features in the
behaviour. 

The present change in attitude was brought about by the work of Feigenbaum
(1978). He showed there exists quantifiable numbers universal to many types of
nonlinear systems. These Universal Scaling Constants were initially obtained by
considering the 1-dimensional logistic map, but they were replicated in
unrelated dynamical systems.  

Referring to Figure~\ref{fig-LogMap} one easily sees the period-doubling route
to chaos in the equation
	\[      x_{k+1} = 4 \mu x_{k} (1 - x_{k}) = f_{\mu}(x_{k})   \]
   \begin{figure}
	\vspace{5cm}
	\caption{Asymptotic iterates of the mapping $f(x)$ for $\mu \,\epsilon
		\,[0.5,1]$ (from Berg\'{e} {\em et al.}, 1984)}
	\label{fig-LogMap} 
	%figure on p201 from Berge, Pomeau etc.
   \end{figure}
The values of $\mu$ at which the bifurcations in the period-doubling cascade
occur have the property that they form an increasing series which rapidly
converges to an {\em accumulation point}\footnote{These figures from Berg\'{e}
{\em  et al.\ } (1984)}
	\[      \mu_{\infty}=0.892 \; 486 \; 418\ldots        \]
The asymptotic rate of convergence towards the accumulation point obeys a
simple and rigorous law: the difference between values of $\mu$ associated with
two consecutive bifurcations is reduced each time by an almost constant factor
    \begin{equation}
	\delta = \lim_{i \rightarrow \infty} \frac{ \mu_{i} - \mu_{i-1} }{
			\mu_{i+1} - \mu_{i} } 
    \end{equation}
Feigenbaum found this reduction factor $\delta$ to be a universal constant,
independent of the details of the function $f$ considered. Present numerical
calculations place its value at
	\[ \delta = 4.669 \; 201 \; 609 \; 102 \; 990 \; 9 \ldots      \]

The distances on the $x$ axis between points on the attractor are reduced by a
scale reduction factor given by a second universal constant:
	\[ \alpha = 2.502 \; 907 \; 875 \; 095 \; 892 \; 84 \ldots     \]
This quantity is more easily measured in the experimental situation from the
Fourier power spectra as it also determines the amplitude of the subharmonics
present. After $n$ period-doublings a subharmonic of frequency $f / 2^{n}$ in
the spectrum lies below the preceding level by a factor\footnote{These figures
from Lichtenberg \& Lieberman (1983)} 
   \begin{equation}
	\gamma^{-1} = {\textstyle \frac{1}{4 \alpha}} 
				[ 2(1 + \alpha^{-2})]^{1/2} = 0.1525
   \end{equation}
so that $\gamma \approx 6.57$. Thus the
amplitude separation of successive, fully developed, subharmonics should be
8.18 dB. 

These constants are universal for 1-dimensional maps and equivalent difference
equation models for dissipative systems.  They differ in the two-dimensional
area preserving maps, but these maps are known to show a period-doubling
cascade to chaos. The scaling constants for these maps are similarly
universal to them. The area preservation property represents the preservation
of volume under flows in phase space, so that the scaling constants also exist
in conservative systems.

Consideration of Hamiltonian bifurcation theory in maps such as 
   \[   x_{i+1} +x_{i-1} = 2 \mu x_{i} + 2 x_{i}^{2}    \]
(which is an adaptation of Hellman's Standard map, see
Section~\ref{subsec-ExpCons}) shows upon lowering $\mu$ from zero the system can
be driven through a series of period-doubling bifurcation into chaos. At each
bifurcation the $\mu_{i}$ geometrically converge to the
accumulation point
	\[      \mu_{\infty} = -1.2656\ldots            \]
with the rate of convergence of $\mu_{i}$ to $\mu_{\infty}$ given by
  \[  \delta = \lim_{i \rightarrow \infty} \frac{ \mu_{i} - \mu_{i-1} }{
		\mu_{i+1} - \mu_{i} } = 8.721 \; 097 \; 2 \ldots        \]
The rescaling parameter $\alpha$ is also different for conservative systems,
its value approximately
	\[      \alpha = 4.018 \ldots                   \]
\\
Summarising, many conservative and dissipative systems show universal behaviour
in both the route to chaos and in the rates of convergence to the
accumulation point $\mu_{\infty}$. However, the rates of convergence are
different for conservative systems and dissipative systems. Correspondingly one
defines the universal scaling constant for conservative systems as 
   \begin{equation}
	\delta_{C} = 8.721 \; 097 \ldots
   \end{equation}
and for dissipative systems as 
   \begin{equation}
	\delta_{D} = 4.669 \; 201 \ldots
   \end{equation}

\subsection{Scaling in Near Conservative Systems}
\label{subsec-ExpCons}
The previous section implies that because a near conservative system is 
strictly dissipative then the universal scaling is still given by $\delta_{D} =
4.669\ldots$. However, these constants are defined in the limit of high order
bifurcations. {\em Very different behaviour occurs in the early bifurcations.}

To understand the behaviour, consider Hellemann's standard map:
   \begin{eqnarray*}
	x_{i+1} & = & y_{i} + 2 \mu x_{i} + 2 x_{i}^{2}         \\
	y_{i+1} & = & - B x_{i}
   \end{eqnarray*}
In this system $B$ is analogous to the Jacobian, so that the system is
conservative for $\mid \! B \! \mid = 1$ and dissipative for $\mid \! B \! \mid
< 1$. The system follows the period-doubling route to chaos as $\mu$ 
is  lowered below zero. Consistent with the Feigenbaum theory detailed above,
upon defining 
   \begin{equation}
    \delta_{i}(B) \: \stackrel{\rm def}{=} \: \frac{ \mu_{i-1}(B) - \mu_{i}(B)}
						{ \mu_{i}(B) - \mu_{i+1}(B)}
   \end{equation}
then one finds
   \begin{equation}
	\lim_{i \rightarrow \infty} \delta_{i}(B) =
		\left\{  \begin{array}{ll}
		   \delta_{D}   & \mbox{ for } \mid \! B \! \mid = 1  \\
		   \delta_{C}   & \mbox{ for } \mid \! B \! \mid < 1
			 \end{array}                    \right.
   \end{equation}
Reick (1991) studies the behaviour of $\delta_{i}$ as a function of $B$. His
findings are of great interest as they imply that the first few bifurcations
in a strictly dissipative system can actually scale at the conservative rate.
This means that potentially, a real physical system with finite dissipation
could be used to measure a quantity associated with conservative systems. 

To see how this occurs, consider the weakly dissipative system by defining the
value of $B$ as:
	\[      B_{m} \stackrel{\rm def}{=}0.999^{2^{(m+1)}}        \]
so that for larger $m$ the system becomes more dissipative. By computing the
scaling constant after $i$ bifurcations, one obtains
Figure~\ref{fig-CrossScal}, where for a given $m$ the scaling is seen to
undergo a rapid crossover from conservative to dissipative scaling. This
crossover occurs within three bifurcations, however the number of previous
bifurcations is seen to differ for the value of $m$. The higher the dissipation
(larger $m$), the more quickly the crossover occurs.  This crossover is
described by Zisook's (1981) scaling function:
   \begin{equation}
	\delta(B_{\rm eff}) \stackrel{\rm def}{=} \lim_{i \rightarrow \infty}
				\delta_{i}(B^{2^{(2-i)}})
   \end{equation}
where \( B_{\rm eff} = B^{2^{(2-i)}} \)
   \begin{figure}
	\vspace{5cm}
	\caption{Crossover of the effective scaling $\delta_{i}(B)$ for
		different, but fixed $B$-values (The lines are only for
		guiding the eye) [from Reick (1991)].}
	\label{fig-CrossScal}
   \end{figure}
This function descends monotonically except for a minimum close to
$\delta_{D}$ and is shown in Figure~\ref{fig-UniScalFunc}. This diagram has
been computationally derived for the Hellemann map.
   \begin{figure}
	\vspace{5cm}
	\caption{The Universal Scaling Function $\delta (B_{\rm eff})$
		(solid curve) compared with $\delta_{2} (B)$ (dashed curve) for
		Hellemann's map [from Rieck (1991)].}
	\label{fig-UniScalFunc}
   \end{figure}
Due to the sharpness of the crossover, the parameter plane $(\mu,B)$ has three
distinct regions: 
   \begin{equation}
	\delta_{i}(B) \approx \left\{   \begin{array}{ll}
						\delta_{D} & \mbox{ for }
						B_{\rm eff} < 0.2   \\
						\mbox{crossover region} &
						\mbox{ for } 0.2 < B_{\rm eff}
						< 0.8   \\ 
						\delta_{C} & \mbox{ for }
						B_{\rm eff} > 0.8
					\end{array}             \right.
   \end{equation}
If this crossover was smeared instead of being sharp then there would not be
any regions of definite conservative scaling in the dissipative case.

Fortunately the scaling function is universal to period-doubling systems with
two dimensional (planar) Poincar\'{e} maps and uniform Jacobian. Physical
systems fulfilling these requirements should be able to measure the
conservative scaling. In practice the experimentally determined value of
$\delta_{C}$ will come from an approximate measure from low order
bifurcations. The lowest order approximation to $\delta(B_{\rm eff})$ is
$\delta_{2}$, as for $i=2$ one has $B_{\rm eff}=B$. The numerical
calculations support $\delta_{2}(B) \approx \delta(B_{\rm eff})$ as can be
seen in Figure~\ref{fig-UniScalFunc}.

From the above considerations, the following conclusion can be reached:
\\
To measure $\delta_{C}$ from the first three period-doubling bifurcations one
has to attain
	\[      \det J  > 0.8  \]
with the experimental system. This system must satisfy the conditions:
   \begin{enumerate}
	\item   Period-doubling bifurcations are inherent to the system.
	\item   The system may be represented by a 2D Poincar\'{e} map.
	\item   The system must have a uniform Jacobian.
   \end{enumerate}

To be able to measure a conservative quantity in an experimental system with
unavoidable dissipation is an exciting prospect. As will be seen, the
experimental system used fulfils the above requirements, so one should be able 
to measure the quantity.

\section{Numerical Methods}
Nonlinear differential equations present a problem for many of the existing
analytical techniques used to solve general differential equations. Despite the
profound occurrence of such equations in nature, existing analytical methods
are usually only suited for linear differential equations. Thus numerical
methods for the solution of ordinary differential equations are a widely used
tool in the study of nonlinear continuous differential equations. However,
the majority of these techniques replace the continuous dynamical system by a
discrete dynamical system. The evolution of these discrete systems is then
modelled on computer. 

Although such methods are very useful, there are pitfalls associated with their
use which must be recognised when employing them. This section describes a
popular numerical method used for solving ordinary differential equations, and
points out the general danger areas involved in using these techniques. 

\subsection{Runge-Kutta Solvers}
The Runge-Kutta method for solving ordinary differential equations is popular
due to its ease of use and speed of computation. It is often used in computer
systems because the method does not require the evaluation of derivatives of
the functions involved (as is the case for methods such as the Taylor Series).

Suppose a differential equation is given in the form
	\[      y' = f(x,y)     \]
with initial conditions \( (x_{\rm o},y_{\rm o}) \) supplied.  Then the solution at  \(
y(x_{\rm o} + h) \) is analytically given by
	\[      y(x_{\rm o} + h) = \int_{x_{\rm o}}^{x_{\rm o} + h} f(x,y) \, dx    \]
Now the Taylor Series expansion of $y(x_{\rm o} + h)$ is
 \[ y(x_{\rm o}+h) = y(x_{\rm o}) + h \, y'(x_{\rm o}) + {\textstyle \frac{h^{2}}{2!}} 
						    \,  y''(x_{\rm o}) + \ldots \]
so that one could determine the value of \( y(x_{\rm o}+h) \) without the need to
evaluate integrals. However, the Taylor Series is difficult to computationally
program due to the need to calculate (complicated) derivatives. The Runge-Kutta
method is essentially a replacement for the Taylor Series, but in a special
form that does not require the evaluation of derivatives. The $4^{\rm th}$
order Runge-Kutta equations agree with the Taylor Series up to terms of order
$h^{5}$ (${\cal O}(h^{5}$)). A particular case is:
   \begin{eqnarray*}
	y_{n+1} & = & y_{n} + {\textstyle \frac{1}{6}} 
				(k_{1} + 2k_{2} + 2k_{3} + k_{4})  \\
	k_{1}   & = & h f(x_{n},y_{n})                             \\
	k_{2}   & = & h f(x_{n} + {\textstyle \frac{1}{2}} h, \:
			y_{n} + {\textstyle \frac{1}{2}} k_{1})    \\
	k_{3}   & = & h f(x_{n} + {\textstyle \frac{1}{2}} h, \:
			y_{n} + {\textstyle \frac{1}{2}} k_{2})    \\
	k_{4}   & = & h f(x_{n} + h, \: y_{n} + k_{3})                    
   \end{eqnarray*}
where $y_{n} = y(x_{n}),\; y_{n+1} = y(x_{n} + h)$.

\subsection{Problems with Numerical Methods}
Despite the usefulness of numerical methods for theoretically investigating
systems governed by nonlinear differential equations, there are problems
associated with them. These problems stem from the fact that the computer
models the continuous system by a discrete system. This has many ramifications,
several of which are described below.

\subsubsection{Round-off errors}
Computer operations can only be carried out to finite precision. Usually the
modelling is done using floating point arithmetic where the number is expressed
in normalized form as 
  \[  s \cdot m \cdot b^{e} \; \mbox { where } \left\{
	\begin{array}{ll}
		s & \mbox{ is the sign}                 \\
		m & \mbox{ is the mantissa}             \\
		e & \mbox{ is the biased exponent}      \\
		b & \mbox{ is the base of the number system}    
	\end{array}                     \right.                 \]
The number of bits used to specify the number in this format depends on the
precision used. Common sizes are:
   \begin{itemize}
	\item   single precision (32 bits)
	\item   double precision (64 bits)
	\item   extended precision (40 or 80 bits)
   \end{itemize}
Obviously, this means for storing the numbers in the computer invariably
produces round-off errors, the consequences of which depends on the nature of
the equations being investigated. Fryska \& Zohdy (1992) have shown that the
normalization-rounding process inherent to floating point arithmetic
has a deterministic character. The normalization-rounding process constitutes
the deterministic dynamics of the computer.

Round-off and associated processes have a general effect on the numerical study
of chaotic systems. In essence the initial conditions are lost as the mapping
is iterated, due to the exponential growth of errors for chaotic orbits, and
linear growth for regular orbits. If the initial conditions can only be
specified to $N$ bit accuracy, then after $n_{\rm max} \sim {\cal O}(N)$
iterations for chaotic orbits and $n_{\rm max} \sim {\cal O}(2^{N})$ iterations
for regular orbits, the initial condition is completely lost. This is to say,
the system will be far from its initial conditions if run forward $n_{\rm max}$
times and then backward $n_{\rm max}$ times (Lichtenberg \& Lieberman, 1983).

\subsubsection{Step Size}
The step size $h$, determining the interval from the previous to the next
solution, is a critical property in numerical methods. The discrete computer
system is connected to the underlying continuous dynamical system in the limit
of zero step size. Thus the usual presumption is that by making the step size
``small enough'' the unwanted, superposed computer dynamics is eliminated. This
is not always the case as shown by Corless {\em  et al.} (1991), so that the
choice of step size is not a trivial exercise. The step size can have two
profoundly different effects on the numerical outcome when compared to the real
physical solution:
   \begin{itemize}
	\item   For some numerical models too large a step size produces a
		``computational chaos''. The numerical simulation displays
		chaotic behaviour while the underlying dynamics of the real
		continuous system shows no such behaviour.
	\item   Lorenz (in Corless {\em  et al.}, 1991) originally put forward 
		the notion that numerical methods may, conversely, suppress 
		chaos. Then for a given step size, the computational model 
		shows a stability not encountered in the real system.
   \end{itemize}
Evidence for both these effects is given in Corless {\em et al.} (1991).





\chapter{Specific Theory}
\label{chpt-specthy}
Now that some key aspects and features of chaotic systems have been introduced,
the situation which has been investigated is analysed. 

\section{Equation of Motion}

The equation of motion governing the system looks comparatively trivial, but as
will be shown, there is a multitude of behaviour ensuing from the `simple'
equation. The experimental set-up can be depicted as follows:

A magnetic dipole is placed in a uniform magnetic field ${\bf B}$.  The dipole
is at an angle $\theta$ to the field and since its rotational axis is
perpendicular to the magnetic field direction, it is be free to rotate.
Letting ${\bf {\cal M}}$ stand for the magnetic moment of the dipole, ${\bf I}$
for the moment of inertia, the situation may be compared to a compass in the
earth's magnetic field. 
   \begin{figure}
	\vspace{4cm}
	\caption{Dipole in a uniform magnetic field.}
   \end{figure}
%Picture of exp set-up.
The dipole will obviously align itself with the magnetic field owing to the
influence of a restoring torque \( \mbox{\boldmath $\tau$} \), given by
\[  \mbox{\boldmath $\tau$} = {\bf {\cal M}} \times {\bf B}  \]
with magnitude 
   \begin{equation}
      \tau = - {\cal M} B \sin \theta            \label{eq:tor1}
   \end{equation} 
where the negative sign is due to the restoring nature of the torque.

Elementary rotational dynamics shows that 
   \begin{equation}
      \tau = I \alpha \equiv I \ddot{\theta}     \label{eq:tor2}
   \end{equation}
so by combining \ref{eq:tor1} and \ref{eq:tor2} yields

   \[   I \ddot{\theta} = - {\cal M} B \sin \theta    \]
or equivalently
   \begin{equation}
      I \ddot{\theta} + {\cal M} B \sin \theta = 0 .       \label{eq:2ononde}
   \end{equation}

If ${\bf B}$ is constant this is a second order nonlinear differential equation
in $\theta$, where  the nonlinearity has arisen due to the $\sin \theta$ term.

However, the fact that the magnetic field is varying in time has been 
ignored. In the experimental situation, the field is sinusoidally 
varying, so
   \begin{equation}
      B = B_{\rm o} \cos \omega t .                  \label{eq:Bcos}
   \end{equation}
Combining \ref{eq:Bcos} with \ref{eq:2ononde} gives 
   \begin{equation}
      I \ddot{\theta} + {\cal M} B_{\rm o} \sin \theta \cos \omega t  = 0 .
					     \label{eq:conseofm}
   \end{equation}

In any real system, dissipation is present.  In deriving
Equation~\ref{eq:conseofm} this factor has been neglected.  The nature of the
dissipation in the experimental set-up shall be assumed to be proportional to
the velocity\footnote{Experimental and numerical evidence suggests the friction
is of this form for constant magnetic fields, so there is no reason to suppose
it is different for an oscillating field.} as is typical of a frictional
component in the system hindering the dipole rotation. Such resistance is
unavoidable in the set-up; there is contact between the bearings supporting the
dipole's rotation and there is air resistance.  In
Section~\ref{sec-SystemParameters} there is little evidence to suggest the
dissipation is of another form. Meissner \& Schmidt (1986) comment that the
exact form of the dissipation is not important if the system follows the
universal period doubling bifurcation sequence. With this duly noted, the
equation of motion proposed is 
   \begin{equation}
      I \ddot{\theta} + k \dot{\theta} + {\cal M} B_{\rm o} \sin \theta  
	 \cos \omega t = 0 ,                            \label{eq:disseofm}
   \end{equation}
where $k$ is the proportionality constant between the friction and the 
velocity ($ \dot{\theta} $). 

\section{Dipole Behaviour}
Equations \ref{eq:conseofm} and \ref{eq:disseofm} are non-integrable. 
However, it is useful to reconsider the nature of the magnetic field described
by  Equation~\ref{eq:Bcos}.  A basic trigonometric identity 
   \begin{equation}
      2 \sin \theta \cos \omega t = \sin(\theta - \omega t) + \sin(\theta + 
				    \omega t)
   \end{equation}
shows that the magnetic field can be thought of as two counter-rotating fields.
Although this may seem to introduce more complexity into the equation of
motion, it enables one to gain an insight into some of the dipole's behaviour.

\subsection{Approximate solutions}
\label{subsec-approxsols}
	In aiming to get an intuitive feel for how the dipole may behave, it is
easiest to return to the frictionless equation of motion, \ref{eq:conseofm}.
Using the trigonometric identity above, and dividing through by $I$, gives
   \begin{equation}
      \ddot{\theta} + {\textstyle \frac{{\cal M} B_{\rm o}}{2 I}} [ \sin(\theta
			- \omega t)  +  \sin(\theta + \omega t) ] = 0 .
   \end{equation}
The equation of motion in either one of the rotating field components is
integrable; it is the presence of the counter-rotating field that makes the
system nonintegrable. 

At this point a very simplistic solution can be obtained. Provided $B_{\rm o}$ is
sufficiently small, the interference of the other half field is
reduced\footnote{Obviously this is a big assumption, but the solution agrees
with characteristics of the dipole behaviour.} so a solution is 
   \begin{equation}
      \theta = \pm \omega t             \label{eq:simple_theta}
   \end{equation} 
where the sign applies to which half field is important due to initial
conditions. The solution indicates that the dipole will corotate with one of
the half fields at the same frequency as the harmonically varying field. Such
behaviour is in qualitative agreement with experimental results. 

Further insight into the dipole's behaviour is obtained if one considers the
equation of motion with one of the fields removed. This gives
   \begin{equation}
      \ddot{\theta} + F \sin(\theta \pm \omega t) = 0    \label{eq:intcase}
   \end{equation}
where 
\[ F = \frac{{\cal M} B_{\rm o}}{2 I}  \]
and one of the fields has been arbitrarily chosen.  To change to a new 
frame of reference which rotates with the half field chosen, one defines the
variable 
   \[ \psi \stackrel{\rm def}{=} \theta \pm \omega t  \]
In this frame \ref{eq:intcase} becomes
   \begin{equation}
      \ddot{\psi} + F \sin \psi = 0                   \label{eq:psintcase}
   \end{equation}
This equation is analogous to the equation of motion for the simple 
pendulum.  Now the equation is completely integrable.  

The solutions of Equation~\ref{eq:psintcase} are of two types, each
corresponding to different motions.
    \begin{description}
	\item[Librational Motion]       This type of motion occurs when the
			dipole is trapped by the half field.  In the $\psi$
			frame this motion corresponds to an oscillation about a
			fixed point \( \psi = 0 \). The frequency of
			oscillation for small $F$ is  
\[ \omega_{\rm o} = \sqrt{F} = \left( \frac{ {\cal M} B_{\rm o} }{2 I} \right)^{1/2} \]
			Returning to the $\theta$ frame, the solution
			\ref{eq:simple_theta} indicates that the dipole will 
			rotate synchronously with one of the half fields, 
			the libration motion present as a superposed feature.
	\item[Rotational Motion]        In this instance the dipole is not
			trapped by the half field, but instead rotates (in the
			$\psi$ frame) with the field. In the simple pendulum
			this corresponds to the infinite rotation of the
			pendulum. For the dipole in the $\theta$ frame, its
			behaviour will be characterised by its rotation at a
			different rate to the half field. 
   \end{description}

Each of these types of motion can be represented in phase space. Since the
$\psi$ frame is relevant, the appropriate phase space is ($\psi,\dot{\psi} $).
Librational motion in this phase space will be closed ellipses, while for
rotational motion one expects an infinitely long trajectory extending to
$\theta = \pm \infty$, but bounded in $\dot{\theta}$. Thus the phase space can
be represented as in Figure~\ref{fig-flat_phase_space}
   \begin{figure}
	\vspace{5cm}
	\caption{Phase space diagram showing the characteristic motions.}
	\label{fig-flat_phase_space}
   \end{figure}

	It is the energy of the system which determines the type of motion
observed. If the energy is insufficient, the trajectory is trapped in the
system's potential well and librational motion is only possible. If sufficient
energy is initially applied, the system's potential can be exceeded and the
motion will be infinite rotational. A hybrid motion exists when the energy
equals the height of the well, so that the motion is neither librational or
rotational.  This is called {\em separatrix} motion. As can be seen from
Figure~\ref{fig-flat_phase_space}, the trajectories on the separatrix pass
through an unstable equilibrium position.\footnote{The trajectory actually
approaches the unstable equilibrium position asymtotically in time.} For the
dipole this occurs when the dipole moment ${\bf {\cal M}}$ is antiparallel to
the magnetic field ${\bf B}$ and at rest, whereas in the simple pendulum
analogy, the unstable equilibrium is when the mass is standing vertically
upright. In general, the trajectory on the separatrix is called a {\em
homoclinic orbit}.  In this simple example the homoclinic orbit\footnote{There
are actually two homoclinic orbits one corresponding to clockwise rotation, the
other anticlockwise.} does not correspond to any complicated motion, it merely
separates the two qualitatively distinct motions, libration inside the
homoclinic orbit, rotation outside.

