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Final Year Project 3P-9Y
John Hawkins PHY 3
Investigation of Chaos in
Electronic and Mechanical Systems
25/4/94
Supervisor Dr. Dan Wolverson
- University of East Anglia
Abstract
This
investigation was involved with the study of Chaos.
Using an electronic circuit it is possible to achieve a
chaotic system. This system has been analyzed to see if the
circuit is truly chaotic. It is also possible to set up a second
circuit, which is in synchronization.
A mechanical system, consisting of a compass needle in an
oscillating magnetic field, is also set up which should exhibit a
cascade of period doubling bifurcations.
Both systems exhibit chaos.
The circuit was tested using the Lyapunov Multiplier method to
see if the output was successful. The values obtained were
correct and so the circuit is chaotic.
The second circuit was also set up which was synchronized with
the first.
Due to lack of time the bifurcations of the compass needle
were not measured.
Index
- 1. Introduction
2. Theory
2.1 Phase Space and Chaotic Attractors
2.2 Bifurcations
2.3 Circuit Theory
2.4 Compass Needle Theory
3. Apparatus
3.1 Circuit and explanation
3.2 Reading Equipment
3.3 Compass Needle Set-up
4. Results
4.1 Electronic Circuit Phase Space
4.2 Electronic Circuit Output and Synchronization
4.3 Lyapunov Multipliers obtained from output analysis
4.4 Compass Needle observed behaviour.
5. Discussion
5.1 Electronic Circuit Phase Space
5.2 Electronic Circuit Output and Synchronisation
5.3 Compass Needle observed behaviour.
6. Conclusion
6.1 Experimental Improvements
6.2 Comments on Further Work
7. Appendices
8. References
1. Introduction
It is often asked, 'Why Study Chaos?'. It is said by most that
chaos is unpredictable, unreliable and therefore unusable. More
and more each day chaos is becoming more controllable and usable.
It is not a study of randomness as is a popular belief, but a
study of order in a seemingly random sea of information. A chaos
scientist does not look at the end product but takes a look
instead at the basic principles and history that make up this
seemingly random action. There are many various reasons for
studying chaos. The fields in which chaos has been put to use are
wide and varied, and new applications are being found all the
time. Some applications of chaos are fluid dynamics, increasing
the power of lasers, synchronising the output of electric
circuits, stabilising erratic animal heartbeats and encrypting
messages for secure communications.
In this study of chaos I hope to tackle three main fields of
chaos.
1. Onset of chaos. This will be achieved in two ways.
Observation of a chaotic circuit as the voltage is increased, and
motions of a compass needle as the frequency of a signal driving
it is decreased.
2. Synchronisation of two chaotic outputs. This is possible by
using one part of a chaotic circuit to drive two identical parts.
The synchronised outputs can be used for encoding signals which
could then be used for communications.
3. Lyapunov multiplier proof of chaotic systems. After enough
data has been gathered I hope to apply Layapunov's method of
perturbations to analyse numerically whether or not the circuit
is truly chaotic.
2. Theory
2.1 Phase Space and Chaotic
Attractors
An important concept of chaos which has been used later in
this report needs to be explained. Conventionally the motion of
an objected is represented as a "time series", a graph
showing the change in position over time. It is far more
convenient to look at the system as a whole as we can gather far
more information. A phase space diagram is one way of doing this.
A phase space diagram is a history of the changing variables of
the system. Any state of the system at a moment in time is
represented as a point in phase space. All the information about
the system is contained within the co-ordinates of that point.
Then as the system changes the point will move to another place
in phase space. As the system changes with time the point in
phase space will trace a trajectory on the phase space diagram .
Consider the motion of a ball attached to a spring that gets
stiffer as it is stretched or compressed (Diagram 2.1.1 a). As
the board moves cyclically back and forth, the spring is pushed
and pulled and causes the ball to move. The movement of the ball
is dependent of the force with which the spring is pushed.
If the force is small the ball moves in a very simple
trajectory, which repeats itself with the same cycle as the
board. Diagram 2.1.1 b shows a time series plot of such a
movement, it's position versus time. Diagram 2.1.1 c shows a
phase space plot, it's position versus it's velocity. Because the
motion is periodic the path in the phase space plot will trace
itself with each period of the driving force. This phase space
diagram is a period one attractor.
If the driving force is increased, at a certain point the ball
will move back and forth in a more complex motion for each period
of the driving force. The time series and the phase space plots
(diagrams 2.1.1 d and 2.1.1 e respectively) change accordingly.
The period one orbit becomes unstable and the system produces a
period two attractor.
Past a certain force the motion of the ball becomes chaotic.
The time series (diagram 2.1.1 f) does not render any useful
information, but the phase space plot (diagram 2.1.1 g) reveals a
chaotic attractor. The path of the system never repeats itself,
yet it only occupies certain regions of phase space.
Diagram 2.1.1
  
2.2 Bifurcations
If we take a simple law like 2x2-1 and perform an iteration on
it, (take the answer and put it back into the equation), and
repeat this operation, the output of this experiment is chaotic.
There does not seem to be any pattern (see diagram 2.2.1).
Diagram 2.2.1 Plot of
iteration of equation 2x2-1. X-axis is iteration number and
Y-axis is the value obtained at that number of iterations.
Lets take it a step further. Our new equation is kx2-1 where k
is a number between 1 and 4. For some values of k the continuous
iterations settle down to one value (use a value of k less than
0.3). If we now look at these results in a different way we can
get a far greater understanding of what is going on. Diagram
2.2.2 shows the output of a programme written by myself in Turbo
Pascal. This programme works out the values to which the
iterations settle down to if any and plots them on a graph. On
the Y-axis is the values to which the equation kx2-1 settles down