	Figure \ref{fig-flat_phase_space} is not the simplest phase space
representation. The rotational motion is still periodic, repeated every $2 \pi$
cycles of the angle variable. Consequently, the phase space trajectories of all
motion types can be plotted on the surface of a cylinder, as shown in
Figure~\ref{fig-cyl_phase_space}.
   \begin{figure}
	\vspace{5cm}
	\caption{Transitions to a cylindrical phase space}
	\label{fig-cyl_phase_space}
   \end{figure}
This method of recognising periodicities in the phase space and appropriately
transcribing to another more convenient representation is applied to the phase
space of the dipole in Section~\ref{sec-PoinSec2}.

\subsection{Dual half field interaction}
Recall that in the real situation the two counter-rotating half
fields result in the system being nonintegrable.  
In the \( \theta \) frame the conservative equation of motion is
   \[ \ddot{\theta} + F[ \sin( \theta - \omega t) + \sin( \theta + \omega t)]
					 = 0    \]
In the rotating \( \psi \)
frame the conservative equation of motion can be written as
   \begin{equation}
	\ddot{\psi} + F \sin \psi = - F \sin( \psi \pm 2 \omega t)  
						\label{eq:psinonint}
   \end{equation}

	The nonintegrability of either equation makes it difficult to predict
how the dipole will behave under varying conditions. However, numerical and
experimental analysis has identified two distinct types of motion.
   \begin{description}
	\item[Periodic] This motion is regular and repeats itself with time.
			There are two classes of periodic motion, rotational
			and vibrational.
			\begin{enumerate}
			   \item Rotational motion is characterised by the
				sustained rotation in one direction of the
				dipole. As detailed  earlier, the librational
				and rotational effects due to a single half
				field may be superposed on the motion, a change
				to the $\psi$ frame revealing these.
			   \item Vibrational motion is associated with the 
				oscillation of the dipole about a fixed point
				($ \theta = 0 \mbox{ or } \pi$), depending on
				the initial conditions.
			\end{enumerate}
	\item[Chaotic]  Irregular and unpredictable motion of the dipole. The
			dipole undergoes a range of very short lived rotations
			and vibrations accompanied by multiple direction
			changes. The \( \theta \) time series for such
			behaviour should appear similar to a random walk. 
   \end{description}

	The reason why chaotic motion is observed, and the condition for
predicting its onset, are dealt with in the next section.

\section{Criteria for Chaos}
	It is necessary to establish that chaotic behaviour is inherent to the
system and to find the conditions when it is prevalent.  The second aim is
common to work with dynamical systems capable of chaos.  However, usually this
onset is determined so that chaotic behaviour is avoided.  In the dipole we
want to know the conditions for chaos so that we can observe such behaviour.  

\subsection{Necessary conditions}
The general theoretical methods outlined in Section~\ref{sec-diffeqs} show
Equation~\ref{eq:disseofm} can be re-expressed as a coupled set of first order
differential equations. Firstly write~\ref{eq:disseofm} as
   \begin{equation}
      \ddot{\theta} + \alpha \dot{\theta} + 2 F \sin \theta \cos \omega t = 0 
						\label{eq:normdisseofm}
   \end{equation}
where $ \alpha = k/I $. Upon defining the variables
   \begin{eqnarray*}
	x_{1} & = &  \theta             \\
	x_{2} & = &  \dot{\theta}       \\
	x_{3} & = &  \omega t           
   \end{eqnarray*}
Equation \ref{eq:normdisseofm} can be expressed as
   \begin{eqnarray}
	\dot{x}_{1} & = & x_{2}                         \nonumber       \\
	\dot{x}_{2} & = & - \alpha x_{2} - 2 F \sin x_{1} \cos x_{3}    \\
	\dot{x}_{3} & = & \omega                        \nonumber  
							\label{eq:coupdiss}
   \end{eqnarray}
By defining $x_{3}$, the nonautonomy that existed in \ref{eq:normdisseofm}
has been removed, at the expense of an extra dimension.
Equation~\ref{eq:coupdiss} is a nonlinear, autonomous equation in 3-dimensional
phase space.  Thus two necessary conditions for chaos have been satisfied for
the dipole system, nonlinearity and more than two degrees of freedom. However,
these are not sufficient conditions for chaos.  Analytically establishing that
chaos is inevitable in a system is a difficult task (for example, see Wiggins,
1988).  Due to this fact, at this stage chaos is expected, but not assured, in
the dipole configuration. However, experimental evidence shows chaos is
inherent to the system. Accepting that chaotic behaviour can occur, there must
be certain conditions for which it becomes prevalent.  The next aim is to
establish these.

\subsection{Predicting the onset of chaos}
\label{subsec-Chir}
	Previous arguments (Subsection~\ref{subsec-approxsols}) showed that an
approximate solution could be obtained under the assumption that the half field
interaction is reduced when $B_{\rm o}$ is sufficiently small.  However, it was not
quantified how small the field must be.  Conversely, this argument can be used
to predict when chaotic behaviour may occur.  As the magnetic field is
increased\footnote{Actually, the system is dependent on several factors, as
will be shown.} the interaction between the half fields becomes more pronounced
and instabilities grow. The method used to predict the onset of the
instabilities is dependent on the greater interaction between the half fields.

	The simplest means by which the onset of instabilities can be
establised is Chirikov's resonance overlap condition (Chirikov, 1979). This is
a generalized method commonly used to ascertain the ``transition to global
stochasticity'' for near-integrable Hamiltonian systems with two degrees of
freedom.  To apply the method to the dipole, the Hamiltonian of the system must
be derived in a form that implies near-integrability.  Since dissipation is
minimised, the dipole is nearly Hamiltonian, but unfortunately it is not nearly
integrable. However, the resonance condition does give a useful estimate.

	The generalized form of a Hamiltonian is \(   H = \frac{ p^{2}}{
2 m} + V  \) where $p$ is the momentum, $m$ the mass and $V$ is the potential
of the system. Thus for our angular system the Hamiltonian will be 
\[ H ( \theta ,p,t) = p^{2} / 2 I + {\cal M} B_{\rm o} \cos \theta \cos \omega t \]
where $p=I \dot{\theta}$ is the angular momentum of the dipole.  Again a
trigonometric identity is used to decompose the potential into two
counter-rotating components. 
   \begin{equation}
	H ( \theta , p , t) = \frac{p^{2}}{2I} + \textstyle{\frac{1}{2}} {\cal
				M} B_{\rm o} [ \cos( \theta + \omega t ) +
				 \cos ( \theta - \omega t )]
   \end{equation}
Previous work has shown that for small enough $B_{\rm o}$, a general solution to
the equation of motion is $\theta = \pm \omega t$, so that rotational motion
ensues. Converting to the $\psi$ frame, equating to the energy of the system
and ignoring the influence of the second half field, the {\em unperturbed}
Hamiltonian is defined as 
   \begin{equation}
	H_{\rm u}( \psi ,p,,t) = \frac{ p^{2}}{2 I} + V_{\rm o} \cos \psi = E ,
						\label{eq:unpertH}
   \end{equation}
where $V_{\rm o} = \frac{1}{2} {\cal M} B_{\rm o}$ is the potential energy
amplitude and $E$ is the conserved energy. To quantify the effect of the other
half field, it is considered as a periodic perturbation to the motion. Thus the
Hamiltonian can be represented as a sum of the unperturbed Hamiltonian and the
periodically perturbing potential
   \begin{equation}
	H( \psi ,p,t) = H_{\rm u}( \psi ,p,t) + \epsilon V_{\rm p}( \psi ,t) ,
   \end{equation}
where $V_{p} = V_{\rm o} \cos( \psi \pm 2 \omega t)$.  Obviously our construction
of a near-integrable Hamiltonian is somewhat artifical as $\epsilon = 1$ for
the dipole system, so it is hardly a perturbation. Nonetheless, Chirikov's
method does give a good estimate for the onset of instability in the dipole
system. His resonance overlap condition uses many approximations to overcome
the numerous difficulties associated with predicting the onset of
instabilities. It is stressed that the following account of resonance overlap
in the dipole system is a further simplification of his treatment. Many of the
approximations and assumptions have been omitted to clarify the
discussion.

The first step is to establish the physical conditions which will lead to
instabilities. Previous work showed that when the magnetic field is small
enough the half fields can be considered as independent of one another and that
the motion of the dipole is governed by only one of these fields. This
phenomenon can be represented in a phase space diagram. 
Figure~\ref{fig-dualFields} describes the situation. 
   \begin{figure}
	\vspace{6cm}
	\caption{Phase space diagram.  Motion is influenced by one of the half
		fields [from Meissner \& Schmidt 1986].}
	\label{fig-dualFields}
   \end{figure}
The approximate solution is $ \theta = \pm \omega t$ so $ \dot{\theta} = \pm
\omega $. Thus the separation between the half fields in phase space is
$\dot{\theta} = 2 \omega $.  As $\omega$ is decreased the separation between
the half field resonances\footnote{These are not strictly resonances, however
the Poincar\'{e} section for the dipole has not been introduced so a formal
definition of the resonance is not possible. The misuse of dialogue does not
cause a change in the outcome.} decreases, so it becomes more and more
incorrect to ignore the other half field.

Now that a physical interpretation of the way in which instabilities may grow
in the system has been established, the aim is to obtain a measureable point at
which the instabilities are enough to cause chaos. To this end, assume the half
field resonances are well apart so that the main influence on the dipole motion
is a single half field. The separatrix trajectory of each resonance has a
half-width. This corresponds to the point on the separatrix trajectory with
maximum velocity (therefore angular position is zero) and is measured from the
centre of the resonance to this point on the separatrix. Chirikov's resonance
overlap condition essentially states that instabilities are significant when
the separatrix of one half field resonance touches the separatrix of the other.
Thus the change from stability to instablity can be pictorially represented as
in Figure~\ref{fig-restouch}.
   \begin{figure}
	\vspace{7cm}
	\caption{Chirikov's resonance overlap condition in the dipole system,
		the behaviour changes from stable to unstable, hence the 
		subscripts.}
	\label{fig-restouch}
   \end{figure}
This shows the criteria for the onset of instability is
\[  2 \omega \leq 2 \Delta \dot{\psi} \]
The criteria is simplified by defining the quantity \( s \stackrel{\rm
def}{=} \frac{\Delta \dot{\psi}}{\omega} \) so that chaos is expected for \( s \geq
1 \). 

Now all that is needed is the separatrix half width so that this criteria
can be used.  To this end, consider the nature of the energy of the system for
homoclinic orbits.  Recall from Subsection~\ref{subsec-approxsols} that
separatrix motion is observed when the energy of the system is equal to the
height of the potential well.  Thus substituting $E = V_{\rm o}$ into
\ref{eq:unpertH}, the unperturbed Hamiltonian, one obtains the following
relation for the separatrix trajectory
\[  \dot{\psi}(\psi) = [2 {\textstyle \frac{V_{\rm o}}{I}}
				 ( \cos \psi + 1 )]^{1/2}  \]
but $V_{\rm o}/I = F$ so
   \begin{equation}
	 \dot{\psi}(\psi) = [2 F ( \cos \psi + 1 )]^{1/2} .
						\label{eq:septraj}
   \end{equation}
The centre of the resonance is at $\psi = 0$ so the separatrix half width
$\Delta \dot{\psi}$ is simply $\dot{\psi} (0)$. Solving from \ref{eq:septraj}
gives 
   \[   \Delta \dot{\psi} = 2 \sqrt{F}      \]
so that chaos is predicted for
   \begin{equation}
	s = \frac{ 2 \sqrt{F}}{\omega} \geq 1 .
   \end{equation}
Note that from the definition
   \[   s^{2} = \frac{ {\cal M} B_{\rm o} }{\frac{1}{2} I \omega^{2} }   \]
so that $s^{2}$ is the ratio of the dipole's maximum potential energy to its
kinetic energy when rotating at the driving frequency.

This simplistic approach has given a condition which predicts the onset of
chaotic behaviour. However, the derivation, although it appears sound, has
assumed a great deal. A few of these assumptions and approximations are:
   \begin{itemize}
	\item   The resonances have been related to the phase space
		representation. 
	\item   The half widths stay $\Delta \dot{\psi}$ the same as the
		`resonances' approach. 
	\item   In the true treatment, there is a width associated with a
		stochastic region near the separatrix. This width is not taken
		into account, either in this or the proper treatment.
	\item   There are also secondary resonances lying between the main
		ones, and their role is conveniently ignored in 
		Chirikov's treatment.
   \end{itemize}
Chirikov's method gives an approximate condition for an abrupt transition to
large scale stochastic motion in systems with two degrees of freedom.  When the
method is applied to systems with more than two degrees of freedom (such as the
dipole) then an abrupt change no longer exists. This is because the stochastic
layers associated with each resonance separatrix are connected throughout phase
space. This ``spiderweb'' formation results in {\em Arnold diffusion}.  The
resultant stochastic motion can, by stepping from layer to layer, carry the
system to any point in phase space consistent with energy conservation
(Lichtenberg \& Lieberman, 1983). Motion associated with Arnold diffusion
would appear as a random sequence of rotations and librations interspersed by
many changes in direction. This prediction is supported by experimental
observation. 

For the above reasons our criterion for the onset of chaos is only approximate.
Application of renormalization and numerical integration techniques by Escande
\& Doveil (1981) indicate that $s = 1$ is a lower bound. The experimental
situation similarly finds the condition too severe, where the presence of
friction significantly changes the point at which chaotic behaviour emerges
(Crouette \& Poitou 1981, Meissner \& Schmidt 1986, Ballico {\em  et al.\ } 1990).

A numerical technique which can be applied to the dipole system is that
developed by Iyengar (1991). The method compares the way in which nearby
solutions evolve under the influence of a system's vector field.\footnote{The
periodically forced nonlinear Duffing oscillator is treated in the paper.}  By
specifying the initial conditions of all the parameters as random variables
with small variances, one can calculate how the differential equation of motion
will affect the evolution of the variances. If in the limit the initial
variances go to zero, the solution of the original dynamical system is
non-chaotic; otherwise the solution is be chaotic.  The technique is useful as
the influence of friction on the onset of chaos can be ascertained.
Unfortunately time restrictions did not allow the implemetation of the
technique.

\section{Poincar\'{e} Section}
\label{sec-PoinSec2}
	In Section~\ref{sec-PoinSec1}, the powerful method of the Poincar\'{e}
section was introduced. Its usefulness was expounded but the exact form of
the section for the dipole system was not derived. This is performed here.

Recall the governing differential equation is nonautonomous due to the
sinusoidally varying magnetic field. This nonautonomy was eliminated in
Equation~\ref{eq:coupdiss} by defining a third phase space co-ordinate
$x_{3}=\omega t$. Although this increases the phase space dimension, it has the
advantage of making the system of equations autonomous. It also has the
advantage of simplifying the choice of the Poincar\'{e} section. 

Consider the hypothetical situation that the the dipole is exhibiting period
one motion. As yet, there is no proof that such motion will occur, but this
assumption enables easy visualization of the appropriate Poincar\'{e} section.
Since the period of the magnetic field is the only means to ``clock'' the
motion, period one motion implies that 
   \[ \theta (t + T) = \theta(t) \] 
and 
   \[ \dot{\theta} (t+T) = \dot{\theta} (t) \]
where $T= \frac{2 \pi}{\omega}$. Thus if one examines the solutions at times $t
= n T = n \frac{2 \pi}{\omega}$ for $n$ integer, only a single point $(\theta
(t),\dot{\theta} (t))$ will be observed in phase space.  This forms the basis
for the Poincar\'{e} section. Period two motion means that the solution repeats
itself every two periods of the driving frequency of the magnetic field.  Thus
if the solution is examined at times $t = n T$, two points in phase space would
be observed, each with a repition time of $2 T$. This form of the Poincar\'{e}
section is called the {\em stroboscopic map}. Effectively the section gives the
solution in equally spaced ``flashes'' of time.

A convenient way of picturing the stroboscopic map is shown in
Figure~\ref{fig-torus}. 
   \begin{figure}
	\vspace{7cm}
	\caption{Transition of phase space to torus. The Poincar\'{e} section
		is a circle cut through the torus [from Berg\'{e} {\em  et al.\ }
		1984].} 
	\label{fig-torus}
   \end{figure}
Here the 3-dimensional phase space, with time as one of the coordinates, is
transcribed to a toroidal phase space.  This is possible since the dipole
system is periodic in $t$. The ``circumference'' of the time component is $t =
\frac{2 \pi}{\omega}$, so the need to sample at times $t = n\frac{2
\pi}{\omega}$ is eliminated. Now the trajectories are easily seen to traverse a
Poincar\'{e} section taken at some point in time.  This type of Poincar\'{e}
section defines a {\em global} cross section since all solutions cross the
plane transversely.\\ 

Following is a summary of the possible Poincar\'{e} sections one can observe
and the implications of each:
   \begin{itemize}
	\item   A finite number of fixed points is indicative of periodic or 
		subharmonic motion. $n$ points in the section means the motion
		is repeated every $n$ periods of the driving period. Thus
		period $n$ motion has been observed.  The attractor is either a
		fixed point or limit cycle.  
	\item   A closed curve, or a finite number of points distributed on the
		curve is associated with {\em quasiperiodic} motion. In this
		instance there is an interaction between two fundamental
		frequencies, the response frequency $f_{r}$ of the system and
		the driving frequency $f$. The attractor of the system is a
		torus. Any trajectory on the surface of this torus is
		associated with these fundamental frequencies: one describing
		the frequency of rotation along the ``circumference'', the
		other, the frequency of rotation about the axis of the
		``cylinder'' forming the torus.  The resultant Poincar\'{e}
		section depends on the ratio $f_{r} / f$.
		\begin{enumerate}
			\item   If it is irrational, the trajectory never
				closes so that the surface of the torus is
				covered. The frequencies are said to be {\em
				incommensurate}. The section through the torus
				would show a closed curve.
			\item   If it is rational, the trajectory does close on
				itself.  The frequencies are said to be {\em
				locked} as the ratio $f_{r}/f$ is equal to that
				of two integers $n_{r}$ and $q$ say. After
				$n_{r}$ ``circuits'' and $q$ ``rotations'' the 
				trajectory closes on itself so that this is
				also a periodic solution of period 
		\[ T = n_{r} / f_{r} = q / f    \]
				In this case the section is a finite set of 
				points on the curve. It is important to note
				that the section will have $q$ points on the
				curve.
		\end{enumerate}
		The curve can be simple such as a circle or ellipse, or
		complicated where the presence of harmonics of both frequencies
		causes the curve to self-intersect (for example a figure
		eight). 
	\item   If the section has a fractal collection of points (complicated
		structure with {\em self similarity} at different length
		scales, so that features are reproduced as one ``zooms in'' on
		the object) then a strange attractor has been sliced through.
	\item   A diffuse scattering of points can mean two things.  It may
		correspond to quasiperiodic motion interaction between {\em
		three} fundamental frequencies, or it may indicate conservative
		chaos (stochastic motion). 
   \end{itemize}
Various forms of Poincar\'{e} section are shown in Figure~\ref{fig-PoinSecs}
   \begin{figure}
	\vspace{18cm}
	\caption{Poincar\'{e} sections. Their ability to readily characterize
			behaviour is displayed in this sequence. Top: section
			for a quasiperiodic regime, (a) irrational and (b) 
			rational relation between fundamental frequencies. 
			Bottom: section indicative of conservative chaos.}
	\label{fig-PoinSecs}
   \end{figure} 

\section{Bifurcations}
\label{sec-bifs(b)}
In Section \ref{sec-bifs(a)} the concept of a bifurcation was introduced as a
structural instability of the flow and it was hinted that a wide range of
bifurcations that occur. The bifurcations that are particular to the dipole
system are now discussed. 

The dipole is an example of a driven dissipative nonlinear oscillator.  Work
by Scheffcyzk {\em  et al.} (1991) has recognised distinctive bifurcation
patterns that occur in such systems. Their results suggest that there is a
universal bifurcation structure for the oscillators. Experimental evidence
for the dipole conforms with their findings so an emphasis on the typical
features of the structure is given.

From Subsection~\ref{subsec-Chir} it is known that dipole behaviour is
dependent on a single parameter $s$. Although $s$ is a function of two
variables $F$ and $\omega$, the motion is replicated for different values of
these so long as the value of $s$ is the same. The fact that the dipole system
is dependent on a single parameter means that the {\em codimension} of any
bifurcation is one. The codimension of a bifurcation is defined as the smallest
dimension of parameter space which contains the bifurcation in a persistent
way. The dipole behaviour is governed by a single parameter, so too is the
bifurcation structure. 

There are several codimension-1 bifurcations ; saddle-node, transcritical,
pitchfork and Hopf bifurcations to name a few. Our aim is to recognise the
types of bifurcations inherent to the dipole.

The fact that the dipole is a strictly dissipative system implies that the 
Poincar\'{e} section is everywhere contracting. Such a property immediately
excludes Hopf bifurcations from the system (Scheffcyzk {\em et al.}, 1991).
These bifurcations are characterised by the transition from a fixed point to
limit cycle attractor when a critical parameter value is crossed. The limit
cycle behaviour is created with zero amplitude at the bifurcation point, and
gradually increases as the control parameter is changed further.

Of the remaining possible codimension-1 bifurcations possible, the types
particular to driven dissipative systems are saddle-node (in anti-symmetric
vector fields) or symmetry-breaking (in symmetric vector fields) bifurcation
and period-doubling bifurcation.  To determine if the dipole vector field is
symmetric or not, so that the bifurcation type(s) is further distinguished, it
is noted that the vector fields under consideration are symmetric if they obey
the following relation: 
   \[   {\bf f}_{\mu}(x_{1},x_{2},x_{3}) = 
		- {\bf f}_{\mu}(-x_{1},-x_{2},x_{3} + T/2)      \]
where 
   \[    {\bf f}_{\mu}     = \left( \begin{array}{c}
				 f_{1,\mu} \\ f_{2,\mu} \\  f_{3,\mu}
			      \end{array}                      \right)
   \] 
is the vector field and \( f_{2,\mu}  \) is periodic with respect to $x_{3}$ so
that 
   \[   f_{2,\mu} (x_{1},x_{2},x_{3}) = f_{2}(x_{1},x_{2},x_{3}+T) ; \;\; 
						 T = 2 \pi / \omega . \]
Recalling that the equation of motion can be expressed as three coupled first
order differential equations as such:
   \begin{eqnarray}
	\dot{x}_{1} & = & x_{2}                         \nonumber       \\
	\dot{x}_{2} & = & - \alpha x_{2} - 2 F \sin x_{1} \cos x_{3}    \\
	\dot{x}_{3} & = & \omega                        \nonumber  
   \end{eqnarray}
then comparing with the general form Equation~\ref{eq:1ode} the dipole vector
field is 
   \begin{equation}
	 {\bf f}_{s}     = \left( \begin{array}{c}
				 f_{1,s} \\ f_{2,s} \\  f_{3,s}
			      \end{array}                      \right)
			   = \left( \begin{array}{c}
						 x_{2}  \\  
				 - \alpha x_{2} - 2 F \sin x_{1} \cos x_{3}  \\
						 \omega
			       \end{array}                      \right)
   \end{equation}
Now 
   \begin{eqnarray*}
{\bf f}_{s}(-x_{1},-x_{2},x_{3}+T/2) &  = & \left( \begin{array}{c}
						-x_{2}                  \\
				  \alpha x_{2} - 2 F \sin (-x_{1}) 
						\cos (x_{3}+T/2)        \\
						 \omega                 
					      \end{array}  \right)      \\
				       &  = & - \left( \begin{array}{c}
						x_{2}                    \\
			     - \alpha x_{2} + 2 F \sin x_{1} \cos x_{3}  \\
						\omega                   
					      \end{array}   \right)      \\
				       &\neq &-{\bf f}_{s}(x_{1},x_{2},x_{3})
   \end{eqnarray*}
so that the dipole vector field is not symmetric. (Note that $s$ is the
relevant control parameter for the dipole system.)
Thus the expected bifurcation types in the dipole system are saddle-node and
period-doubling. 

The saddle-node bifurcation is similar to the period-doubling bifurcation. From
a fixed point two branches of steady states emerge as the control parameter is
changed. However, only one branch is stable in the saddle-node bifurcation and
thus it is distinctly different to the period-doubling bifurcation which has
two stable branches. Saddle-node bifurcations are known to have important
global effects on the bifurcation structure of some systems (Guckenheimer \&
Holmes, 1983).

The saddle-node and period-doubling codimension-1 bifurcation points constitute
curves in the parameter plane formed by $F$ and $\omega$. Scheffczyk's {\em
 et al.\ } (1991) investigation numerically confirms experimental evidence that
different oscillators show characteristic patterns of bifurcation curves in the
parameter plane. These bifurcation curves recur with the resonances of the
system where the response frequency becomes mode locked by some integer to the
driving frequency. Every bifurcation curve can be described by two integers,
the torsion number $n_{r}$ and the period number $n$ of the oscillation. As
before, $n_{r}$ counts the number of rotations of a neighbouring orbit around
the given periodic trajectory during one period of the oscillation. The
investigation supports the experimental findings that parameter planes for
general oscillators show similar sequences of bifurcation curves (``resonance
horns'') with increasing torsion numbers as displayed in
Figure~\ref{fig-modToda1} for the modified Toda oscillator. This is a
nonlinear, driven dissipative oscillator described by the equation
	\[       \ddot{x} + d \dot{x} - e^{-x} + x + 1 = f \cos(\omega t) . \]
The bifurcation curves are labelled by the numbers
$(n_{r},n)$ and the bifurcation type (saddle-node, sn; period-doubling, pd).
   \begin{figure}
	\vspace{8cm} 
	\caption{Bifurcation curves in the parameter plane for the modified
		Toda oscillator with $d=0.05$ (Note $f$ is forcing amplitude).
		The
		solid and dotted lines correspond to period-doubling (pd) and
		saddle-node (sn) bifurcation curves, respectively. [from
		Scheffcyzk {\em et al.}, 1991]} 
	\label{fig-modToda1}
   \end{figure} 

For the modified Toda oscillator, the pd(1,2) and sn(1,2) bifurcation curves
have a common segment. It becomes smaller for the pd(3,2) and sn(3,2) curves
and eventually vanishes for corresponding curves [pd($n_{r}$,2) and
sn($n_{r}$,2)] for $n_{r} \geq 5$. The size of the common segment of two
bifurcation curves within a given resonance horn is also dependent on the
damping parameter $d$. As $d$ increases the overlap is reduced and the two
curves separate when the segment shrinks to zero. This dual dependence can be
explained by considering the contraction rate of the associated Poincar\'{e}
map. As will be shown in Section~\ref{sec-DipJac}, where the Jacobian for the
dipole will be derived, the contraction rate of the Poincar\'{e} map for driven
dissipative nonlinear oscillators is given by $e^{-2 \pi d / \omega}$ where $d$
is the dissipation parameter. Thus an extra parameter for the systems is the
effective damping $d / \omega$. Subsequently, the size of the common segment
deceases as the effective damping grows, resulting from either an increase in
$d$ or a decrease in $\omega$. 

To fully comprehend the evolution of the common segment it is necessary to
consider the bifurcation surfaces in three-dimensional parameter space with
damping as the third parameter. The bifurcation surfaces of each resonance horn
have a common shape. A model of these bifurcation surfaces for the modified
Toda oscillator is given in Figure~\ref{fig-modToda2}.
   \begin{figure}
	\vspace{5cm}
	\caption{ Model of pd($ n_{r} $,2) and sn($ n_{r} $,2) bifurcation 
		surfaces [from Scheffcyzk {\em et al.}, 1991].} 
	\label{fig-modToda2}
   \end{figure}

If the bifurcation surfaces intersect one another then a common segment is
formed. The codimension-3 bifurcation point $P$ is where the sn($n_{r}$,2)
surface touches the pd($n_{r}$,2) surface. The coalescence of saddle-node and
period-doubling bifurcations means that {\em subcritical} period-doubling
bifurcations take place. In this instance, two stable solutions exist at the
same time and the period-doubling bifurcation is not immediately realised.  

Subcritical bifurcations occur where the nonlinear terms in the equation of
motion tend to amplify the structural instability in the flow. This is opposite
to supercritical bifurcations in which the solution changes gradually due to a
stabilising effect in the nonlinear terms. Often the change from supercritical
to subcritical bifurcation is bought about by a change in the sign of the
nonlinear terms. Since the nonlinear term in the dipole system is $\sin
\theta$, sign changes regularly occur.  Thus a simplistic treatment also
predicts subcritical bifurcations in the dipole system. 

Subcritical bifurcations have a physical characteristic that may be observed
experimentally. Referring to Figure~\ref{fig-subcrit1} one sees the stable
solution is always at a finite distance from the unstable solution arising from
the subcritical bifurcation. 
   \begin{figure}
	\vspace{5cm}
	\caption{Subcritical bifurcation. In the interval
		$ [ \epsilon_{c}',\epsilon_{c} ] $ two solutions are stable; here,
		one is stationary and the other periodic [from Berg\'{e} {\em
		et al.\ }, 1984]. } 
	\label{fig-subcrit1}
   \end{figure}
As $\epsilon$ is decreased, upon reaching $\epsilon_{c}$, an unstable fixed
point gives rise to a stable fixed point and to an unstable limit cycle. For
$ \epsilon_{c}' \leq \epsilon \leq \epsilon_{c}$ two stable solutions coexist.
However, it is emphasised that:
  \begin{itemize}
	\item   The two solutions cannot be simultaneously observed. 
		The basin of attraction for each coexisting attractor and thus
		the initial conditions will determine which solution emerges. 
	\item   At onset the stable limit cycle will have a finite amplitude.
		It similarly disappears with a finite amplitude.
   \end{itemize}
These points show that a {\em hysteresis} should be observed as one varies the
control parameter of the system in one direction and then the other as shown in
Figure~\ref{fig-subcrit2}.
   \begin{figure}
	\vspace{5cm}
	\caption{Hysteresis phenomenon associated with a subcritical
		bifurcation [from Berg\'{e} {\em et al.}, 1984].}
	\label{fig-subcrit2}
   \end{figure}

Scheffcyzk {\em et al.} (1991) performed more work into the universal aspects
of the bifurcation structure. The experimental results are discussed in
relation to this universal structure in Chapter~\ref{chpt-Discussion}. The 
full details of this universal structure are not given here simply due to the
extensive explanation needed to understand their findings and results. Since
the implications of their work is easily related to the experimental findings,
a lengthy discussion is not warranted here. 

\section{Jacobian}
\label{sec-DipJac}
Chapter~\ref{chpt-genthy} discussed many important aspects of chaos theory. One
which regularly appeared in this discussion was the Jacobian, a quantity used
to quantify the difference between conservative and dissipative systems. The
present aim is to derive the Jacobian for the dipole system. From this many
aspects of the dipole system can be characterised.

Recall from Section~\ref{sec-vol_els} that the Jacobian is related to the
divergence of the vector field in the following manner (compare with
Equation~\ref{eq:dotJ,divf}): 
   \begin{equation}
	 \det \dot{J} (t) = \nabla \cdot {\bf f}_{s} \; \det J (t)
							\label{eq:dotJ=divfJ}
   \end{equation}
From Section~\ref{sec-bifs(b)} the vector field for the dipole system is 
   \[
	 {\bf f}_{s} = \left( \begin{array}{c}
						 x_{2}  \\  
				 - \alpha x_{2} - 2 F \sin x_{1} \cos x_{3}  \\
						 \omega
			       \end{array}                      \right)
   \]
with the divergence of the vector field given by
   \begin{equation}
\nabla \cdot {\bf f}_{s}  \equiv \frac{\partial f_{1,s}}{\partial x_{1}}
				    + \frac{\partial f_{2,s}}{\partial x_{2}}
				    + \frac{\partial f_{3,s}}{\partial x_{3}}
			   = - \alpha                  
   \end{equation}
Upon substitution into Equation~\ref{eq:dotJ=divfJ}, solving gives
	\[      \det J(t) = e^{-\alpha t}       \]
so from Equation~\ref{eq:dVdt}
	\[      \delta V(t) =  e^{-\alpha t} \: \delta V(0)         \]
However, the dipole is a periodically forced system. Thus over one period the
Jacobian is uniform with value
   \begin{equation}
	\det J = e^{-2 \pi \alpha / \omega}
					\label{eq:dipoleJacobian}
   \end{equation}
Since $\alpha$ is finite (and positive), the system is always strictly
dissipative. The result \ref{eq:dipoleJacobian} shows that the contraction rate
for the Poincar\'{e} map of the system is $e^{-2 \pi \alpha / \omega}$,
qualifying the statement given in Section~\ref{sec-bifs(b)}. 

It has been established that:
   \begin{enumerate}
	\item   period-doubling bifurcations are expected in the dipole system,
	\item   the Poincar\'{e} map is two dimensional,
	\item   the Jacobian is uniform,
   \end{enumerate}
so that the conditions of Subsection~\ref{subsec-ExpCons} are satisified for
the dipole system. Thus from Equation~\ref{eq:dipoleJacobian}, the condition 
   \begin{equation}
	e^{-2 \pi \alpha / \omega} > 0.8
   \end{equation}
determines the experimental observability of conservative scaling in the first
three bifurcations of the dissipative system.

\section{Lyapunov Exponents}
A brief discussion of the Lyapunov exponents of the dipole system is now given.
Despite the ease with which they are derived, it is shown that they have
a complex dependence on the state of the system. However, some interesting
conclusions can be drawn from the results.

The Lyapunov exponents are the eigenvalues of the matrix $L$
(Section~\ref{sec-LyapExps}) where $L$ has elements given by 
	\[      L_{ij} = \frac{\partial f_{i}}{\partial x_{j}}  \]
Evaluating this matrix for the dipole system, one finds
   \begin{equation}
	L({\bf x}) = \left(   \begin{array}{ccc}
			0       &       1       &       0                    \\
      -2F \cos x_{1} \cos x_{3} &   - \alpha    &  +2F \sin x_{1} \sin x_{3} \\
			0       &       0       &       0
			      \end{array}               \right)
   \end{equation}
with characteristic polynomial 
   \begin{equation}
	p( \lambda ) = - \lambda ( \lambda^{2} + \alpha \lambda + 2F \cos x_{1} 
			\cos x_{3} )
   \end{equation}
The zeroes of the characteristic are the eigenvalues of $L$. Thus the
eigenvalues are
   \begin{eqnarray*}
	\lambda_{1} & = & 0                             \\
	\lambda_{2} & = & -{\textstyle \frac{\alpha}{2}} + [{\textstyle
		\frac{\alpha^{2}}{4}} - F \cos \theta \cos (\omega t)]^{1/2} \\
	\lambda_{3} & = & -{\textstyle \frac{\alpha}{2}} - [{\textstyle
		\frac{\alpha^{2}}{4}} - F \cos \theta \cos (\omega t)]^{1/2}
   \end{eqnarray*}
This shows that the eigenvalues are dependent on $s$, $\alpha$, time (but there
is a periodicity) and angle. Such dependence means one cannot analytically
determine the points of bifurcation. \\
Note that 
	\[      \sum_{i=1}^{3} \lambda_{i} = - \alpha   \]
so  
	\[      \delta V(t) = \delta V(t_{\rm o}) e^{-\alpha (t-t_{\rm o})}     \]
complying with the strictly dissipative nature of the system.  

Despite the difficulty in determining the points of bifurcation from the
eigenvalues, there are some generalizations which can be made from the form of
the $\lambda_{i}$.
   \begin{itemize}
	\item   Since there is always one zero eigenvalue then motion is
		confined to either a limit cycle, torus or strange attractor
		(Subsection~\ref{subsec-LyapSpec}). 
	\item   The uncertainty in the initial conditions for the dimension of
		phase space corresponding to this zero eigenvalue neither grows
		nor diminishes (from Equation~\ref{eq:delxisol}). 
	\item   It is inappropriate to study the local behaviour of the
		linearized flow about the fixed points in the hope that insight
		of the global behaviour of the nonlinear system can be gained
		(the zero eigenvalue contradicts the conditions of the
		Hartman-Grobman theorem in Guckenheimer \& Holmes, 1983). 
   \end{itemize}
The final point has been included as often nonlinear systems are studied by
linearizing the system near the fixed points. This can be a useful technique
since a linear system is much easier to understand. Once again, the dipole
system is seen to be particularly hard to investigate analytically. 

\section{Effects of Noise}
\label{sec-Noise}
Since the dipole set-up is a real physical system, noise is inherent. This
may be in the form of apparatus vibrations, air disturbances, signal
irregularities or stray magnetic fields. Much effort has been made to minimise
these effects, but they are unavoidable to some extent. It is important to
understand how noise changes the various motions and observations. 

Extensive work has been done in this area and the influence of noise in chaotic
systems is well understood. The reader is referred to Schuster (1988) for a
readable discussion on the implications of noise. Other work such as Sun \& Hsu
(1991) and Kapitaniak (1988) are more mathematical. From these peoples' work
the following conclusions can be drawn:
   \begin{enumerate}
	\item   Noise fluctuations induce early transitions to chaos in
		period-doubling systems (Figure~\ref{fig-EarlyChaos}). This is
		related to the ``blurring'' of the bifurcation structure as
		shown in Figure~\ref{fig-NoiseBif}. 
	\item   The subharmonics of the universal power spectra are eventually
		suppressed by any finite white noise.
	\item   Noise fluctuations in systems possessing more than one
		attractor (not necessarily coexisting) increase the probability
		of switching between them. This explains the sudden change from
		regular to stochastic behaviour in many time periodically
		driven systems (Stiller {\em et al.}, 1992).
	\item   For systems with coexisting steady state solutions (and
		therefore coexisting attractors) noise destabilizes one of the
		solutions. The solution which is destabilized first corresponds
		to that which is closest to the boundary of its basin of
		attraction, or equivalently the furthest from its attractor
		(Sun \& Hsu, 1991).
   \end{enumerate}
   \begin{figure}
	\vspace{5cm}
	\caption{Suppression of the periodic regime by the presence of external
		noise $\sigma$ for the Logistic map [from Schuster 1988].}
	\label{fig-EarlyChaos}
%Diagram from page 61, Schuster
   \end{figure}
   \begin{figure}
	\vspace{5cm}
	\caption{Bifurcation diagrams of the Logistic map with different noise
		levels. Left : white noise level $\sigma_{s}=0.001$. Right :
		white noise level $\sigma_{s}=0.01$. [from Sun \& Hsu, 1991]}
	\label{fig-NoiseBif}
%Diagram from Sun, Hsu p 189, 190
  \end{figure}
Note that many of these effects are expected in the dipole
system.

\section{Dimensionless Equation}
\label{sec-dimlessequn}
Following the work of Tisch (1989), the equation of motion is given in a
dimensionless form. Firstly, consider the full equation of motion in the
rotating frame
	\[ \ddot{\psi} + \alpha \dot{\psi} + F \sin \psi =
		- F \sin(\psi \pm 2 \omega t) \mp \alpha \omega \]
Now there are two characteristic times for the dissipative case:
   \begin{enumerate}
	\item   ${\cal T}_{1}=\alpha / F$ which is the exponential damping time
		for highly damped motion.
	\item   ${\cal T}_{2}=1/ \omega_{\rm o}$ which is proportional to the small
		undamped period of oscillation.
   \end{enumerate}
D'Humieres {\em et al.} (1982) show that these lead to lead to two possible
dimensionless equations upon a change of time scale $t \! \rightarrow \!
t/{\cal T}_{\rm i}$.
For very low friction the time scale ${\cal T}_{2}$ is applicable. Thus upon
defining the {\em quality factor}
   \begin{equation}
	Q \equiv \sqrt{F} / \alpha = \omega_{\rm o} / \alpha
   \end{equation}
and normalizing the time $t$ by ${\cal T}_{2}$ then the dimensionless equation
of motion is given by 
   \begin{equation}
	\ddot{\psi} + {\textstyle \frac{1}{Q}}\dot{\psi} + \sin \psi =
		-\sin(\psi \pm 2 \omega t) \mp \gamma      \label{eq:dimeofm}
   \end{equation}
where $\gamma = \alpha \omega / F$.

This dimensionless representation has several advantages
   \begin{itemize}
	\item   The results are independent of the units used for measurement
		so that the mathematical essence of the problem is captured in
		this form.
	\item   Equation~\ref{eq:dimeofm} is suitable for reduction to a
		2-dimensional circle map.
	\item   An analytic form of the Poincar\'{e} section can be
		constructed. 
   \end{itemize}
Tisch (1989) pursues the two latter points. His work leads to the definition of
a variable $b$ such that 
	\[      b=1-T/Q=1-\pi \alpha / \omega   \]
where $T=\frac{\pi}{\omega {\cal T}_{2}}$ is the rescaled relative driving
period. Now for the low dissipation case
	\[      0 \leq b \leq 1         \]
and it is the value of $b$ which determines the type of mapping the system is
equivalent to. Firstly consider the limiting cases.

As $b \! \rightarrow \! 0$ then the 1-dimensional sine circle map is recovered.
The limit is obtained when $T \rightarrow Q$ or $\alpha \rightarrow \omega /
\pi$ and the angular velocity becomes a ``slave'' of the angle.

The limit $b \! \rightarrow \! 1$, corresponding to the frictionless case,
means that the {\em standard area preserving map} is obtained (Chirikov, 1979).

\subsection{Mode Locking}
For intermediate $b$ the {\em dissipative standard map} is retrieved (Jensen
{\em et al.}, 1984). Resonance mode-locking is possible in which case the motion
is quasiperiodic. Recall there are two characteristic frequencies associated
with the motion, the response frequency and the driving frequency, related
through 
	\[ f_{r}/f = n_{r}/q    \]
Tisch (1989) shows that the average value of $\dot{\psi}$ over the $q$-cycle
(quasiperiodic cycle) is
	\[      < \! \dot{\psi} \! > = {\rm w} \; 2 \omega       \]
where ${\rm w}=n_{r}/q$ is the {\em winding number} of the mode locked state.
The stability of the $q$-cycle can be determined and upon plotting the winding
number ${\rm w}$ as a function of the control parameter a {\em Devil's staircase} is
constructed. Figure~\ref{fig-Devil} is such a diagram. The steps indicate the
stable regions for a particular winding number. The diagram, produced from the
results of Croquette {\em et al.} (1981), shows that the $q$-cycle is expected
to follow a hierarchical structure.
   \begin{figure}
	\vspace{5cm}
	\caption{Winding number of mode locked states versus stochasticity
		parameter. [from Tisch, 1989] }
	\label{fig-Devil}
   \end{figure}




\chapter{Experimental Setup}
\label{chpt-expset}
	The experimental apparatus used was that constructed by John Tisch
(1989). Full details of the construction of the entire apparatus is embodied in
his Honours Thesis. Tisch (1989) recognised improvements to the setup had to be
made, but time restrictions did not allow the implementation of these. Thus a
major requirement of the Honours project was to evaluate the various problems
with the previous design and employ appropriate solutions where neccessary.
 
	Since Tisch (1989) describes in great clarity the construction of the
apparatus only a summary of important features of the setup shall be given.
More emphasis will be placed on the improvements undertaken, and any warnings
shall be given to future users (supervisors of third year students) as what not
to do to the existing setup.

\section{Experimental Apparatus}
	An overview of the apparatus will now be given.  In doing so only those
points essential to the operation of the device shall be elucidated.  The
apparatus is composed of five essential components:
\begin{itemize}
   \item The dipole and surrounding equipment (sensors, Helmholtz coils, and
	 strobes) hereafter referred to as the {\em motor} and {\em
	 assembly}. 
   \item The signal generator which is the source for the alternating current
	 essential for the magnetic field generation.
   \item A direct current power supply used for investigation of the
	 frictional damping constant.
   \item A standard cathode ray oscilloscope (CRO) used for measurement of
	 the voltage being supplied to the coils.
   \item The interface box comprised of digital circuitry for coding the
	 information suitable for interpretation by the computer, as well as an
	 amplifier used to boost the sinusoidal current in the coils.
   \item An IBM compatible personal computer, with an installed 8255 interface
	 card. 
\end{itemize}
Specific details of each component are now given: 

\subsection{Motor and Assembly}
	In essence this component consisted of the dipole centred between two
square section Helmholtz coils, surrounded by sensors to measure angular
position and velocity, complemented by a series of strobing lights used to
characterise the motion.  
	
	To support these elements of the component, a three tier system of
perspex plates was used.  The top two plates contained the dipole, and strobing
lights. In addition, the coils were wound around these plates.  The area
between the two lower plates comprised the motion sensors. The design is shown
in Figures \ref{fig-moa} and \ref{fig-mob}

\begin{figure}
   \vspace{20cm}
   \caption{Details of motor and assembly.}
   \label{fig-moa}
\end{figure}
 
\begin{figure}
   \vspace{20cm}
   \caption{Details of motor and assembly from different side angle.}
    \label{fig-mob} 
\end{figure}

\subsubsection{Coils and Dipole}        
The coils are in a square section, Helmholtz configuration, yielding as near as
possible a uniform field at the centre. The dipole consists of two circular
alloy magnets mounted on a perspex cylinder.  This cylinder is mounted
horizontally on the motor axle which is vertically aligned through the centre
of the assembly.  Thus the dipole is free to rotate in the horizontal plane.
The axle is supported at the top and bottom by a pair of bored out screws. The
bottom point of contact is similar to a needle point bearing, while the top
contact is quite unique from other designs of the same apparatus. This design
is a modification from Tisch's setup, refer to
Subsection\ref{subsec-axlecontacts} for details. These contacts are easy to use
and have very low friction. Since roller bearings have not been used there is
little chance of static friction, as found by Ballico {\em et al.} (1990) in
their device.
	
\subsubsection{Motion Detectors}
	Measurement of the angular position of the magnet is performed in a
unique way by the setup. A light sensitive photogate device is used in
conjunction with an encoder disc attached to the motor axle.  This disc,
consisting of equally spaced alternate opaque and transparent sectors, rotates
with the dipole. The photogate devices, two commercial slotted optoswitches,
have an air gap of sufficient size as to allow for the passsage of the encoder
wheel. An infrared beam is produced at the bottom of the slot in the
optoswitch. It passes upwards, through the encoder wheel, and is incident on a
phototransistor.  A dark sector in the wheel prevents the passage of the beam,
while clear sections have no effect.  With the appropriate electronics and
computer software, the sequence of digital pulses produced by the optoswitch
from the rotation of the encoder wheel can be interpreted to give the angular
position and velocity of the dipole.

This sytem has several advantages over those implemented by others :
\begin{itemize}
   \item It is unobtrusive.  There is no contact between the wheel and the
	 optoswitches, so the system exerts no load on the rotation.
   \item It enables direct measurement of the angular position and velocity
	 simultaneously. 
   \item The information is digitally stored, a practice usually preferred to
	 analysis and storage of analogue signals.
\end{itemize}
There are several disadvantages of the system, the major one being that real
time analysis is not possible. This is due  to several reasons, mainly
software, and these will be elaborated upon in future passages.

In addition to the motion optoswitches, a third optically sensitive device is
contained in the assembly.  It is a reflective optoswitch. This contains a LED
(Light Emitting Diode) to one side of the device and a phototransistor to the
other. A voltage pulse is produced when the infrared beam from the LED is
detected by the phototransistor after it reflects off a strip of aluminium foil
on the underside of the encoder wheel. The device was included to provide a
reference angular position and to prevent errors accumulating which often
occurs with encoder wheel setups.

\subsubsection{Strobing LEDs}

	A circular array of 16 LEDs was arranged on the middle perspex plate.
They were symmetric about the axle and flashed simultaneously, at a
frequency equal to that for the alternating current supplied to the coils. A
viewing disc, of the same diameter as the encoder wheel, was attached to the
axle just above the array of LEDs. This disc was totally opaque except for a
small radial sector of sufficient size so as to allow the viewing of the LEDs.
Such an arrangement is analogous to a real time Poincar\'{e} section. By
viewing from above, one can tell if the motion has ceased transient behaviour,
or discern the periodicity of the motion.  For example, transient or chaotic
motion would appear as a random (or near random)  sequence of the strobe
position, whereas period 1 motion would be signified as a single, stationary
strobe position, period 2 as two stable positions, and so on. 

The LEDs are powered by a 20 volt supply from the interface box. There is
additional circuitry on the bottom plate of the assembly used to allocate the
correct current to each.

\subsubsection{Connections}
	There are four 5-way terminals on the assembly, two on each of the
middle and top plates. They are connected to one another by means of a solid
wire. The top terminals are used to connect the alternating current from the
signal generator (via the interface box) to the coils. They are similarly used
to connect the direct current power supply.  The solid wires between the two
sets of terminals are a convenient means to connect the probe from the CRO. The
bottom terminals are no longer used in the setup.

\subsection{Signal Generation}
	The requirement for a sinusoidally varying current in the coils,
necessitates use of a signal generator. The interface box has a BNC connector
on its front panel for connection to a signal generator, so almost any signal
generator can be used. However, due to the sensitive nature of the dynamics
being investigated, the signal generator must have a stable output and fine
adjustments for amplitude control.  It is not necessary that the signal
generator have a frequency display, as the computer software is capable of
determining the driving frequency very accurately. Two signal generators were
used in the course of the investigation: an inexpensive signal generator made
within the Physics Department, and a Rockland Model 5100 Programmable Frequency
Synthesizer. The performance of the cheap model was compared to that of the
Rockland in order to determine its suitability for use in a third year
experiment. These findings will be given later in
Subsection~\ref{subsec-siggens}.

\subsection{D.C.\ Power Supply}
	The means by which the frictional damping constant is determined, as
well as calculation of the relation between the amount of forcing and the
current in the Helmholtz coils, is best undertaken by observation of the
dipole's motion in a constant magnetic field (produced by direct current in the
coils).  Thus a d.c.\ power supply is an essential piece of equipment.  The
power supply used was  manufactured in the Physics Department but any supply
that is capable of delivering a potential of 15-17 volts is adequate. The
output from the supply is connected directly to the coils, not via the
interface box as is the case for the signal generator.

\subsection{Cathode Ray Oscilloscope}
	A standard CRO is an essential piece of equipment. This is used to
obtain the peak voltage in the coils, and with known coil impedance, the peak
current in the coils is calculable using Ohm's Law. A one channel CRO was
sufficient for all measurements.

\subsection{Interface Box}      
	This box comprises much of the electronics needed to interface between
the various components, hence its name. It contains two major circuits; an
amplifier for boosting the signal generator output, and digital circuitry for
converting the various sensors' output to TTL (Transistor Transistor Logic)
levels and assigning them to various bits to be read by the computer interface.
The circuits will be treated separately but they are not totally independent.

\subsubsection{Signal Amplifier}

	The output from the signal generator must be amplified to obtain
sufficient current in the coils.  The coils are a large inductive load, and
supplying a low frequency, high power signal to them requires a high
performance amplifier. Tisch employed a LM1875 20 watt amplifier, the
advantages of which are expounded by Tisch. Important features to note are:
\begin{itemize}  
   \item The amplifier is highly stable and provides low distortion
	 amplification. 
   \item It is powered off $\pm$ 20 volt supplies.
   \item Gain is controlled by an adjustable trimpot. This must be set so that
	 the maximum amplitude setting on the signal generator does not lead to
	 a distorted sine wave, as is the case when amplifier saturation is
	 reached.
   \item Only alternating currents are amplified; the d.c.\ power supply should
	 not be fed through the amplifier. 
\end{itemize}

\subsubsection{Zero-phase Detection}

	The circuit incorporated to control the flashing of the LEDs and to
produce digital pulses from which the driving frequency could be calculated, is
the zero-phase detection circuit.  The output from the signal amplifier was not
only fed to the coils, but was also diverted to this circuit. The circuit uses
zero-crossing detector and a one-shot monostable multivibrator to produce a
series of pulses.  The pulses correspond to the sine signal when it has zero
amplitude on the rising curve of the wave. Thus the pulse separation
corresponds to the frequency(period) of the signal. The circuit contains a
trimpot which is capable of adjusting the pulse width, but such action is
not advised. The multivibrator has two output channels, one the logic NOT of
the other. The normal output was connected to the stobing LED system, while the
NOT output was fed to the 8255 interface.

\subsubsection{Direction Determination}

	The use of two motion optoswitches enables one to measure the direction
of rotation of the encoder wheel.  The sensors were located almost
diametrically opposite one another of the encoder wheel. One detector was
slightly to one side, so that when an edge of a sector of the wheel was over an
optoswitch, the other optoswitch was at the centre of a sector.  This meant
that the pulse train outputs from the detectors are $90^{\rm o}$ out of phase.
Analysis of the output from the optoswitches by a direction detection circuit
relied on this phase difference.  Using a logic circuit, this phase information
was converted to a digital signal which the computer interpreted to mean either
clockwise or anticlockwise rotation of the dipole.

\subsection{Computer Interface}

	The output from the interface box had to be inputed to the computer by
an appropriate computer interface.  Since an IBM-XT compatible PERICOM computer
was used to interpret the data, a suitable interface was the INTEL 8255
Programmable Peripheral Interface (PPI).  This is a powerful Integrated Circuit
which when mounted on a standard interface board is capable of many complicated
tasks. Its 24 digital input/output (I/O) lines can be used for a variety of
purposes, as far reaching as controlling a floppy disk drive.  The 24 digital
I/O lines are divided into three 8 bit ports: A,B and C. For the data needs of
the experiment, only 4 bits were needed, so the lower 4 bits of port A were
chosen (PA0-3). The data was allocated to the following bits:
\begin{description}
   \item [Bit 0] : {\em Motion}. Output from a single motion optoswitch 
	 is of suitable TTL levels to be allocated this bit.  Low TTL level
	 (0) indicates a dark sector is blocking the optoswitch, while high 
	 (1) occurs when a clear sector is not hindering the optoswitch.
   \item [Bit 1] : {\em Strobe}. The zero-phase detection circuit 
	 corresponds to this bit.  A pulse from low to high occurs whenever
	 a zero on the rising curve of the current sine wave is detected.
   \item [Bit 2] : {\em Index}. As is the case for the motion optoswitch,
	 the index detector output can be directly used as a bit. Normal level
	 is high (1), detection of the aluminium strip causes the bit to pulse
	 to low (0).
   \item [Bit 3] : {\em Direction}. The output from the direction 
	 determination circuit corresponds to this bit. A high TTL level (1) 
	 is indicative of clockwise rotation of the dipole (when viewed from 
	 above), low (0) means anticlockwise rotation.
\end{description}
The construction and installation of the interface board with a 8255 IC was 
performed by the author. Figure~\ref{fig-electr} is an overview of the
electronic components of the  system and displays the way in which they are
connected. 
   \begin{figure}
	\vspace{20cm}
	\caption{Block Diagram of all the electronics}
	\label{fig-electr} 
   \end{figure}

\subsection{Computer Software}
\label{subsec-compsoft}

	Supplied with the experimental apparatus were ten Turbo Pascal (version
4) programs.  These were written by Tisch (1989) and ranged from data
acquisition, analysis and display programs.  With the aim of producing a
working third year practical experiment, it was essential these programs be
reliable, free of bugs, robust and user friendly. As it stood, the supplied
programs did not meet these requirements.

The upgrade of the software produced two programs, comprised of the ten
original programs assimilated together. One of these is specifically concerned
with the acquisition of the raw data encoder8.pas, while the other chaos3.pas
is able to read the raw data, analyse it and produce graphical displays of the
data.  Each of the programs is user friendly with a emphasis on operating menus
as the main means for using the software. The program code is robust against
operator misuse and errors.

In addition to these programs, an extra one was written by the author. This is
designed to analyse data concerned with determination of the frictional damping
constant and forcing-current relation. The program, called Damp.pas, is
similarly user friendly and robust.

Damp.pas enables the investigator to analyse many decays at different coil
current levels. This greatly increases the accuracy of the experimental
parameters, as will be seen. Previously, these measurements had to be carried
out by hand, a laborious process which did not readily give error estimates of
results.

Full, annotated listings of the computer programs are supplied in a Supplement.
This step has been taken so that any future work on the programs is assisted by
a clear description of their working. Also many remedies to problems in the
software were tried and these are mentioned to prevent future workers wasting
time.

\section{Improvements}

	Apart from the improvements made in the computer software as outlined
above, many improvements were made to the mechanical apparatus, particularly
the motor assembly.  Tisch (1989) failed to observe period 4 in the dipole
motion. This he put down to several factors, all of which fall into two
categories; lack of control over physical parameters, and the presence of noise
in the system. These various problems have been evaluated and solutions have
been instigated where necessary. A summary of this is now given.

\subsection{Minimization of Noise in the System}

	Building vibrations were a possible source of noise. A simple remedy
was to place the motor and assembly on a bench designed to damp out floor
vibrations.  The bench had 3 large concrete bricks placed on a 5cm thick
section of Coolite. Cork tiles were placed between the table and Coolite, the
Coolite and bricks, and the bricks and apparatus. This sandwich type
arrangement worked successfully, eliminating any possible problems.

	Air disturbances did effect the dipole motion, although only minorly.
Once again a simple solution was possible so the problem was rectified.  A
perspex enclosure was constructed that surrounded the assembly. Vision of the
dipole was not impaired, but it was made indifferent to air currents.

\subsection{Precise control over parameters}

	It was suggested to build a fixed frequency oscillator with precision
amplitude control. However, this was not carried out, mainly because in the
experimental method adopted, frequency needs to be an adjustable parameter.

	The construction of a peak responding ammeter as an improved means of
measuring the coil current was evaluated.  Apart from the large cost of such
an item, an oscilloscope is probably more accurate and useful.

	The degree to which the coil impedance varies with temperature was
quantified. Since the coils dissipate energy, some in the form of heat, it was
thought the coil impedance may be temperature dependent. Investigations into
the claim were made, results indicating that the impedance changes very little
with temperature.  However, at large voltages, direct current in the coil
causes more pronounced heating, resulting in a change of coil {\em resistance}.
Subsequently, d.c.\ investigations were carried out so that large voltages were
applied to the coils for the least amount of time.

\subsection{Axle Contacts}
\label{subsec-axlecontacts}

	The motor assembly initially had a pair of needle-point bearings at the
top and bottom of the dipole axle. The axle was tapered to a fine point at
either end, with the screws in the centre of the top and bottom perspex plates
providing the support for the axle's rotation. Although this method provided
very little frictional resistance to motion, it had several disadvantages.  The
pressure on the contacts was determined by the `depth' of each screw, the
greater they were screwed in, the greater the amount of pressure exerted on the
axle contacts. However, this change was very quick and abrupt, if the top screw
was tightened by a tiny amount, the increased pressure on the contacts could be
enough to prevent dipole rotation. Also the axle points have to be centred on
the screw heads to get the axle vertical, a task made more complicated by the
fact that minimal pressure had to be applied. Obviously the arrangement was
unsuitable for a third year practical, where ease of use and the need to
replicate results are essential requirements.

	The final method employed to overcome the problem uses a pair of screws
with bored out centres in which the axle ends fit. The diameter and depth of
the bore is the smallest possible. The depth is such that the taper is fully
enclosed, and the diameter is the least which allows rotation. The
configuration is shown in  Figure~\ref{fig-axles}, alongside the previous
setup. 
  \begin{figure}
     \vspace{5cm} 
     \caption{Previous and existing axle contact} 
     \label{fig-axles}
  \end{figure}
 Although this appears as though it will increase friction, there is not a
significant difference. The configuration has many advantages over its
predecessor:
  \begin{itemize}
     \item   Since no pressure is exerted on the axle by the top screw, the
	     setup is insensitive to small changes in screw depth.
     \item   The axle is automatically in vertical alignment.
     \item   Greater depth and diameter bores in the bottom screw allow for the
	     insertion of damping oil in the `well'. Changing the depth and
	     diameter of the bore varies the amount of damping. Now the system
	     can be studied at different levels of damping. 
\end{itemize}

\subsection{Signal Generation}
\label{subsec-siggens}

	In the course of collating data, some shortcomings of the Physics
Department signal generator became apparent. Frequency and amplitude drifts in
the output were believed responsible for the inability to observe period 4. A
more stable signal generator was required. For this part of the experiment a
Rockland Frequency Synthesizer was used, a greater precision device than the
cheap signal generator.  The computer software quickly established the
pronounced difference in frequency stability of the two devices. From this, a
quick investigation into the stability of other parameters of the Physics
Department signal generator seemed appropriate. Consultation of the appropriate
electronic data sheets  gave much of the information needed. The Physics
Department signal generators have at their heart a XR--2206 Monolithic Function
Generator IC. Following is a summary of their important features (from the data
sheets): 
   \begin{itemize}
	\item   The frequency drift is of the order of 0.01\%
	\item   The amplitude stability is typically 0.5dB, corresponding to
		approximately 0.01\%.
	\item   The typical distortion of the sine wave output is 0.5\%, with a
		maximum distortion of 1.5\%.  Thus higher order harmonics are
		present in the output. 
   \end{itemize}
Several operational characteristics were identified with experimental use of
the signal generator. These were:
   \begin{itemize}
	\item   The computer software indicates the frequency can be
		ascertained to a minimum uncertainty of 0.007\%.
	\item   Heating of internal components (resistors) leads to a net
		frequency drift. This is most pronounced just after the turning
		on of the signal generator. After approximately 30 minutes the
		exponential heating reaches a plateau, and the net drift is
		eliminated. 
	\item   Changing the output amplitude leads to a short lived change in
		the frequency. Small adjustments in the frequency by using the
		fine frequency control are associated with a short lived
		oscillation in frequency about the desired setting.
   \end{itemize}
These factors show the Physics Department signal generator is not stable enough
for observing the sensitive period 4 motions and above in the low dissipation
case. The various frequency drifts and oscillations must be taken into account
when using the generator.

The Rockland Frequency Synthesizer uses high precision timing mechanisms and
digital electronics to synthesize the sine wave.  Amplitude and frequency
are controlled more precisely in the device as can be seen in the following
notes on its operational features:
  \begin{itemize}
	\item   The amplitude is controlled by a level potentiometer and a
		multitude of attenuatation settings. This enables precise
		control and variation of the output amplitude.
	\item   The frequency can be set from 2 MHz to 0.001 Hz. The computer
		software indicated the frquency drift was typically 0.0015\%.
	\item   Stability is instantaneous (or nearly). Component heating or
		amplitude changes do not cause any problems.
  \end{itemize}
Obviously, this generator is preferred to the Physics Department generator for
high precision measurements but the cheap generator suffices, particularly in
the high damping regime.

\subsection{Encoder Wheels}

	Although an encoder wheel was supplied with the apparatus, some higher
resolution ones were made. This was achieved by plotting the model wheel on a
computer and making a photographic reduction of the print-out. To obtain the
required 46mm diameter the prints were done on `bromides', or photographic
negative plastic. Table~\ref{tab-encwheels} gives details of the encoder 
wheels, where the R value corresponds to that used in the computer program
chaos.pas (line number 131). Any change of encoder wheels means the R value
must be changed in the computer program. 
 \begin{table} 
  \begin{center}
   \begin{tabular}{|c|c|c|}      \hline
	\# Dark Sectors  &  Angular Resolution  &  R value   \\   \hline
		16       &       $11.25^{\rm o}$    &    32      \\   
		18       &       $10^{\rm o}$       &    36      \\
		36       &       $5^{\rm o}$        &    72      \\   \hline
   \end{tabular}
  \end{center}
  \caption{Details on the encoder wheels available}
  \label{tab-encwheels}
 \end{table}




\chapter{Experimental Procedure}
\label{chpt-expmeth}
	A discussion of the experimental method for the various stages will be
given. This is necessary as the experiment is to be used in the undergraduate
coursework. A successful means for carrying out the experiment and associated
warnings of problem areas provides the future users with helpful information
that will enable them to obtain quality results in the minimum of time.

\section{Coil Impedance}
The Helmholtz coils present an inductive load. The impedance of this load must
be known so that the current in the coils can be calculated from measurement of
the potential applied across them. Since standard methods can be used to
determine the coil impedance, these are not detailed here. Instead they are
given in Appendix~\ref{appx-CoilImp}.

\section{Friction and Forcing}
\label{sec-FricForce}
	As already hinted at, the frictional damping constant and the relation
between the forcing term $F$ and the current in the Helmholtz coils are
obtained by study of the motion of the dipole in a constant magnetic field.
This field is produced by a d.c.\ voltage applied to the coils, hence the need
for a d.c.\ power supply. To understand how the unknowns are obtained using this
technique, consider the equation of motion for the situation. Recall the
treatment for the dipole in the oscillating field. The following relation was
derived:
   \begin{equation}
	I \ddot{\theta} + {\cal M} B \sin \theta = 0    \label{eq:eofmdc}
   \end{equation}
Previously, the sinusoidal variation of the magnetic field $B$ was then
introduced, but in the case for a d.c.\ field Equation~\ref{eq:eofmdc} describes
precisely the equation of motion for the dipole in a constant field.
Re-expressing the equation and introducing the frictional term at the same time
gives
   \begin{equation}
	\ddot{\theta} + \alpha \dot{\theta} + 2 F \sin \theta = 0  
						\label{eq:truedc}
   \end{equation}
where
\[ F = \frac{ {\cal M} B_{d.c.} }{ I }   \]
The factor 2 was not present in the previous equations of motion as the field
was decomposed into two counter-rotating components.

Equation~\ref{eq:truedc} describes the exact motion of the compass in a constant
field. In obtaining a solution of \ref{eq:truedc}, consider small deviations in
$\theta$ from the equilibrium position $\theta = 0$. Then the equation
simplifies to
   \begin{equation}
	\ddot{\theta} + \alpha \dot{\theta} + 2 F  \theta = 0  
						\label{eq:approxdc}
   \end{equation}
under the assumption \(  \sin \theta \approx \theta  \) for small \( \theta
\) .
The general solution for the linear equation \ref{eq:approxdc} is 
   \begin{equation}
	\theta = A e^{-i \omega_{\rm d} t}          \label{eq:thdampsol}
   \end{equation}
which when substituted into \ref{eq:approxdc} gives a condition on the angular
frequency $\omega_{\rm d}$ as
   \begin{equation}
	( \omega_{\rm d} )^{2} + i \alpha \omega_{\rm d} - 2 F = 0
   \end{equation}
Solving this quadratic formula gives
   \[   \omega_{\rm d} = \pm ( 2 F - \alpha^{2}/4 )^{1/2} 
				- i ( \alpha/2 )           \]
Returning to the solution for $\theta$, given by Equation~\ref{eq:thdampsol},
substitution of $\omega_{\rm d}$ yields
   \begin{equation}
	\theta  =  A e^{-[ \alpha/2 \mp i (2 F - \alpha^{2}/4)^{1/2} ]t}
							\label{eq:thetasol}
   \end{equation}
Now consider the form of $\dot{\theta}$, as this show how the parameters may be
determined. Taking the time derivative of \ref{eq:thetasol} leads to
   \begin{eqnarray*}
     \dot{\theta}  & = & - A e^{- \frac{\alpha}{2} t}
			[ \alpha/2 \mp i (2 F - \alpha^{2}/4)^{1/2} ]  \\
		   &   & [ \cos(2 F - \alpha^{2}/4)^{1/2} t 
			 \pm i \sin(2 F - \alpha^{2}/4)^{1/2} t ]    
   \end{eqnarray*}
with expansion giving
  \begin{eqnarray}
      \mbox{Re}( \dot{\theta} ) & = & - A e^{- \frac{\alpha}{2} t}
				[(\alpha/2) \cos(2 F - \alpha^{2}/4)^{1/2} t 
					\nonumber       \\
				&   & + (2 F - \alpha^{2}/4)^{1/2}
				\sin(2 F - \alpha^{2}/4)^{1/2} t ]  
						\label{eq:rethdot}
  \end{eqnarray}
This cumbersome relation is readily simplified.  Comparing the solution of the
damped equation to its conservative equivalent one sees for
\[  \ddot{\theta} + 2 F \theta = 0  \]
the natural frequency of undamped oscillation is \( \omega_{\rm u} = \sqrt{2 F} \).
Under the assumption that $\alpha$ is small, in particular $ 2 F \gg
\frac{\alpha^{2}}{4} $, then \ref{eq:rethdot} becomes
   \begin{equation}
	\mbox{Re}( \dot{\theta} )  =  - A' e^{- \frac{\alpha}{2} t}
					\sin ( \omega_{\rm u} t + \phi )
						\label{eq:rethedot}
   \end{equation}
where \( A' = \sqrt{2 F} \; A \) and \( \phi = \arctan( \alpha / 2 \omega_{\rm u})
\).
Equation~\ref{eq:rethedot} shows that the dipole's angular velocity in a
constant magnetic field will vary sinusoidally with an exponentially decaying
amplitude. From observations of the $\dot{\theta}$ time series, the frequency
of oscillation gives $F$ via \( F = 2 \pi^{2} f_{\rm u}^{2} \) while a
plot of ln(Max $\dot{\theta}$) against time has a slope
of $ - \alpha / 2$. Performing the investigation of the $\dot{\theta}$ time
series at a range of currents gives the relation between the forcing and
the current. Recall $F \propto B_{\rm o} \propto i$, so a linear relation is
anticipated. Note this method obtains the value of $F$ for a given $i$ without
the need to measure the dipole moment $\cal M$ or the moment of inertia $I$.

Although several assumptions have been made, the numerical and experimental
results indicate they are quite reasonable.

\section{Data Collection}
	The method employed for the collation of data was systematic in its
approach. It allowed for quick acquisition of quality data, so it is worthwhile
detailing the procedure applied.

	There are two important factors one has to consider when using the
apparatus. Data of high stability is only obtained when
   \begin{itemize}
	\item   Frequency drifts are minimal.
	\item   Transient behaviour has ceased.
   \end{itemize}
With the following points taken into consideration during the course of 
investigation, these criteria are reached in the minimum of time.
   \begin{enumerate}
	\item   The signal generator should be running for at least thirty
		minutes before data is collected.
	\item   For the desired $s$ one wishes to obtain, the corresponding 
		voltage is applied to the coils. This gives transients time
		to decay while frequency stability is established.
	\item   The computer software responsible for obtaining the driving 
		frequency gives the mean and standard deviation of the 
		calculation. Stability in the mean and low deviation indicates 
		little frequency drift. Thus several determinations of the 
		driving frequency should be carried out before the actual
		collection of data, so that these conditions are satisfied.
	\item   Highly accurate determination of the driving frequency is 
		necessary for rotational locked states. In the analysis 
		of the superposed librational motion one will wish to convert
		the data to a rotating frame. Recall \( \psi = \theta \pm
		\omega t \) so for large t, an error in the driving frequency
		(and therefore $\omega$~) will result in an incorrect
		conversion. Stability in the mean frequency and low standard
		deviation indicate accurate determination, but in addition the 
		software analyses the frequency with which the index mark is
		detected.  In the rotational states one expects the strobe and 
		index frequency to be equivalent; any inconsistency between
		them indicates inaccuracies.
	\item   During the time taken to establish frequency stability,
		transient behaviour usually dies out. This is checked by 
		examining the strobing position of the circular array of LEDs
		on the assembly. If the place(s) where the LEDs flash through
		the viewing disc appears stationary, transients are most likely
		to have ceased. A quick data sample can confirm this but such
		a step is not usually required.
	\item   Data stability is further improved by checking for the index
		mark during the acquisition of the motion data. The software
		makes it optional as to whether index detection is on. Under 
		certain conditions it is disadvantageous to be
		checking for the index mark. For rapid rotation and frequent
		direction changes, `spikes' in the \( \dot{\theta} \) time
		series can occur due to the extra monitoring of the index 
		bit.\footnote{The computer timing mechanism restricts the rate
		of data acquisition. Having index detection on can lead to loss
		of motion data.}
   \end{enumerate}
This method for ensuring accurate data can be represented as a flow chart.
Included in the chart is the actual means by which data is collected and
analysed. Such a chart is given in Figure~\ref{fig-flowchart}.
   \begin{figure}
	\vspace{20cm}
	\caption{Flow chart for experimental procedure.}
	\label{fig-flowchart}
   \end{figure}
	There are additional experimental procedures which are particular to
the dissipation regime under investigation.  These are now detailed.

\subsection{Low Dissipation Case}

	Earlier theoretical discussion indicated that, depending on the values
of the damping and driving frequency, it was possible to achieve conservative
scaling in the convergence ratio of the stochasticity parameter. In the hope of
using the apparatus to experimentally measure the conservative scaling
Feigenbaum constant $ \delta_{C} = 8.72109 \ldots $ it was essential that the
friction was minimised. Thus the bottom axle well contained no oil.
Correspondingly, the dipole rotation will be basically unhindered, so rapid
changes in rotation direction and speed are anticipated.  To prevent the data
rate exceeding the acquisition capabilities of the computer, a low resolution
encoder wheel was fitted to the axle (the R=32 wheel).

	If the conservative scaling regime is reached, one expects the high
period doubling bifurcations to be difficult to observe. Thus minimization of
noise is essential and high precision must be exerted on the control
parameters. Thus the Rockland generator was used to supply the alternating
current and experimentation was carried out at night so that building
vibrations were minimal.\footnote{Due to the high sensitivity of the equipment
at the low friction limit, it is envisaged that the third year practical will
consist of measurements made with high dissipation.}

	Once the degree of damping was established, by the method outlined
above, the data were collected. The stochasticity parameter $s$ is a function
of both $F$ and $\omega$ ( $ s \propto \sqrt{i} / f $ ). Thus a wide range of
the $s$ parameter space can be examined by varying both these variables. This
method differs from that of Tisch (1989). He only changed one variable, the
current, but this prevented him from seeing some interesting dynamical
features.  Subsequently the dynamics were investigated at a range of currents
and driving frequencies.

\subsection{High Dissipation Case}
	To investigate the dipole behaviour in the high dissiaption regime, oil
placed in the screw bores increases the damping to levels where the dissipative
Feigenbaum scaling constant $ \delta_{D} = 4.469\ldots $ is approached. This
should facilitate easier observation of higher period doubling bifurcations.
Thus  noise levels and parameter stability are not as crucial. However, they
still represent experimental goals.

	The higher dissipation means rotation speeds and changes are slower, so
a higher resolution encoder wheel can be used. The actual wheel used depends on
how much extra damping the oil exerts. Large damping should be investigated
with the R=72 wheel, while for intermediate damping, the R=36 wheel should
suffice. 



\chapter{Experimental Results}
\label{chpt-expres}
A review of the experimental results obtained shall now be given. Extensive
amounts of data were collected in the course of investigating the dipole
dynamics. This data is both rich in its variety and the information it gives
about the dynamics. Tedium will hopefully be avoided by concentrating on the
key aspects of the results. The stability and accuracy of the data should be
apparent, displaying the suitability of the system for use in the undergraduate
laboratory. 

\section{System Parameters}
\label{sec-SystemParameters}
Subsection~\ref{subsec-Chir} dealt with an analytically derived condition which
predicts the onset of chaos. The Chirikov resonance overlap criterion was for  
	\[      s \geq 1        \]
where \( s \equiv \frac{2 \sqrt{F}}{\omega} = [\frac{2 {\cal M}
B_{\rm o}}{ I \omega^{2}}]^{1/2}  \) then chaos is expected. Due to the
number of assumptions and approximations made in the derivation, it was only a
tentative criterion. 

Subsequently it is crucial to know the values of these parameters so that one
can ensure chaotic behaviour is observable. This is the present aim of this
section. It is also necessary to know the degree of damping in the system so
that the value of the Jacobian can be calculated. Only when this is known can
one determine the rate at which the first few bifurcations scale (refer to
Subsection~\ref{subsec-ExpCons}).

Following the method outlined in Section~\ref{sec-FricForce} these results were
obtained:
   \begin{equation}
	\alpha = (0.2831 \pm 0.0006) \; {\rm s}^{-1}
   \end{equation}
for the low dissipation case and 
   \begin{equation}
	F = (682 \pm 22)i + (5.0 \pm 2.4) \; {\rm s}^{-2}   \label{res:FvsI}
   \end{equation}
by weighted least squares fitting, where $i$ is the current in the Helmholtz
coils. The offset term arises from the least square fit and is equivalent to a
slight calibration required. The details of the calulations for these results
are given in Appendix~\ref{appx-fric&forcing}. These results compare with those
quoted by Tisch (1989) using the same apparatus of: 
	\[      \alpha = (0.27 \pm 0.02) \; {\rm s}^{-1}        \]
	\[      F = (653 \pm 8)i \; {\rm s}^{-2}                \]
However, the author derives the following results from the data given by Tisch
(1989): 
	\[      \alpha = (0.277 \pm 0.028) \; {\rm s}^{-1}      \]
	\[      F = (652 \pm 21)i \; {\rm s}^{-2}               \]
showing that much better accuracy has been achieved in obtaining the damping
constant, and comparable accuracy in the $F$ against $i$ relation.

Using the result given in \ref{res:FvsI} the stochasticity parameter is related
to the coil current by 
   \begin{equation}
	s = \frac{[69.1 \, i + 0.5]^{1/2}}{f}
   \end{equation}
where $f=\omega / 2 \pi$ is the driving frequency. The uncertainty in the
stochasticity parameter is derived by considering error propogation in the
uncertainty of the values of $F$ and $i$. The value is calculated from the
relation: 
   \begin{equation}
	\sigma_{s} = \frac{[12.33 \, i^{2} + 0.15]^{1/2}}
			{f \,[682\, i + 5]^{1/2}} 
   \end{equation}
derived in Appendix~\ref{appx-fric&forcing}

\subsection{Chaos Region in the Parameter Plane}
It is now possible to designate an area in the $(f,i)$ parameter plane in which
chaos is expected. Chirikov's resonance overlap condition in terms of $i$ and
$f$ is
	\[      \frac{[69.1 \, i + 0.5]^{1/2}}{f} \geq 1        \]
Noting that this is most likely an under-estimate, a contour-plot of the
parameter plane has been made. The $i$ axis is restricted to 0.75 Amps as this
is the maximum current that can be delivered to the coils. The
contours of Figure~\ref{fig-ParaPlane} correspond to constant $s$ levels. The
rapid increase in $s$ as $f$ goes to zero is obvious. The adjoining
3-dimensional plot with $s$ as the vertical coordinate displays this as well. 
   \begin{figure}
	\vspace{4cm}
	\caption{Dipole parameter plane. The contours correspond to constant
		$s$ values. Ajoining this is a 3D plot with $s$ as the vertical
		axis.} 
	\label{fig-ParaPlane}
   \end{figure}

Calculations show that for a maximum current of 0.75 Amps, the frequency for
which $s \geq 1$ is 7.2Hz. However, previous experience with the dipole system
indicates $s \sim 2$ is a more typical condition. This gives a
frequency of approximately 3.6Hz. Since $s \propto i^{1/2} / f$ then greater
control is anticipated in varying the coil current. However,in some instances
it is necessary to adjust the frequency as well. 

\subsection{Other system parameters}
Since a procedure has been used which relates $F$ to the coil current, 
then it is not necessary to measure ${\cal M}$, $B_{\rm o}$ or $I$. The
experimental method is being tested to establish its suitability for an
undergraduate investigation so the dipole's magnetic moment and moment of
inertia, and the magnetic field strength have not been determined. One
would not expect the undergraduate to perform unnecessary measurements
when they can easily derive the $F$ - $i$ relation. However, for the interested
reader the following values were obtained by Tisch (1989) using the same
apparatus: 
	\[      B_{\rm o} = (3.4 \pm 0.1) \times 10^{-3} \, i 
			+ (8 \pm 5) \times 10^{-5} \; \mbox{ Tesla}     \]
	\[  {\cal M} / I = (3.84 \pm 0.12) \times 10^{-5} \; \mbox{ Akg}^{-1} \]
Importantly, he found the magnetic field to be uniform at the centre, where the
dipole was positioned. 

\section{Low Dissipation Case}
The central aim of examining behaviour at very low dissipation is to measure
the conservative universal scaling constant and to recognise any
characteristics in the bifurcation scheme of the dipole system which are
universal to driven, dissipative oscillators. 

\subsection{Summary of Results}
In the course of experimentation a diverse range of behaviour was encountered.
These are listed below: 
   \begin{enumerate}
	\item   Mode-locked rotational.
	\item   Mode-locked vibrational.
	\item   Period-doubled vibrational.
	\item   Period-tripled vibrational.
	\item   Chaotic.
   \end{enumerate}
It is emphasised that previous work by Tisch (1989) with the apparatus did not
observe several of these motions. The existence of chaotic bands
interspersed by periodic bands as reported by Ballico {\em et al.} (1990) was
confirmed. The present investigation established that a highly uniform nature
is associated with these bands, a result not previously reported. 

Tisch (1989) concentrated on studying the rotational mode-locked states. These
resonances have been observed and a summary of the data will be given. However,
the present emphasis is on the prospect of measuring the conservative scaling
and detailing the bifurcation structure.

\subsection{Format of Results}
The computer software is capable of representing the data in many ways. A brief
description of the form of these results is given here for easy reference.
   \begin{description}
	\item[Time Series]      These plots show either $\theta$ or
			$\dot{\theta}$ as a function of time. Since most data
			samples are typically 200 to 300 seconds only a few
			seconds of the data is shown so that detail can be
			resolved. 
	\item[Phase Space Diagrams]     Since the phase space is 3-dimensional
			($\theta,\dot{\theta},t$) it is only convenient to plot
			$\theta$ as a function of $\dot{\theta}$ for the phase
			plot. These plots are equivalent to looking down the
			time axis as shown in Figure~\ref{fig-ExpPhaseDia}.
	\item[Poincar\'{e} Section]     By sampling the data at times $t=nT$
			where $T=2 \pi / \omega$ and plotting
			($\theta,\dot{\theta}$) the Poincar\'{e} section is
			obtained. 
	\item[Power Spectra]    Displays the amount of energy per unit time
			(power) as a function of the frequency. Power
			spectra are very useful for the visualization of
			periodic and quasiperiodic phenomena. For a periodic
			system with frequency $\omega$ then the spectrum has
			peaks at $\omega$ and its harmonics $2\omega, 3\omega,
			\ldots$ . A quasiperiodic system with basic frequencies
			$\omega_{1}, \ldots, \omega_{k}$ has peaks at these
			positions and also at all linear combinations with
			integer coefficients. 
	\item[Return Mappings]  Since data is analysed at regular time
			intervals one ends up with a series of $\theta$ and
			$\dot{\theta}$ values. Given the time series
			$\theta_{j}=\theta(j \Delta t)$ and
			$\dot{\theta}_{j}=\dot{\theta}(j \Delta t) \; \;
			j=(1,2,\ldots,P)$ where $P$ is the number of data
			points and $\Delta t$ is the time interval between
			samples, then return maps can be plotted of
			$\theta_{j}$ against $\theta_{j+1}$ or
			$\dot{\theta}_{j}$ against $\dot{\theta}_{j+1}$ for
			$j=1,\ldots,P-1$. Such plots are only qualitative and
			no measurements can be made from them. However, they
			are quite spectacular so some examples have been
			included.
   \end{description}
   \begin{figure}
	\vspace{6cm}
	\caption{Left: full 3-dimensional phase space. Right: effect of
		looking down the time axis to produce a 2-dimensional
		representation.} 
	\label{fig-ExpPhaseDia}
   \end{figure}

\subsection{Mode-locked rotational}
This motion is characterised by the synchronous rotation of the dipole with the
magnetic field. Despite the bland appearance of this motion, the superposed
librational motion can be very complicated. To resolve this motion one must
convert to the rotating frame $\psi$. The computer analysis software undertakes
this transformation easily, allowing for the direction of rotation so that the
correct $\psi$-frame is chosen from the possible
	\[      \psi = \theta \pm \omega \, t .   \]
In the $\psi$-frame the angular velocity corresponds to that in the
$\theta$-frame through
	\[      \dot{\psi} = \dot{\theta} \pm \omega    \]
Thus the net effect of the transformation is to remove any constant rotational
component from the dipole's angular motion. Non-uniformities in the angular
position are easily observed in the $\psi$ component. The $\dot{\psi}$ time
series is identical to that of the $\dot{\theta}$ except for a vertical
translation which makes $< \! \dot{\psi} \! > = {\rm w}\,2\,\omega$.

The superposed librational motion can be quite complicated. Often it is locked
into a resonance where a quasiperiodic cycle is observed. From
Section~\ref{sec-PoinSec2} this is associated with motion on a toroidal
attractor. The theoretical foundations accounting for this frequency locking
haven't been included as the investigation has centred on other aspects of the
motion. The reader is referred to Tisch's (1989) work for a comprehensive
explanation of the phenomena. Necessary information regarding these resonance
lockings is given in the present discussion of these so-called {\em
$q$-cycles}. 

\subsubsection{$q$-cycles}
A $q$-cycle implies the motion of the dipole repeats itself after q periods of
the driving torque. The $q$-cycle can be detected by use of the Poincar\'{e}
section and power spectra by the following manner.

In the rotating frame the relative driving period is half that of the magnetic
field variation. This is because of the lock-in with one of the half-fields.
The equation of motion in the rotating frame
	\[       \ddot{\psi} + \alpha \dot{\psi} + F \sin \psi =
		- F \sin(\psi \pm 2 \omega t) \mp \alpha \omega       \]
shows that the relative frequency is double that of the magnetic fields
variation, so that the period is halved. Thus the Poincar\'{e} section is
formed by sampling the solution at times $t=m \pi / \omega $ for $m$
integer. Then a $q$-cycle appears as $q$ distinct points in the section - each
point only visited once in $q$ iterations. The power spectra of both $\psi$ and
$\dot{\psi}$ will have peaks at frequencies $\omega / (\pi q)$ as well as
other harmonics at integral multiples of this. Note that period 2 in the
rotating frame is period 1 in the stationary ($\theta$) frame.

The results are given in Figures \ref{fig-qcyc1}, \ref{fig-qcyc2} and
\ref{fig-qcycspec}. As with all displayed data, the opposite page has an easy
reference caption system which shows the value of the stochasticity parameter
$s$ and the driving frequency $f$ for each diagram. The driving frequency has
been included as the effective damping ($\alpha / \omega$) and the Jacobian
depends on its value ($\alpha = 0.2831 \pm 0.0006\; {\rm s}^{-1}$ for all these
graphs). The values given are the best values. Errors have not been included
but are given at the end.  

The time series plots display the nature of the $q$-cycle.  The minima or
maxima of the $\dot{\theta}$ time series are separated by $\pi / \omega =
1 /(2 f)$ seconds, where $f$ is the driving frequency of the coil current.
Thus the time separation is typically $\sim 0.17$ seconds. By sampling the
solution at the appropriate time interval, the time series show that the
solution repeats itself after an integer number of such intervals, the exact
number corresponding to the $q$ value. Phase portraits have also been included
to show the variety of behaviour, and because they readily show how the stable
solution changes as the $s$ parameter is changed.
   \begin{figure}
	\vspace{18cm}
	\caption{Top: 14-cycle behaviour, readily seen in the $\dot{\psi}$
		time series. Lower 4 diagrams: 10-cycle behaviour. The
		$\dot{\psi}$ return map has been included to show its
		interesting form. Note the correct labels are given in the
		opposite diagrams.}
	\label{fig-qcyc1}
   \end{figure}
   \begin{figure}
	\vspace{18cm}
	\caption{Top 4 diagrams: 10-cycle behaviour. Bottom: 6-cycle
		behavior. Note the evolution of the phase portrait and thus
		solution as the $s$ parameter is increased.}
	\label{fig-qcyc2}
   \end{figure}
   \begin{figure}
	\vspace{20cm}
	\caption{Power spectra diagrams. The peaks have been labelled by their
		frequency in relation to the driving frequency.}
	\label{fig-qcycspec}
   \end{figure}
From the data shown in the diagrams and from that collected in the course of
investigation the following table was constructed, detailing the nature of the
$q$-cycle observed at a given $s$-value.
   \begin{center}
	\begin{tabular}{|c|c|c|c|} \hline
$q$-cycle :& 6-cycle            & 10-cycle              & 14-cycle   \\ \hline 
	   & 0.638 $\pm$ 0.023  & 1.155 $\pm$ 0.022     & 1.203 $\pm$ 0.022 \\
$s$        &0.769 $\pm$ 0.020  & 1.227 $\pm$ 0.022     & 1.250 $\pm$ 0.022 \\
	   & 0.878 $\pm$ 0.020  & 1.494 $\pm$ 0.025     & 1.367 $\pm$ 0.024 \\
value      & 1.028 $\pm$ 0.021  & 1.588 $\pm$ 0.027     &               \\
	   & 1.239 $\pm$ 0.022  & 1.639 $\pm$ 0.027     &               \\
	   & 1.706 $\pm$ 0.028  & 1.719 $\pm$ 0.028     &               \\ \hline
	\end{tabular}
   \end{center}
This shows that the $q$-cycle behaviour does not follow a hierarchical
structure as found by Croquette \& Poitou (1981). Tisch (1989) similarly did
not observe the structure with the apparatus, but he did find that each
$q$-cycle was confined to a range of $s$. Since extensive study of the
rotational locked states was not undertaken the conflicting nature of the
results outlined above was left unresolved.  Interestingly, the low dissipation
meant that the variable $b$ defined in Section~\ref{sec-dimlessequn} had a
value of
	\[      b=1-\pi \alpha / \omega \approx 0.953   \]
Recall that for $b$ between 0 and 1 mode-locking is anticipated, but for $b
\rightarrow 1$ the phenomenon becomes less pronounced. This may explain the
anomalous results.

Finally, the range over which rotational locked states were observed is given.
The lowest parameter value for which sustained rotational motion occurred was
$s = 0.404 \pm 0.034$ while the largest value was $s = 1.806 \pm 0.030$.

\subsection{Vibrational States} 
Period 2 vibrational states were first observed at $s \sim 1.8$. However,
further investigation identified period-doubled and period-tripled vibrational
states in the $s \sim 1$ region. These states were difficult to find as they
were sensitive to disturbances and were only realised for certain initial
conditions. They were found when the low rotational $q$-cycles were being
studied. In fact {\em either} vibrational or rotational motion can be achieved
at the same $s$ value in these regions depending on the initial conditions.

Some of the results are given in Figures~\ref{fig-p2vlw1} and \ref{fig-p3vlw1}.
The period 3 motion is unusual and several plots have been given, all verifying
its nature. This motion has also been observed by Meissner \& Schmidt (1986)
working with a similar setup.
   \begin{figure}
	\vspace{15cm}
	\caption{Period 2 motion in the low $s$ region. Note the regular
		$\theta$ time series with period $\approx 0.7$ secs compared
		with $T=0.34$ secs driving period. The power spectrum shows a
		distinct peak at $f \approx 1.5$Hz, which is half the driving
		frequency. The Poincar\'{e} section shows two points, as
		expected for such motion.}
	\label{fig-p2vlw1}
   \end{figure}
   \begin{figure}
	\vspace{17cm}
	\caption{Period 3 motion in the low $s$ region. Both time series have
		periods $\approx 1$ sec. The Poincar\'{e} section has three
		distinct points, while both power spectra show the fundamental
		peak at $f \approx 1$Hz. All these features confirm the motion
		is indeed period 3.}
	\label{fig-p3vlw1}
   \end{figure}
The range of stable period 2 motion in the low $s$ region was from $s = 0.997
\pm 0.020$ to $s =1.090 \pm 0.021$. 

Period 2 vibrational states were also observed at larger $s$ values as expected
for the period-doubling route to chaos. Figure~\ref{fig-p2vfw1} displays some
of the behaviour. There was a hysteresis associated with the bifurcation as
period 2 motion could be sustained down to $s = 1.776 \pm 0.029$. Thus a
subcritical bifurcation has taken place. More will be said on this matter in
Subsection~\ref{subsec-ExpBifStruc}. The period 2 state extended to $s = 1.977
\pm 0.032$ from which noise and other effects caused an early transition to
chaos without observation of a period 4 or any higher order motion. The
possible reasons for not observing period 4 motion will be given in
Subsection~\ref{subsec-period4discuss}. This period 2 motion is different from that
observed by Tisch (1989). He encountered a {\em period 2 rotational} state
which is very different to the vibrational state recorded in this instance. A
full comparison between the results of this investigation and all other work
performed with similar setups is given in Chapter~\ref{chpt-Discussion}.
   \begin{figure}
	\vspace{18cm}
	\caption{Period 2 vibrational motion. The time series, Poincar\'{e} and
		power spectrum plots all indicate period 2 type motion. Note
		the difference in the phase portrait from period 1 motion.}
	\label{fig-p2vfw1}
    \end{figure}

\subsection{Band Structure}
Stimulated by the work of Ballico {\em et al.} (1990), the possibility that
periodic behaviour occurred past the chaotic region was examined. Indeed this
was the case, and surprisingly a regular band structure exists where chaotic
regions are interspersed by periodic bands. The periodic bands occupy a very
similar width in the $s$ parameter plane to that of the first region reported
above. They also have features in common.

The characteristics of these bands is discussed presently, and treatment of the
chaotic motion is delayed until Subsection~\ref{subsec-ExpChaosMotion}. 

\subsubsection{Second Periodic Band}
As the stochasticity parameter was increased through the first chaotic region
the dipole was eventually observed to return to stable period 2 vibrational
motion. Figure~\ref{fig-p2vlw2} displays the characteristics of the motion.
The side lobes in the phase portrait are due to a ``wobble'' in the dipole
motion. The numerical results confirm the existence of these lobes, they are
not an artifact of the computer analysis. 
   \begin{figure}
	\vspace{16cm}
	\caption{Period 2 vibrational motion observed after the first
		chaotic band. The side lobes in the phase portrait, correspond
		to short lived direction changes as shown in the $\dot{\theta}$
		time series. Note the change in frequency required to access
		these higher $s$ value motions. The frequency spectrum shows a
		peak at 1 Hz; half that of the driving frequency.}
	\label{fig-p2vlw2}
   \end{figure}
This motion type was observed from $s~=~2.597 \pm 0.023$ to $s = 2.852 \pm
0.047$. 

As the $s$ value was increased beyond this upper limit {\em period 1
vibrational} motion resulted. Figure~\ref{fig-p1vw2} shows several phase
portraits and time series of the motion. The effect of increasing $s$ is
clearly displayed in the sequence where the $\dot{\theta}$ time series shows a
dramatic change. This change corresponds to a small superposed oscillation
about the fixed point. As the $s$ value is increased the speed of this
oscillation increases, but the amplitude remains the same (there is no evidence
of increasing deformation in the $\theta$ motion).
   \begin{figure}
	\vspace{20cm}
	\caption{Period 1 vibrational motion in the second band. Note the
		change in $\dot{\theta}$ as $s$ is increased. This deformation
		is not replicated in $\theta$, however the magnitude of the
		large oscillation about a fixed point is seen to change.}
	\label{fig-p1vw2}
   \end{figure}
A sizeable hysteresis was associated with this area of parameter space as the
motion was stable from $s = 2.797 \pm 0.046$ to $s = 3.653 \pm 0.066$. Thus a
subcritical bifurcation has occurred from the initial period 2 motion to the
period 1 vibrational.

Beyond the period 1 behaviour another stable period 2 vibrational motion was
observed as shown in Figure~\ref{fig-p2vfw2andp2vlw3}. The $\theta$ and
$\dot{\theta}$ time series are very complicated and lead to a very interesting
phase portrait.  This particular behaviour existed from $s = 3.891 \pm 0.069$
to $s = 3.963 \pm 0.047$. Between the end of period 1 vibrational motion and
the observed start of period 2 motion there was a small unstable region where
no steady state solution was achieved.

\subsubsection{Third Periodic Band}
Further increase in $s$ resulted in the discovery of a second band of chaotic
motion. This band was similar to the first with motion appearing random and
sensitive to initial conditions (full discussion in
Subsection~\ref{subsec-ExpChaosMotion}).  When the coil current was further
increased the motion returned to period 2 behaviour at $s = 4.621 \pm 0.078$,
marking the onset of the third periodic band. The characteristics of the
trajectory are also shown in Figure~\ref{fig-p2vfw2andp2vlw3}. One recognises
similarities between the motions in the different periodic bands. It is
emphasised that these time complicated series are stable and replicate
themselves for the duration of the sampling run. Only 8 second time series are
given here. The phase portraits show that the system is becoming more sensitive
to noise. This is because the region of $(f,i)$ parameter space under
investigation has very close $s$ contours (refer to Figure~\ref{fig-ParaPlane})
so that small drifts in current or frequency cause sizeable changes in $s$.
   \begin{figure}
	\vspace{19cm} 
	\caption{Top: Period 2 vibrational motion in the upper regions
	of the second periodic band. Note the unusual, but replicated time
	series, causing the inner loop in the phase portrait.
	Bottom: Period 2 vibrational motion in the lower regions of the third
	periodic band.}
	\label{fig-p2vfw2andp2vlw3}
   \end{figure}

The third periodic band had the same structure as the previous bands. At the
end of period 2 motion ($s = 4.658 \pm 0.078$), period 1 (rotational) behaviour
ensues (from $s = 4.761 \pm 0.080$ to $s = 5.513 \pm 0.091$), followed by a
return to period 2 vibrational (from $s = 5.848 \pm 0.096$ to $s = 6.039 \pm
0.098$). This is displayed in Figure~\ref{fig-p1and2w3}.

The period 1 motion is comparatively simple again, however the phase portrait
and time series of the period 2 trajectory is the most complicated yet seen.
The influence of noise is even more pronounced in these diagrams. 
   \begin{figure}
	\vspace{19cm}
	\caption{Top: period 1 rotational motion.
		Bottom: period 2 vibrational motion. Note the complex time
			series and phase diagrams.}
	\label{fig-p1and2w3}
   \end{figure}

\subsubsection{Uniformities in Band Structure}
From the investigation of the band structure the following table can
constructed, showing the range of $s$ over which a particular motion type is
observed: 
  \begin{center}
   \begin{tabular}{|c|l|c|}     \hline
Band No.& Motion Type           & $s$ range             \\      \hline
	& period 2 vibrational  & $0.997 \pm 0.020 \;$ to $\;1.090 \pm 0.021$\\
1       & period 1 rotational   & $0.404 \pm 0.034 \;$ to $\;1.806 \pm 0.030$\\
	& period 2 vibrational  & $1.783 \pm 0.022 \;$ to $\;1.977 \pm 0.032$\\ \hline
	& period 2 vibrational  & $2.597 \pm 0.023 \;$ to $\;2.852 \pm 0.047$\\
2       & period 1 vibrational  & $2.797 \pm 0.046 \;$ to $\;3.653 \pm 0.066$\\
	& period 2 vibrational  & $3.891 \pm 0.069 \;$ to $\;3.963 \pm 0.047$\\ \hline
	& period 2 vibrational  & $4.621 \pm 0.078 \;$ to $\;4.658 \pm 0.078$\\
3       & period 1 rotational   & $4.761 \pm 0.080 \;$ to $\;5.513 \pm 0.091$\\
	& period 2 vibrational  & $5.848 \pm 0.096 \;$ to $\;6.039 \pm 0.098$\\ \hline
   \end{tabular}
  \end{center}
The values show a surprising uniformity between the points at which a
particular motion type exists in each band. For example period 2 motion in the
upper regions of the first band at $s \approx 1.78$, in the upper regions of
the second band at $s \approx 3.89$, and in the upper regions of the third band
at $s = 5.85$. Thus the motion types are roughly separated by $ s \sim 2$. This
separation is common to other motion types. This feature will be discussed in
more depth in Subsection~\ref{subsec-ExpBifStruc} after the characteristics of
the chaotic motion are detailed.

\subsection{Chaotic Motion}
\label{subsec-ExpChaosMotion}
As already mentioned chaotic motion was observed in the dipole system. Recall
that analytical conditions did not guarantee the existence of such motion,
it was experimental evidence from other studies which suggested the system was
inherently chaotic. The present investigation gives strong proof that
stochastic motion emerges in the lightly damped dipole system.

The three Poincar\'{e} sections of Figure~\ref{fig-ExpChaosMotion} were
produced in each chaotic band. They all show a diffuse scattering of points,
indicative of conservative chaos (Section~\ref{sec-PoinSec2}). The long
$\theta$ time series shows the random-walk nature of the variable in the
chaotic regime. The erratic and unpredictable behaviour of the dipole is
reflected in the $\dot{\theta}$ time series, but note $< \! \dot{\theta} \! >
\rightarrow 0$, a feature associated with random noise which also has equal
probabilty of being either positive or negative. The power spectrum further
reflects this point with characteristics of broad band noise indicating the
signal is {\em aperiodic} (no periodicity).
   \begin{figure}
	\vspace{18cm}
	\caption{Dipole chaotic motion in each of the three bands.}
	\label{fig-ExpChaosMotion}
   \end{figure}

Motion in the chaotic regime was consistent with Arnold diffusion, where the
dipole system `explores' the energetically available phase space by
undergoing random sequences of rotations and vibrations.

Since the experimental setup required changing the driving frequency so that
particularly large values of $s$ could be achieved, it was appropriate to
confirm that the onset of chaos did follow the constant $s$ contours in the
$(f,i)$ parameter plane. It also meant that the borders between the periodic
and chaotic bands could be more accurately defined through greater number of
measurements. The following results were obtained:
   \begin{center}
	\begin{tabular}{|l|c|c|c|}      \hline  \hline
Band number and & Driving       & Coil current  & Stochasticity         \\
which edge      & frequency (Hz)& (Amps)        & parameter     \\ \hline \hline
Onset of        & 3.001093      & 0.520         & 2.011 $\pm$ 0.033     \\
first           & 2.500912      & 0.339         & 1.956 $\pm$ 0.033     \\
chaotic         & 2.000729      & 0.221         & 1.985 $\pm$ 0.035     \\
band            & 1.500546      & 0.126         & 2.022 $\pm$ 0.041     \\
		& 1.000365      & 0.0577        & 2.117 $\pm$ 0.066     \\ \hline
End of          & 2.500913      & 0.642         & 2.678 $\pm$ 0.043     \\
first           & 2.000730      & 0.375         & 2.569 $\pm$ 0.043     \\
chaotic         & 1.500548      & 0.209         & 2.576 $\pm$ 0.046     \\
band            & 1.000366      & 0.0849        & 2.522 $\pm$ 0.062     \\ \hline
Onset of second & 1.500548      & 0.507         & 3.973 $\pm$ 0.065     \\
chaotic band    & 1.000367      & 0.219         & 3.952 $\pm$ 0.069     \\ \hline
End of second   & 1.500543      & 0.697         & 4.649 $\pm$ 0.075     \\
chaotic band    & 1.000367      & 0.298         & 4.591 $\pm$ 0.077     \\ \hline
Onset of third  & 1.000364      & 0.521         & 6.039 $\pm$ 0.098     \\ 
			\hline  \hline
   \end{tabular}
  \end{center}
Thus the $s$ contours do reproduce the observed behaviour. From the data it can
be seen that it gets progressively harder to study the higher $s$ regions as
they are only reached for low frequencies and high currents. Performing a
weighted means analysis on the data yielded the following values for the
borders between the periodic and chaotic bands:
   \begin{center}
    \begin{tabular}{lc}
Description of border           & Weighted mean value of $s$    \\ \hline
First periodic to chaotic       & 1.999 $\pm$ 0.017             \\
First chaotic to periodic       & 2.597 $\pm$ 0.023             \\
Second periodic to chaotic      & 3.963 $\pm$ 0.047             \\
Second chaotic to periodic      & 4.621 $\pm$ 0.054             \\
Third periodic to chaotic       & 6.039 $\pm$ 0.098             \\ \hline
   \end{tabular}
  \end{center}

These results show that the chaotic bands are uniformly spaced as was the case
for the changes in periodic motion. By plotting these $s$ values as contours in
the $(f,i)$ parameter space Figure~\ref{fig-RevisParaPlane} has been
constructed, showing the revised version of parameter space where chaotic
motion is anticipated.
   \begin{figure}
	\vspace{6cm}
	\caption{New $s$ contours in $(f,i)$ parameter space which mark
		transition between periodic and chaotic behaviour.}
	\label{fig-RevisParaPlane}
   \end{figure}

\subsection{Bifurcation Structure}
\label{subsec-ExpBifStruc}
During the course of investigation the $\theta$ values were recorded at various
points in the parameter space with the aim of producing a bifurcation diagram
for the dipole system. The values used in the diagram were obtained in the
following manner:
   \begin{itemize}
	\item   Period 1 rotational states obviously do not produce a constant
		$\theta$ value due to the synchronous rotation. The values used
		in the bifurcation diagram correspond to the amplitude of the
		superposed librational motion, normalised above zero.
	\item   Period 1 vibrational states correspond to the oscillation of
		the dipole about a fixed point with period one. The amplitude
		of this oscillation varied with $s$ so this was ideal for use
		as the $\theta$ bifurcation value. These amplitudes have
		similarly been normalised above zero.
	\item   For period 2 vibrational motion the {\em peak-to-peak}
		amplitude was used and normalised {\em about} zero.
   \end{itemize}
These steps had to be carried out by hand as the computer software is
incapable of determining the initial start angle. 

The data obtained by this method is given in Appendix~\ref{appx-ExpBifData}.
Figure~\ref{fig-ExpBifDia} shows the bifurcation diagram for each periodic
band, and a combined diagram for the whole region of parameter space
investigated. 
   \begin{figure}
	\vspace{19cm}
	\caption{Bifurcation diagrams determined from the experimental data.
		Top Left: first periodic band. Top Right: second periodic
		band. Bottom Left: third periodic band. Bottom Right:
		combined plot of whole $s$ parameter plane.}
	\label{fig-ExpBifDia}
   \end{figure}

These diagrams dramatically show the uniform nature of the band structure. To
confirm this phenomenon the $s$ spacing between analogous motion types has been
calculated and the results are represented in Table~\ref{tab-UniSpacing}. The
method for describing which difference has been taken is as follows. Each
motion type has been designated a code of the form p$n$RTb$o$. The p$n$ tells
the periodicity of the motion (p1:period 1, p2:period 2), R
describes the region of the band (L:lower, M:middle, U:upper), T the
transition type (N:minimum of the motion type, X:maximum), while b$o$ is for
the band number (b1:band 1, b2:band 2, b3:band 3). This code had to be
introduced due to the number of possible linear combinations arising from the
data. 
   \begin{table}
    \begin{center}
	\begin{tabular}{|c|c|}  \hline
Motion type             &       $\Delta s$              \\      \hline
p2LNb2 - p2LNb1         &       1.600 $\pm$ 0.030       \\
p2LXb2 - p2LXb1         &       1.762 $\pm$ 0.052       \\
p1MNb2 - p1MNb1         &       2.393 $\pm$ 0.057       \\
p1MXb2 - p1MXb1         &       1.847 $\pm$ 0.072       \\
p2UNb2 - p2UNb1         &       2.108 $\pm$ 0.072       \\
p2UXb2 - p2UXb1         &       1.986 $\pm$ 0.057       \\
p2LNb3 - p2LNb2         &       2.024 $\pm$ 0.081       \\
p2LXb3 - p2LXb2         &       1.806 $\pm$ 0.047       \\
p1MNb3 - p1MNb2         &       1.964 $\pm$ 0.092       \\
p1MXb3 - p1MXb2         &       1.860 $\pm$ 0.112       \\
p2UNb3 - p2UNb2         &       1.957 $\pm$ 0.118       \\
p2UXb3 - p2UXb2         &       2.076 $\pm$ 0.109       \\      \hline
	\end{tabular}
    \end{center}
	\caption{Uniform $s$ spacing between similar bifurcation points in
		different bands. The coding system is described in the text.}
	\label{tab-UniSpacing}
    \end{table}

The close proximity of many of these values to $\Delta s \sim 2$ (particularly
between the chaotic bands) implies there must be a theoretical basis for the
phenomenon. Some explanations will be offered once the numerical results have
been detailed (Section~\ref{sec-ExpandNumRes}). This uniform nature of the band
structure has not been previously reported.

\subsection{Scaling Constant}
\label{subsec-LowDissScalCons}
What is immediately apparent is that period 4 motion was not observed, and thus
the central aim of measuring the universal scaling constant could not be
carried out. Every effort was made to try and observe the motion, warranting
a minimum of 40 hours of experimentation, but no method was successful in
producing the motion. The possible reasons for the inability to resolve the
behaviour are discussed in Subsection~\ref{subsec-period4discuss}, after the numerical
results of attempting to model the motion have been given.

In the low dissipation regime, the Jacobian for the dipole system had the value
	\[      \det J = 0.910  \]
at a driving frequency of 3Hz, so the first few bifurcations scale at the
conservative rate. It was unfortunate that period 4 (and period 8) motion could
not be observed as the use of a real physical system, in which friction is
unavoidable, to measure a quantity associated with conservative systems is an
exciting prospect. An approximate value for the conservative scaling constant
can be derived from the experimental data in the manner outlined below.

Recall that the $s$ contours were studied to confirm that different points in
$(f,i)$ parameter space with the same $s$ value produced the same behaviour. In
particular the point at which chaotic behaviour occurred in the first band was
determined to substantial accuracy, with value $s = 1.999 \pm 0.017$. This
differs from the value at which maximum period 2 motion is sustained in this
band ($s = 1.977 \pm 0.032$). In addition, the lower point of this period 2
motion is known, with value $s = 1.783 \pm 0.022$. These points form the basis
of the calculation.

Approximating $s = 1.999$ to $s_{3}$ (valid for rapid convergence), $s=1.977$
to $s_{2}$ and $s=1.783$ to $s_{1}$ then the value for the scaling constant is
	\[ \delta \approx \frac{s_{3} - s_{2}}{s_{2} - s_{1}} = 8.818   \]
which is surprisingly close to the expected value of 8.721\ldots in the
conservative regime. However, the method is very approximate so little weight
can be given to the result.

\section{High Dissipation Case}
Unfortunately time restrictions did not allow for data collection in the high
dissipation regime. However, preliminary work with the apparatus indicates
damping as high as 
	\[ \alpha \sim 2.3 \; {\rm s}^{-1} \]
is achievable. At such levels the susceptibility of the dipole system to noise
fluctuations was reduced, so the apparatus is ideally suited for use in the
undergraduate laboratory. More details about the use of the apparatus in an
undergraduate experiment are discussed in Chapter~\ref{chpt-Discussion}.


\chapter{Numerical Analysis}
\label{chpt-numres}
This chapter details the numerical work that has been performed. The reader is
aware of the results obtained in the last chapter. As was shown these were
quite different from Tisch's (1989) work with the same apparatus. Fortunately
the numerical results comply with the experimental results obtained by the
author. 

\section{Computer Software}
The differential equation solver used was a program called {\sf ODE} ({\sf
O}rdinary {\sf D}ifferential {\sf E}quations) version 2.6, written by K. Briggs
(1990). This is a powerful package, capable of solving up to 18 first order
equations using a variety of solving methods. The wide choice it offers the
user in both set-up, solving and graphing method made it far superior for
solving ordinary differential equations than generalised packages such as {\em
Mathematica} Version 2 (developed by Wolfram Research Inc., 1991). However,
the latter has been used to plot the the bifurcation diagrams and several other
figures in the thesis.

\subsection{Method of Solution}
A variety of solution methods are possible, ranging from 1st order Euler's to
Burlirsch-Stoer's method. The step-size, a crucial feature, is also specified
by the user. The $4^{\rm th}$ order Runge-Kutta technique with step size of
$h=0.005$ seconds was found to be adequate for the dipole system (the time
interval for the solution was 5 seconds with 1000 steps). {\sf ODE} works to
extended double precision (19 decimal places) so that round-off errors are
minimised.

\subsection{Equations of Motion}
{\sf ODE} is only capable of solving first order differential equations so the
dipole equation of motion had to be programmed in its coupled first order form.
Also, the dipole could be in a rotational state with a superposed libration. To
resolve this libration it is necessary to convert to the rotating frame, as was
done experimentally. Thus two versions of the equation of motion had to be
used, one for vibrational states, the other for rotational states. This meant
two files for the equations of motion had to be made, the one used depending on
the type of motion encountered. For both equations of motion the time
dependence was removed by defining a variable $z=\omega \, t$. From previous
work it is known this makes the equations autonomous, but in using the solving
package this may seem an unnecessary step as the computer can solve the
nonautonomous equation. However, this inclusion is very useful as it allows for
the removal of transient behaviour from the solution in the following manner.

\subsubsection{Solution Procedure}
{\sf ODE} requires the initial conditions to be specified before the
differential equations. Since the conditions which place the trajectory
immediately on the attractor are not known, several solutions over the
specified time interval must be carried out before the trajectory settles down
onto the attractor. This is the numerical equivalent of the decay of transient
behaviour in the basin of attraction to periodic behaviour when on the
attractor. The time taken to ``find'' the attractor varied on the initial
conditions and on the location in parameter space where a stable solution was
being sought. A very helpful feature of {\sf ODE} is that it can display the
current solution values. When a stable trajectory is finally found the
variable values (including time) can be substituted as the initial conditions.
Then from the onset of solving, a stable trajectory is observed. This has two
extra advantages:
    \begin{itemize}
	\item   For small changes in the dependent parameters, the substituted
		initial conditions are usually near the new attractor so that
		only a few solutions over the time interval have to be computed
		before totally stable behaviour is recovered. Only when the
		solution is near a bifurcation point does this become harder to
		do.
	\item   Computer print-outs can be readily obtained without transient
		behaviour included in the plot.
   \end{itemize}
This method of finding a stable trajectory for some parameter values ($F$ or
$\omega$) and then successively using the stable variable values
($\theta,\dot{\theta},z$) as the new initial conditions for slightly different
parameter values, was used throughout the numerical investigation. 

{\sf ODE} could also give the $\theta$ or $\psi$ values used for construction
of the bifurcation diagram as was done for the experimental results.

\section{Numerical Results}
The numerical solutions show remarkable similarity to the experimental results.
All motion types observed experimentally were replicated, except for the period
3 motion (although there was some evidence for this). In addition a period 4
attractor was observed, but only tentatively. This attractor was highly
sensitive to perturbations (in the form of round-off and discrete step sizes),
mimicking the highly sensitive nature found in the experimental situation.

Selected numerical results have been included in the following pages.
Transitions between different mode-locked states and different periodic
behaviour are shown. Importantly, the numerical data supports the existence of
a periodic band structure with bands of chaotic motion separating them. The
experimental and numerical value of the stochasticity parameter for the onset
of chaos is also similar. The reader should be familiar with the types of
motion being discussed from the treatment of the experimental results in the
last chapter.

The numerical results have been presented in a similar fashion to the
experimental results. For all results $\alpha = 0.2831 \; {\rm s}^{-1}$.
	\begin{figure}
		\vspace{19cm}
		\caption{Period 1 rotational motion in the lower regions of the
			first periodic band. This motion occurs below the
			period 2 vibrational motion, complying with the
			experimental results. Note the lower sequence displays
			the need to convert to the rotating frame.}
	\end{figure}
	\begin{figure}
		\vspace{19cm}
		\caption{Period 2 vibrational motion in the lower regions of
			the first periodic band. The lower phase portraits
			show the change from lock-in with one of the
			half-fields to the oppositely rotating half-field.}
	\end{figure}
	\begin{figure}
		\vspace{19cm}
		\caption{Period 1 rotational motion in first periodic band.
			The dramatic change in the phase portraits is due to a
			resonance.}
	\end{figure}
	\begin{figure}
		\vspace{18cm}
		\caption{Period 1 rotational motion in first periodic band.
			More resonance phenomena is shown here.}
	\end{figure}
	\begin{figure}
		\vspace{18cm}
		\caption{Top phase portraits: period 1 rotational motion in
			first periodic band. These are different to the
			previous diagrams as they have been produced from
			selecting initial conditions which produce rotation in
			the other direction. Correspondingly the equation of
			motion had to be changed to account for this.
			Lower diagrams: period 2 vibrational motion in upper
			regions of the first periodic band. The time series
			reflect the change in period.}
	\end{figure}
	\begin{figure}
		\vspace{18cm}
		\caption{Top: Period 2 vibrational motion in the upper regions
			of the first periodic band.
			Middle: Period 2 resonance.
			Bottom: Unstable period 4 vibrational motion in the
			upper regions of the first periodic band. This
			attractor was often observed in the transient behaviour
			of the dipole, but computational noise made it
			difficult to sustain motion on the attractor.}
	\end{figure}
	\begin{figure}
		\vspace{18cm}
		\caption{Top 3 diagrams: period 2 motion in the lower regions
			of the second periodic band. Resonances were also
			observed (second down).
			Bottom: period 1 vibrational motion in the second
			periodic band.}
	\end{figure}
	\begin{figure}
		\vspace{19cm}
		\caption{Top 2 left diagrams: period 1 vibrational motion in
			the second periodic band.
			3rd down: period 1 vibrational resonance in the second
			periodic band. Note this is in the other direction to
			the top diagrams.
			Bottom and 2nd down, right: period 2 motion in the
			upper regions of the second periodic band.}
	\end{figure}
	\begin{figure}
		\vspace{17cm}
		\caption{Top two levels: period 2 vibrational in the lower
			regions of the third periodic window.
			Bottom: period 1 rotational motion in the third
			periodic band.}
	\end{figure}
	\begin{figure}
		\vspace{18cm}
		\caption{Top 3 levels: period 1 rotational motion in the third
			periodic band. Note the resonance instability that
			leads to period 2 motion.
			Period 2 vibrational motion in the upper regions of the
			third periodic band.}
	\end{figure}

The borders between the various motion types are given, as was done for the
experimental results.
   \begin{center}
   \begin{tabular}{|c|l|c|}     \hline
Band No.& Motion Type           & $s$ range             \\      \hline
	& period 2 vibrational  & 0.9432 to 1.0260      \\
1       & period 1 rotational   & 0.2466 to 1.8783      \\
	& period 2 vibrational  & 1.8515 to 2.0354      \\ \hline
	& period 2 vibrational  & 2.7336 to 2.9987      \\
2       & period 1 vibrational  & 2.8868 to 3.8396      \\
	& period 2 vibrational  & 4.0545 to 4.1533      \\ \hline
	& period 2 vibrational  & 4.7052 to 4.9620      \\
3       & period 1 rotational   & 5.0001 to 5.7890      \\
	& period 2 vibrational  & 6.1133 to 6.1927      \\ \hline
   \end{tabular}
  \end{center}  

In exactly the same manner as carried out with the experimental data, the $s$
spacing between similar points in the bands can be determined.
Table~\ref{tab-NumUniSpacing} shows the results for the numerical data. The
same coding system has been used as that for the experimental data equivalent
(see Subsection~\ref{subsec-ExpBifStruc} for description).
   \begin{table}
    \begin{center}
	\begin{tabular}{|c|c|}  \hline
Motion type             &       $\Delta s$      \\      \hline
p2LNb2 - p2LNb1         &       1.7904          \\
p2LXb2 - p2LXb1         &       1.9727          \\
p1MNb2 - p1MNb1         &       2.6402          \\
p1MXb2 - p1MXb1         &       1.9613          \\
p2UNb2 - p2UNb1         &       2.2030          \\
p2UXb2 - p2UXb1         &       2.1179          \\
p2LNb3 - p2LNb2         &       1.9716          \\
p2LXb3 - p2LXb2         &       1.9633          \\
p1MNb3 - p1MNb2         &       2.1133          \\
p1MXb3 - p1MXb2         &       1.9494          \\
p2UNb3 - p2UNb2         &       2.0588          \\
p2UXb3 - p2UXb2         &       2.0394          \\      \hline
	\end{tabular}
    \end{center}
	\caption{Uniform $s$ spacing between similar bifurcation points in
		different bands. The coding system is described in the text.}
	\label{tab-NumUniSpacing}
    \end{table}
The numerical modelling confirms a uniform spacing.

\subsection{Numerical Bifurcation Diagram}
From the data shown, and during the course of the numerical investigation,
values of $\theta$ (or $\psi$) were collected for different parameter values.
These have been used to construct a bifurcation diagram. The technique used is
the same as that described for the experimental results.
    \begin{figure}
	\vspace{18cm}
	\caption{Bifurcation diagrams determined from the numerical data.
		Top Left: first periodic band. Top Right: second periodic
		band. Bottom Left: third periodic band. Bottom Right:
		combined plot of whole $s$ parameter plane.}
	\label{fig-NumBifDia}
   \end{figure}
The numerical modelling is seen to produce many more values for the plotting of
the bifurcation diagrams. Figure~\ref{fig-NumBifDia} displays much greater
detail of the bifurcation than that determined from the experimental results.
The numerical values are of much higher quality, but the experimental values
do comply with these.



\chapter{Discussion}
\label{chpt-Discussion}
The results of both the experimental and numerical work are now collected and
discussed. In addition a summary of the results of all other research with the
dipole system is given and they are compared with values determined in the
investigation. 

\section{Numerical and Experimental Results}
\label{sec-ExpandNumRes}
The previous chapter, detailing the computer modelling of the equations of
motion, dramatically showed the excellent correlation between the experimental
and numerical results. Not only were the many complicated motions observed
experimentally replicated in the numerical situation, but the threshold $s$
values corresponding to changes in behaviour were very similar. Hysteresis
phenomena were encountered in both methods, indicative of subcritical
bifurcations arising from the coalescence of saddle-node and period-doubling
bifurcations. The existence of the periodic band structure was confirmed
numerically, and a regular bifurcation pattern was recognised where the motion
was period-doubled vibrational in the low region of each band, period 1
(rotational or vibrational) in the middle of the band, and a return to
period-doubled vibrational at the upper regions of the band. Such a structure
is predicted by Scheffcyzk {\em et al.} (1991) and is universal to
dissipative, driven, nonlinear oscillators. The period-annihilating cascade
(the opposite of the period-doubling cacade) has also been reported in other 
nonlinear systems (Dawson {\em et al.}, 1992). 

The uniform nature of the band structure found in both cases, has been
commented upon, but no explanation has been given. The interaction of the
resonances must play a crucial role in explaining the phenomenon as
consideration of the way the resonances change as the stochasticity parameter
varies led to the Chirikov condition. Since the width between similar motion
types is approximately $\Delta s = 2$ which is also the threshold for the onset
of chaos in the first band, it appears the resonances undergo a regular
overlapping and separation process. Thus the band structure occurs when the
resonances uniformly move through and apart from one other.  The motion
replicates itself when they tend to re-converge and overlap again. 

This is a very simplistic explanation of the uniform nature of the band
structure. A more thorough treatment would examine the exact way in which the
resonances change with the stochasticity parameter. Due to the complexity of
such a study (Chirikov avoids this due to the difficulty), the simplistic
explanation is the best that can be done.

\section{Comparison of Results}
There have been a number of similar investigations carried out on the dipole
system by other researchers. The opportunity is presently taken to compare
their results with those of the experimental and numerical study performed by
the author.

\subsection{First Periodic Band}
There has only been one other study which has reported the existence of the
band structure, so the comparison of the results in the higher regions is
delayed until Subsection~\ref{subsec-CompBandStruct}. 

The following table assimilates all the known results for dipole behaviour in
the first periodic band. The numerical values in the table correspond to the
$s$ value.
\scriptsize
   \begin{center}
	\begin{tabular}{l|c|c|c|c|c|c}  \hline  \hline
Motion  & \multicolumn{2}{c|}{Jones ($\alpha$ = 0.2831)}  & Tisch  & Croquette &
		\multicolumn{2}{c}{Meissner}                            \\
type    & Exp.  & Num.  &  $\alpha$=0.277 & $\alpha$=0.47 & $\alpha$=0.15 &
						$\alpha$=1.38   \\ \hline
period 3&0.895  & *     & *     & *     & 1.3 to 1.32 & *       \\
period 2 (low)& 1.06 to 1.09 & 0.94 to 1.03 & * & * & * & *     \\
period 1&0.40 to 1.81&0.25 to 1.88& to 1.59& to 1.74& to 1.65& to 1.76  \\
period 2 (far)&1.78 to 1.98&1.85 to 2.04&1.59 to 1.83&1.74 to 1.93&1.65 to
						1.87&1.76 to 1.97       \\
period 4& * & * & * & 1.93 to 1.94& * & 1.97 to 2.01            \\ 
chaos   &above 2&above 2.04&above 1.83&above 1.94&above 1.87&above 1.97 \\
							\hline  \hline 
	\end{tabular}
   \end{center}
\normalsize
where an asterix means that the motion type was not observed. From the table it is
obvious the present results are more similar to those of Croquette \& Poitou
(1981) and Meissner \& Schmidt (1986) than those of Tisch (1989). The numerical
work supports the experimental results, so confidence is placed on the most recent
findings. 

The reason why such different behaviour has been reported by Tisch (1989) when
working with the same apparatus is probably the suppression of the periodic
regime by the influence of noise. He recognised that noise levels in the
previous setup were unacceptably high. The reduction of these effects has
resulted in more accurate data and results. 

The apparatus of Meissner \& Schmidt (1986) can apparently achieve very low
dissipation values. Unfortunately they did not utilize this as they performed
measurements at a driving frequency of 1 Hz. Thus the effective damping is
$\alpha / \omega = 0.024$ while the effective damping in the present
investigation was 0.015 at 3 Hz. In any case the method by which they
determined the damping constant was only approximate as they relied on the fact
that a finite field amplitude is required to sustain rotation, relating the
minimum amplitude to the damping present.

\subsection{Band Structure}
\label{subsec-CompBandStruct}
Ballico {\em et al.} (1990) were the first to report the existence of the band
structure. They performed a study into the dynamics of the dipole using a 
device with damping $\alpha=0.9 \;{\rm s}^{-1}$ at a fixed driving frequency of
2 Hz. Their results have been compared with the present ones in 
Table~\ref{tab-BallicoComparison}.
 \begin{table}
   \begin{center}
	\begin{tabular}{|l|c|c|c|}        \hline \hline
Motion type  & \multicolumn{2}{c|}{Jones $(\alpha = 0.2831 \; {\rm s}^{-1})$} 
							     & Ballico \\
and/or band no& Exp.             & Num.             & $\alpha = 0.9 \; 
						       {\rm s}^{-1}$    \\ \hline
first band    & 0.40 to 1.98     & 0.25 to 2.04     & * to 1.95         \\
period 1      & rot 0.40 to 1.81 & rot 0.25 to 1.88 & rot up to *    \\
period 2 (far)& vib 1.78 to 1.98 & vib 1.85 to 2.04 & * to 1.95  \\ 
chaotic       & above 2          & above 2.04       & above 1.95        \\ \hline 
second band   & 2.60 to 3.96     & 2.73 to 4.15     & vib 2.68 to 3.68  \\      
period 2 (low)& vib 2.60 to 2.85 & vib 2.73 to 3.00 & *             \\       
period 1      & vib 2.80 to 3.65 & vib 2.88 to 3.84 & vib *             \\
period 2 (far)& vib 3.89 to 3.96 & vib 4.05 to 4.15 & *             \\  
chaotic       & above 3.96       & above 4.15       & above 3.68        \\ \hline
third band    & 4.62 to 6.04     & 4.71 to 6.19     & 4.88 to *         \\
period 2 (low)& vib 4.62 to 4.66 & vib 4.71 to 4.96 & *     \\
period 1      & rot 4.76 to 5.51 & rot 5.00 to 5.78 & rot *            \\
period 2 (far)& vib 5.85 to 6.04 & vib 6.11 to 6.19 & *             \\
chaotic       & above 6.04       & above 6.19       & *         \\ \hline \hline
	\end{tabular}
    \end{center}
  \caption{Comparison of the experimental and numerical results for $s$
	obtained in this investigation with those of Ballico {\em et al.}
	(1990).} 
   \label{tab-BallicoComparison}
 \end{table}
where an asterix means that the $s$ value has not been given by Ballico 
{\em et al.} (1990). Their thresholds are in good agreement with those obtained
by the author, both experimentally and numerically. Unfortunately, they do not
detail any values for the transitions between periodic and period-doubled 
behaviour so that comparisons cannot be made. Note that they also observed the 
period 1 vibrational motion in the second band. 

From Table~\ref{tab-BallicoComparison} the numerical and experimental values 
diverge as one moves to the higher bands. This may be attributed to several 
factors:
   \begin{enumerate}
	\item   The uncertainty in the experimental results has not been given
		in the Table~\ref{tab-BallicoComparison}. This uncertainty is 
		quite large in the higher bands because the system is studied at
		high currents and low frequencies. When these are taken into 
		account greater agreement between numerical calculations and 
		experimental results is achieved (for example the third chaotic 
		band begins at $s=6.19$ numerically, and $s=6.04 \pm 0.10$ 
		experimentally.
	\item   There appears to be a {\em general} increase in the transition values 
		as one studies the system at higher dissipation. The experimental 
		values may be lower than the values obtained from the numerical 
		modelling because the real dissipation was in fact lower than 
		$\alpha = 0.2831 \; {\rm s}^{-1}$. This is unlikely due to the 
		accuracy with which the damping was derived.
   \end{enumerate}

\section{Shortcomings}
Despite the quality of the data which can be readily obtained and reproduced
by the apparatus, there are several problem areas associated with its
performance and inherent characteristics. Both 
experimental and numerical work have shown this.  By recognising these 
shortcomings, future work will not be hindered by the restrictions of the
apparatus. If the device is to be used in the undergraduate laboratory it is
essential that clear experimental goals are laid down. It is also reasonable 
to expect these goals be achievable so that work which is inevitably futile
is not pursued.

\subsection{Period 4}
\label{subsec-period4discuss}
The most signifcant problem is the inability to achieve stable period 4 motion.
The system is particularly sensitive to noise in the low dissipation limit, 
this feature being replicated in the numerical modelling where only tentative 
period 4 motion was observed. In the higher dissipation regime, period 4 motion has
been found by other workers. However, to observe the motion, highly
precise equipment was used. The only researchers to 
report the period 4 behaviour in their studies were Croquette \& Poitou (1981)
and Meissner \& Schmidt (1986). Interestingly, both their setups had an
additional pair of large Helmholtz coils arranged about the dipole. These were
used to compensate for the Earth's magnetic field as this is a small, but constant 
perturbation to the dipole motion. Since the inclusion of such a set of coils 
in the experimental appartus would have made it difficult to use, this step was
not taken. The fact that the numerical modelling, which did not have this 
constant perturbation, was unable to produce stable period 4 motion shows that
in the low dissipation regime the period 4 motion is {\em inherently} difficult
to observe. Thus a compensating set of Helmholtz coils would not have helped 
resolve the motion.

There are several reasons as to why the period 4 motion is so difficult to
observe in the system
    \begin{enumerate}
	\item   The period-doubling bifurcation from period 2 motion to period
		4 motion may be subcritical as many of the other transitions
		are. This would mean that numerous attractors coexisted in
		a small region of $s$. Thus noise fluctuations cause multiple
		transitions between them resulting in a sudden and early 
		onset to stochastic behaviour (Section~\ref{sec-Noise}). 
	\item   Noise is also responsible for the destabilization of one of the
		steady state solutions when attractors coexist in the same region
		of parameter space. The period 4 steady state is most likely 
		destabilized first since the trajectories of period 2 motion
		are not near the period 4 attractor.
	\item   These effects are further pronounced in the low dissipation case 
		as the first few bifurcations scale close to the conservative rate.
		Not only does this increase the probability of transition between
		the attractors, but transients take much longer to die out so that
		the steady state period 4 motion is only achieved if the
		initial conditions are very close to the attractor.
    \end{enumerate}
Due to these factors, it is unreasonable to expect the undergraduate to 
be able to measure the universal scaling constant with the device, 
particularly in the low dissipation regime.\footnote{In the high dissipation case the
equipment they use will not have the required stability to enable observation
of the period 4 motion either.}

\subsection{Real Time Analysis}
The existing setup, whereby the information from the motion detectors is first
digitally stored and analysed later does have several advantages. However, there
are some disadvantages such as the stability of the system is not known for the
duration of an experimental observation, nor is the frequency spectrum. Ballico
{\em et al.} (1990) give details on how real time signal analysis can be done 
with the apparatus. The implementation of a real-time frequency analyser would
be particularly useful for study of the $q$-cycles as one would immediately
know if mode-locking had occurred and the type of cycle.

\subsection{Stability of higher bands}
The periodic motion in the higher bands of parameter space were very unstable.
These bands can only be studied if stable and precise equipment is used, or
if the dissipation is increased.

\section{Future Work}
Apart from the use of the apparatus in the undergraduate laboratory, there
are many more aspects of the system's dynamics yet to be studied. Specifically,
the strange attractor of the dipole system, reported by Croquette \& Poitou (1981)
and Meissner \& Schmidt (1986), warrants greater investigation 
(Figure~\ref{fig-DipoleSAttr}). Recent advances
in the study of strange attractors means that many of the statistical measures
of such objects can now be performed with experimental data.\footnote{Most work is
limited to numerical study.} The reader is referred to following papers which
detail these methods: 
   \begin{description}
	\item[Abraham {\em et al.}] (1986) for obtaining the dimension of the attractor 
		from comparatively small amounts of data (between 2 and 3 for the dipole - 
		strange attractors are fractal objects with intermediate dimension).
	\item[Schreiber \& Grassberger] (1991) who detail an algorithm capable of
		removing some of the noise from data sets.
	\item[Zeng {\em et al.}] (1991) for an algorithm to estimate the 
		Lyapunov exponents from a short time series of low precision.
		This technique is a major breakthrough in the study of
		strange attractors (since most data for complex dynamical
		systems is experimental; numerical simulations of a system
		are limited by how well the governing equations are known).
   \end{description}
Much time was spent examining the possible implementation of these methods
for the dipole system. Unfortunately, the computing power needed for many
of the algorithms excluded these techniques from the present study. The interested
reader is also referred to Eckmann \& Ruelle's (1985) review on strange 
attractors and their statistical properties as derived from {\em ergodic}
theory.
   \begin{figure}
	\vspace{5cm}
	\caption{Poincar\'{e} section of the strange attractor for the dipole system
		system, as found by Croquette \& Poitou (1981) [from Berg\'{e}
		{\em et al.}, 1984)].}
	\label{fig-DipoleSAttr}
   \end{figure}



\chapter{Conclusion}
\label{chpt-conclus}

Having collated and discussed the many results of the investigation in the
previous chapter, the following is an account summarising the direction that
has been pursued and its relevance to the study of chaotic dynamical systems.

A comprehensive discussion on dynamical systems, particularly the means by
which chaos can arise in some classes of these, was given in
Chapter~\ref{chpt-genthy}. This background theory should enable a newcomer to
the field to quickly grasp many general concepts involved with such systems and
the study of these. 

Chapter~\ref{chpt-specthy} applied many of the analytical tools introduced in
the background theory to a specific situation, the bipolar motor. Upon
derivation of the nonlinear equation of motion and an approximate description
of some of the behaviour the system should display, the way in which
instabilities arise in the system was examined. This process led to definition
of a tentative criterion for the onset of chaos. The universal features of
the bifurcation structures for systems of the type investigated were
discussed, leading to an understanding of how subcritical bifurcations
may occur in the system, and importantly how they are manifested in the
experimental observations. The condition for which the universal conservative
scaling constant can be measured by the system was derived. The fact that a
conservative quantity can be measured by a strictly dissipative system is an
exciting prospect, and obviously formed one of the experimental aims. 

The details of the apparatus were given in Chapter~\ref{chpt-expset}. Emphasis
was placed on the improvements which have been undertaken to make it suitable
for use in the undergraduate laboratory. Additional information on some
operational characteristics of various components was also outlined. Several of
these features must be taken into consideration when using the device. Only
then can quality data can be obtained. Chapter~\ref{chpt-expmeth} gave an
experimental method which achieves the most stable performance in the minimum
time. This chapter is very useful to the future undergraduate researchers
working with the device.

The experimental results obtained in this investigation were given in
Chapter~\ref{chpt-expres}. A wide variety of behaviour was observed and the
results were seen to be of high quality. Periodic, period-doubling,
period-tripled and stochastic motion were observed in the system. Many of the
results were not encountered by Tisch's \cite{bib-Tis} study. Other
experimental work (\cite{bib-Bal,bib-Cro,bib-Mei}) has reported all the
behaviour observed by the investigation. However, the uniform nature of the
band structure has not received substantial comment, nor have the subcritical
bifurcations. This may be due to the extremely low effective damping achieved
in the study. A value for the conservative scaling was derived from the
experimental results. Despite closeness to the expected value, significant
weight cannot be given to the result due to the approximate means by which it
was obtained.

Chapter~\ref{chpt-numres} displayed compliance between the numerical solutions
of the equations of motion, modelled by computer, and the experimental
behaviour. The excellent accordance between the experimental and numerical
results indicates a high degree of accuracy has been achieved in the
experimental setup.

The problems associated with the system have been recognised in
Chapter~\ref{chpt-Discussion}. Importantly, recommendations have been made for
the experimental aims of an undergraduate study into chaotic dynamics in which
the apparatus would play a central role. Time and equipment restrictions will
necessarily limit such a study to the first band. 

There remains a multitude of advanced studies which could be performed with the
system. Recent advances in the experimental determination of measures of
strange attractors could be applied to the system, signifying an exciting
course that could be taken.

The bipolar motor is an interesting example of a nonlinear system. The advances
which have been made in the analytical study of such systems are
still far from useful for the situation investigated. Only when the
analytical techniques have been improved to handle such equations as the
dipole's will many aspects of its unusual behaviour be accounted for. The
investigation has found additional behaviour which must be considered in the
development of these techiques.

\appendix

\chapter{Coil Impedance Measurements}
\label{appx-CoilImp}

\section{Procedure}
The coils can be represented as an inductor,
with the impedance defined as the sum of the resistance and reactance of the
coils: 
	\[      Z_{L} = R_{L} + X_{L}   \]
where \( R_{L} \) is the resistance and \( X_{L} = j \omega L \) is the complex
quantity reactance ($j=\sqrt{-1}, \;\; \omega$ is the angular frequency of an a.c.\ 
voltage and $L$ is coil inductance). Thus the magnitude of the impedance is:
	\[ \mid \! Z_{L} \! \mid = [ R_{L}^{2} + X_{L}^{2} ]^{1/2}      \]
and is a function of the frequency in the 

Two means can be applied to measure the coil impedance. The first measures the
resistance and inductance of the coil, while the second simply measures the
impedance at a fixed frequency.  Both methods use the circuit shown in 
Figure~\ref{fig-CoilCirDia} to perform the necessary measurements.
   \begin{figure}
	\vspace{10cm}
	\caption{Circuit diagram for obtaining the coil impedance.}
	\label{fig-CoilCirDia}
   \end{figure}

\subsection{Coil Resistance \& Inductance}
A d.c.\  power supply is sent through the coils so that they represent a purely
resistive load. By setting the rheostat so that the CRO displays $V_{2} =
\frac{1}{2} V_{1}$ then the potential divider theorem gives $R_{L} = R_{\rm
rheo.}$. Simple measurement of the rheostat resistance by an accurate ohmmeter
gives one the coil resistance.

It is important to note that a high current capacity rheostat is necessary as
typical currents are 0.1 to 0.8 Amps. The author used two $14\;\Omega$, 1 Amp
capacity potential dividers configured in series so that one operated as a
rheostat, the other as a resistor. The coil impedance is $\approx \, 20 \;
\Omega$, so the rheostat must be able to match this.

The coil inductance was measured using a LCR resonance meter. This is a highly
accurate device which is placed across the terminals of the coils. 

The coil impedance is then:
	\[      Z_{L} = R_{L} + j \omega L      \]

\subsection{Coil Inductance}
Since a LCR resonance meter is not readily available to the undergraduate
investigator, a more practical approach is to measure the coil impedance. In
this method an a.c.\  potential of fixed frequency is applied across the coils.
Once again the rheostat is set so that $V_{2} = \frac{1}{2} V_{1}$. When this
condition is achieved the a.c.\  voltage is turned off so that the resistance of
the rheostat can be measured with an ohmmeter. In this case, the rheostat
resistance corresponds to the coil impedance. This method can be carried out
for different frequencies of alternating current, but as will be seen, a single
investigation near 3Hz suffices.


\section{Resistance and Inductance}
The following table refers to the data obtained for measurement of the
resistance of the coils. A d.c.\  potential was passed through the coils and the
rheostat resistance value $R$ was adjusted so that the potential across the
coils $V_{2}$ was half the input voltage $V_{1}$. The rheostat resistance
values were measured using an accurate multimeter (on ohmmeter setting).
   \begin{center}
	\begin{tabular}{|c|c|c|c|}      \hline
 $V_{1}$ (volts)  &$V_{2}$ (volts)  &  $R$ (ohms)  &  $R_{L}$ (ohms) \\ \hline
	2         &        1        &   21.5       &    21.5            \\
	6         &        3        &   21.5       &    21.5            \\
	8         &        4        &   21.5       &    21.5            \\
	16        &        8        &   21.5       &    21.5    \\      \hline
	\end{tabular}
   \end{center}
It is stressed that at each change of input voltage, the rheostat setting was
changed and then re-positioned so that the condition $V_{2} = \frac{1}{2}V_{1}$
was attained. Thus the similarity in $R_{L}$ values is due to experimental
accuracy, not because the rheostat setting was kept the same.

The ohmmeter reading of the rheostat resistance took into account the resistance
of the probes and contacts of the meter. The device had an error of $\approx
0.1 \; \Omega$. Using weighted means where for $N$ observations: 
	\[      x_{1} \ldots x_{i} \ldots x_{N} \]
with weights
	\[      w_{1} \ldots w_{i} \ldots w_{N} \]
(corresponding to \( w_{i} = 1 / \sigma_{i}^{2} \) ) then the mean result is
	\[      \bar{x}_{\rm best} = \frac{\sum_{i=1}^{N} w_{i} x_{i}}
						{\sum_{i=1}^{N} w_{i}} \]
with standard deviation
	\[      \sigma_{s_{\rm best}} = (\sum_{i=1}^{N} w_{i} )^{-1/2} \]
then the coil resistance is given by:
	\[      R_{L} = (21.5 \pm 0.05) \; \Omega       \]

The coil inductance measuremnt using a LCR resonance meter gave the result:
	\[      L = 15.74 \mbox{ mH}    \]
Thus the impedance is given by
	\[      Z_{L} = 21.5 + j \omega (15.74 \times 10^{-3})  \]
At 3Hz the magnitude is \( \mid \! Z_{L} \! \mid = 21.502 \; \Omega \) so that
at low frequencies the inductance has a neglible effect.

\subsection{Coil Impedance}
In this procedure an a.c.\  potential of frequency 3Hz is applied to the coils.
The rheostat is set so that the peak-to-peak voltage across the coils is half
the input value. The following measurements were taken in a similar method to
above. 
   \begin{center}
	\begin{tabular}{|c|c|c|c|}      \hline
 $V_{1}$ (volts)&$V_{2}$ (volts)&  $R$ (ohms)   &  $Z_{L}$ (ohms) \\ \hline
	1       &       0.5     &       22.2    &       22.2            \\
	2       &       1       &       21.8    &       21.8            \\
	3       &       1.5     &       21.4    &       21.4            \\
	4       &       2       &       21.2    &       21.2            \\
	5       &       2.5     &       21.1    &       21.1            \\
	6       &       3       &       21.6    &       21.6            \\
	7       &       3.5     &       21.2    &       21.2            \\
	8       &       4       &       21.1    &       21.1            \\
	9       &       4.5     &       21.8    &       21.8            \\
	11      &       5.5     &       21.4    &       21.4            \\
	13      &       6.5     &       21.9    &       21.9    \\      \hline
	\end{tabular}
   \end{center}
From these values the coil impedance is
	\[      Z_{L} = (21.52 \pm 0.37) \; \Omega      \]
at 3Hz. This is a direct mean value. Associating a weight to each,
corresponding to a measurement uncertainty of $\approx 0.1 \; \Omega$ then the
following result is obtained
	\[      Z_{L} = (21.52 \pm 0.03) \; \Omega      \]
Note for both values, the results agree with impedance obtained by the d.c.\ 
method. 

\chapter{Friction and Forcing Data}
\label{appx-fric&forcing}
\section{Frictional damping constant}
The computational software which automates the determination of the damping
constant enables one to accumulate a large number of values. The least-square
fit between the natural logarithm of the maxima of $\dot{\theta}$ and time,
also gives the standard error associated with each damping value. Thus a
weighted mean can be calculated as described above. From 85 data
sets\footnote{The data has not be given due to the size and simplicity of it.}
the following value was obtained in the low friction regime:
	\[      \alpha = (0.2831 \pm 0.0006) \; {\rm s}^{-1}    \]
Supplied are several diagrams showing the $\dot{\theta}$ decay time series, as
well as the least-square fit to the data. Included are numerically obtained
decay curves, displaying the reaonable nature of the approximations made in
deriving the analytical expression describing the time series. 
   \begin{figure}
	\vspace{20cm}
	\caption{$\dot{\theta}$ decay time series for several coil currents. To
		the right are numerical simulations of the expected time
		series.} 
   \end{figure}
   \begin{figure}
	\vspace{18cm}
	\caption{Examples of data used in the least square fit. Note the large
		number of points the time series produces, allowing accurate
		determination of the damping constant.}
   \end{figure}

Unforseen complications and time restrictions meant that damping values for the
high dissipation regime were not obtained. Preliminary results indicated that
values as high as
	\[      \alpha \sim 2.3 \; {\rm s}^{-1}                 \]
are achievable. Intermediate values can be obtained by varying the wells used.

\section{Forcing-Current Results}
The computational software also automates the procedure for obtaining this
important relation. Following is a table of the results used. The
uncertainities in the quantities have been given as the software performs a
weighted least-square fit. The uncertainty in the current is of the order of
1\% (this has not been ncluded, but the software accounts for it).


\[      \begin{array}{|c|c|c|} \hline
i $(Amps)$      & f_{u} \pm \sigma_{f_{u}}$(Hz)$& F \pm \sigma{F}($s$^{-2})
									\\ \hline
0.0456          & 1.37 \pm 0.06         & 37.0 \pm 3.1                  \\
0.0456          & 1.36 \pm 0.07         & 36.5 \pm 3.8                  \\
0.0456          & 1.31 \pm 0.09         & 33.9 \pm 4.8                  \\
0.0456          & 1.37 \pm 0.09         & 37.3 \pm 5.0                  \\
0.093           & 1.91 \pm 0.08         & 72.1 \pm 6.3                  \\
0.093           & 1.87 \pm 0.16         & 69 \pm 11.6                   \\
0.093           & 1.83 \pm 0.12         & 66.1 \pm 9.0                  \\
0.140           & 2.29 \pm 0.14         & 103 \pm 13                    \\
0.140           & 2.20 \pm 0.20         & 96 \pm 17                     \\
0.140           & 2.23 \pm 0.18         & 99 \pm 16                     \\
0.186           & 2.60 \pm 0.22         & 133 \pm 23                    \\
0.186           & 2.57 \pm 0.23         & 131 \pm 24                    \\
0.186           & 2.58 \pm 0.21         & 131 \pm 22                    \\
0.233           & 2.91 \pm 0.23         & 167 \pm 26                    \\
0.233           & 2.80 \pm 0.24         & 154 \pm 26                    \\
0.233           & 2.84 \pm 0.29         & 159 \pm 33                    \\
0.279           & 3.21 \pm 0.10         & 203 \pm 13                    \\
0.279           & 2.95 \pm 0.25         & 171 \pm 30                    \\
0.279           & 3.19 \pm 0.11         & 200 \pm 14                    \\
0.326           & 3.47 \pm 0.18         & 238 \pm 25                    \\
0.326           & 3.35 \pm 0.24         & 222 \pm 32                    \\
0.326           & 3.25 \pm 0.45         & 208 \pm 58                    \\
0.372           & 3.73 \pm 0.14         & 274 \pm 20                    \\
0.372           & 3.64 \pm 0.29         & 261 \pm 42                    \\
0.372           & 3.61 \pm 0.24         & 258 \pm 35                    \\
0.465           & 3.90 \pm 0.22         & 301 \pm 34                    \\
0.465           & 3.81 \pm 0.53         & 288 \pm 80                    \\
0.465           & 3.90 \pm 0.39         & 299 \pm 60                    \\
0.558           & 4.38 \pm 0.47         & 379 \pm 82                    \\
0.558           & 4.14 \pm 0.54         & 340 \pm 88                    \\
0.651           & 4.75 \pm 0.24         & 446 \pm 46                    \\
0.651           & 4.20 \pm 0.45         & 348 \pm 73                    \\
0.744           & 4.82 \pm 0.53         & 460 \pm 100                   \\
0.744           & 5.27 \pm 0.28         & 549 \pm 58              \\    \hline
   \end{array}          \]
Recall $F$ is related to $f_{u}$ by 
	\[ F = 2 ( \pi f_{u} )^{2}      \]
consideration of the propagation of errors gives
	\[ \sigma_{F} = 4 \pi^{2} \, f_{u} \, \sigma_{f_{u}}    \]
The weighted least-square fit for the above values yields:
	\[ F = (682 \pm 22)i + (5.0 \pm 2.4) \; {\rm s}^{-2}    \]
Since $s=2 \sqrt{F} / \omega$ then
	\[ s = [69.1 \, i + 0.5]^{1/2} / f      \]
The uncertainty in the value of $s$ is now derived as it is not trivial.
Approximations can be made which simplifies the procedure, but a comprehensive
derivation is useful for displaying the means by which errors propagate in the
equation. Beginning at the $F$ vs $i$ relation, expressing this in the form
   \begin{eqnarray*}
	F & = & (b \pm \sigma_{b})(i \pm \sigma_{i}) + (a \pm \sigma_{a})  \\
	  & = & (P \pm \sigma_{P}) + (a \pm \sigma_{a})                    \\
   \end{eqnarray*}
where $P = b \times i$ and 
  \[     \frac{\sigma_{P}}{P} = \sqrt{
				\left( \frac{\sigma_{b}}{b}\right)^{2} 
				+\left( \frac{\sigma_{i}}{i}\right)^{2}}   \] 
using $\frac{\sigma_{b}}{b} = \frac{22}{682}$ and assuming 1\% error in current
measurement so that $\frac{\sigma_{i}}{i} = 0.01$ then 
	\[       \frac{\sigma_{P}}{P} = 0.0324      \]
Now the uncertainty in $F$ from the linear combination of uncertainties in $P$
and $a$ is given by:
	\[      \sigma_{F} = \sqrt{ \sigma_{P}^{2} + \sigma_{a}^{2} }   \]
and substituting values $\sigma_{P} = 0.0324 P = 22.06 i$ and $\sigma_{a}=2.4$
then    \[      \sigma_{F} = \sqrt{ 486.84 i^{2} + 5.76}        \]
Now $s = 2\sqrt{F} / \omega$, so to calculate the effect of the square root on
the error in $F$, define the variable $R=\sqrt{F}$. Then statistical analysis
yields  
  \[  \frac{\sigma_{R}}{R} = \frac{1}{2} \frac{\sigma_{F}}{F}  \]
so that 
	\[ \sigma_{R} =  \frac{1}{2} \frac{\sigma_{F}}{\sqrt{F}} \]
upon multiplication through by $\sqrt{F}$. 
Since $s = \sqrt{F} / \pi f$ then define another variable $Q=\sqrt{F} / \pi =
R/ \pi$. Then
	\[      \sigma_{Q} =  \frac{ \sigma_{R}}{\pi} =
			 \frac{\sigma_{F}}{2 \pi \sqrt{F}}  \]
Finally the relation $s = Q / f$ indicates that the uncertainty in $Q$ and $f$
should be taken into account. However, the typical uncertainty in $f$ is
0.002\% so this can be ignored (especially in the quadratic sum which is the
method applied for quantities derived from a ratio. That is
$\frac{\sigma_{s}}{s} = \sqrt{ (\frac{\sigma_{Q}}{Q})^{2} +
(\frac{\sigma_{f}}{f})^{2} }$ and so typically $\frac{\sigma_{f}}{f} \sim
10^{-10}$). So the uncertainty in $s$ is proportional to the uncertainty in $Q$
yielding
	\[  \frac{\sigma_{s}}{s} = \frac{\sigma{Q}}{Q}      \]
so
   \begin{eqnarray*}
	 \sigma_{s} & = &  \frac{\sigma_{F}}{2 \pi \sqrt{F} }  \\
		    & = & \frac{ \sqrt{ 486.84 i^{2} + 5.76 }}{2 \pi f 
				\sqrt{ 682 i + 5 } }                    \\
		    & = & \frac{ \sqrt{12.33 i^{2} + 0.15} } 
				{f \, \sqrt{682 \, i + 5} }             
  \end{eqnarray*}
 
\chapter{Experimental Bifurcation Data}
\label{appx-ExpBifData}
The experimental data used to construct the bifurcation diagram is given here.
The values were obtained from the $\theta$ time series at various points in 
parameter space. From the data obtained, a means of normalizing the data had
to be employed. This was because the apparatus is incapable of determining the 
initial start angle, so actual $\theta$ values can vary greatly between samples.
However, the range over which $\theta$ varies in each case was dependent on the
position on the $s$ parameter plane. This formed the basis for the bifurcation 
diagram. By measuring the range of $\theta$ in the experimental sample, 
appropriate values could be obtained by consideration of the motion type. This 
is easily seen in the following table of results, where one column gives the 
range of $\theta$ for the stated $s$, and the next column gives the actual
values that were used in the plot. 

The same method was used in the construction
of the bifurcation diagram for the numerical results, but since these are so
extensive they have not been given. 

Note that if the motion was period 1 
rotational then obviously $\theta$ was unbounded. The values which have been 
used here correspond to the size of the superposed libration, measured by 
transforming to the $\psi$ frame. The (low/far) in the table corresponds to 
which end of the 
band the motion occurs.
\small
   \begin{center}
	\begin{tabular}{|l|c|c|c|}      \hline \hline \hline
Band number and & $s$ value             & $\Delta \theta$ & Values used      \\
motion type     & and error             & (radians)     & in diagram    \\ \hline \hline
Band 1          &                       &               &               \\  
period 2 (low)  & 1.055  $\pm$  0.021   & 2.007         & $\pm$  1.0035  \\
vibrational     & 1.090  $\pm$  0.021   & 2.369         & $\pm$  1.1845  \\ \hline
period 3 (vib)  & 0.895  $\pm$  0.021   & 3.880         & 0, $\pm$ 1.293   \\ \hline
period 1        & 0.638  $\pm$  0.023   & 0.103         & 0.103         \\
		& 0.769  $\pm$  0.021   & 0.112         & 0.112         \\
rotational      & 0.878  $\pm$  0.020   & 0.168         & 0.168         \\
		& 1.028  $\pm$  0.021   & 0.274         & 0.274         \\
		& 1.155  $\pm$  0.022   & 0.285         & 0.285         \\
		& 1.203  $\pm$  0.022   & 0.273         & 0.273         \\
		& 1.227  $\pm$  0.022   & 0.315         & 0.315         \\
		& 1.239  $\pm$  0.022   & 0.399         & 0.399         \\
		& 1.250  $\pm$  0.022   & 0.310         & 0.310         \\
		& 1.494  $\pm$  0.025   & 0.473         & 0.473         \\
		& 1.588  $\pm$  0.027   & 0.783         & 0.783         \\
		& 1.639  $\pm$  0.027   & 1.197         & 1.197         \\
		& 1.706  $\pm$  0.028   & 1.340         & 1.340         \\
		& 1.719  $\pm$  0.028   & 1.936         & 1.936         \\ \hline
period 2 (far)  & 1.776  $\pm$  0.029   & 10.02         & $\pm$   5.01    \\
vibrational     & 1.850  $\pm$  0.030   & 9.64          & $\pm$   4.82    \\
		& 1.939  $\pm$  0.032   & 9.53          & $\pm$   4.765   \\
		& 1.967  $\pm$  0.032   & 9.27          & $\pm$   4.635   \\
		& 1.967  $\pm$  0.032   & 9.07          & $\pm$   4.535   \\
		& 1.970  $\pm$  0.032   & 9.09          & $\pm$   4.545   \\
		& 1.977  $\pm$  0.032   & 6.747         & $\pm$   3.3735  \\ \hline \hline
Band 2          &                       &               &               \\
period 2 (low)  & 2.664  $\pm$  0.044   & 9.06          & $\pm$   4.53    \\
vibrational     & 2.769  $\pm$  0.045   & 8.22          & $\pm$   4.11    \\
		& 2.813  $\pm$  0.046   & 2.387         & $\pm$   1.1935  \\ \hline
period 1        & 2.797  $\pm$  0.046   & 6.56          & 3.28          \\
vibrational     & 2.867  $\pm$  0.047   & 6.36          & 3.18          \\      
		& 3.305  $\pm$  0.063   & 6.14          & 3.07          \\
		& 3.528  $\pm$  0.065   & 6.01          & 3.005         \\
		& 3.605  $\pm$  0.066   & 6.36          & 3.18          \\
		& 3.653  $\pm$  0.066   & 6.33          & 3.165         \\ \hline
period 2 (far)  & 3.891  $\pm$  0.069   & 10.56         & $\pm$   5.28    \\
vibrational     & 3.952  $\pm$  0.069   & 10.33         & $\pm$   5.165   \\ \hline \hline
Band 3          &                       &               &               \\
period 2 (low)  & 4.591  $\pm$  0.077   & 10.45         & $\pm$   5.225   \\ \hline
period 1        & 4.826  $\pm$  0.081   & 4.53          & 4.53          \\
rotational      & 5.083  $\pm$  0.084   & 3.69          & 3.69          \\
		& 5.335  $\pm$  0.088   & 3.429         & 3.429         \\ \hline
period 2 (far)  & 5.906  $\pm$  0.096   & 9.15          & $\pm$   4.575   \\
			\hline \hline \hline
     \end{tabular}
  \end{center}

\normalsize

\begin{thebibliography}{99}
	\bibitem{bib-Abr} Abraham, N.B., Albano, A.M., Das, B., De Guzman, G.,
		Yong, S., Gioggia, R.S., Puccioni, G.P., Tredicce, J.R. (1986)
		{\em Calculating the dimension of attractors from small data
		sets} Phys. Lett. A {\bf 114}, 217
	\bibitem{bib-Bal} Ballico, M.J., Sawley, M.L., Skiff, F.(1990) {\em The
		bipolar motor: a simple demonstration of deterministic chaos}
		Am. J. Phys. {\bf 58}, 58
	\bibitem{bib-Ber} Berg\'{e}, P., Pomeau, Y., Vidal, C. (1984) {\em
		Order within Chaos, towards a deterministic approach to
		turbulance} (John Wiley \& Sons, Paris)
	\bibitem{bib-Chi} Chirikov, B.V. (1979) {\em A universal instability of
		many-dimensional oscillator systems} Phys. Rep. {\bf 52}, 463
	\bibitem{bib-Cor} Corless, R.M., Essex, C., Nerenberg, M.A.H. (1991)
		 {\em Numerical methods can suppress chaos} Phys. Lett. A {\bf
		157}, 27
	\bibitem{bib-Cro} Croquette, V., Poitou, C. (1981) {\em Cascade of 
		period doubling bifurcations and large scale stochasticity in 
		the motion of a compass} J. Phys (Paris) Lett. {\bf 42} L537-L539  
		(in Tisch, 1989)
	\bibitem{bib-Daw} Dawson, S.P., Grebogi, C., Yorke, J.A., Kan, I.,
		Ko\c{c}ak, H. (1992) {\em Antimonotonicity: inevitable
		reversals of period-doubling cascades} Phys. Lett. A {\bf 162},
		249 
	\bibitem{bib-D'H} D'Humieres, D., Beasley, M.R., Huberman, B.A.,
		Libchaber, A. (1982) {\em Chaotic states and routes to chaos in
		the forced pendulum} Phys. Rev. A {\bf 26}, 3483
	\bibitem{bib-Eck} Eckmann, J.P., Ruelle, D. (1985) {\em Ergodic theory
		of chaos and strange attractors} Rev. Mod. Phys. {\bf 57}, 617
	\bibitem{bib-Fei} Feigenbaum, M.J. (1978) {\em Quantitative
		universality for a class of nonlinear transformations} J. Stat.
		Phys. {\bf 19}, 25
	\bibitem{bib-Fry} Fryska, S.T., Zohdy, M.A. (1992) {\em Computer
		dynamics and shadowing of chaotic orbits} Phys. Lett. A {\bf 
		166}, 340
	\bibitem{bib-Gle} Gleick, J. (1987) {\em Chaos} (Viking Penguin Inc.,
		New York)
	\bibitem{bib-Guc} Guckenheimer, J., Holmes, P. (1983) {\em Nonlinear
		Oscillations, Dynamical Systems, and Bifurcations of Vector
		Fields} (Springer-Verlag, New York)
	\bibitem{bib-Hol} Holden, A.V. (1986) {\em Chaos} (Princeton University
		Press, Princeton)
	\bibitem{bib-Iye} Iyengar, R.N. (1991) {\em Stochastic characterization
		of chaos in a nonlinear system} Phys. Lett. A {\bf 154}, 357
	\bibitem{bib-Jen} Jensen, M.H., Bak, P., Bohr,T. (1984) {\em Transition to
		chaos by interaction of resonances in dissipative systems} {\rm
		I,II} Phys. Rev. A. {\bf 30} 1960 (in Tisch (1989), p42)
	\bibitem{bib-Kap} Kapitaniak, T. (1988) {\em Chaos,in systems with
		noise} (World Scientific, Singapore)
	\bibitem{bib-Lic} Lichtenberg, A.J., Lieberman, M.A. (1983) {\em
		Regular and Stochastic Motion} (Springer-Verlag, New York)  
	\bibitem{bib-McC} McCauley, J.L. (1988) {\em An introduction to
		nonlinear dynamics and chaos theory} Physica Scripta vol T20
		(in Tisch (1989), p8)
	\bibitem{bib-Mei} Meissner, H., Schmidt, G. (1986) {\em A simple
		experiment for studying the transition from order to chaos} Am.
		J. Phys. {\bf 54}, 800
	\bibitem{bib-Rei} Reick, C. (1991) {\em On the experimental
		observability of conservative scaling in the Feigenbaum
		transition to chaos} Phys. Lett. A {\bf 160}, 223
	\bibitem{bib-Tis} Tisch, J.W.G. (1989) {\em An experiment in
		deterministic chaos} Honours Thesis, University of Tasmania
	\bibitem{bib-Sag} Sagdeev, R.Z., Usikov, D.A., Zaslavsky, G.M.
		(1988) {\em Nonlinear Physics, from the pendulum to turbulence
		and chaos} (Harwood Academic publishers, Chur)
	\bibitem{bib-Sche} Scheffczyk, C., Parlitz, U., Kurz, T., Knop, W.,
		Lauterborn, W. (1991) {\em Comparison of bifurcation structures
		of driven dissipative nonlinear oscillators} Phys. Rev. A {\bf
		43}, 6495 
	\bibitem{bib-Schr} Schreiber, T., Grassberger, P. (1991) {\em A simple
		noise-reduction method for real data} Phys. Lett. A {\bf 160},
		411
	\bibitem{bib-Schu} Schuster, H.G. (1988) {\em Deterministic Chaos: an
		introduction} 2nd edition (VCH Verlagsgesllscaft, Vienheim)
	\bibitem{bib-Sti} Stiller, O., Becker, A., Kramer, L. (1992) {\em
		Noise-induced transitions between attractors in time
		periodicaly driven systems} Phys. Rev. Lett. {\bf 68}, 3670
	\bibitem{bib-Sun} Sun, J.Q., Hsu, C.S. (1991) {\em Effects of small
		random uncertainties on non-linear systems studied by the
		generalized cell mapping method} J. Sound Vibr. {\bf 147}, 185
	\bibitem{bib-Wig} Wiggins, S. (1988) {\em Global Bifurcations and
		Chaos}  (Springer-Verlag, New York)
	\bibitem{bib-Zen} Zeng, X., Eykholt, R., Pielke, R.A. (1991) {\em
		Estimating the Lyapunov-exponent spectrum from short time
		series of low precision} Phys. Rev. Lett. {\bf 66}, 3229
	\bibitem{bib-Zis} Zisook, A.B. (1981) {\em Universal effects of
		dissipation in two-dimensional mappings} Phys. Rev. A {\bf 24},
		1640
   \end{thebibliography}



\end{document}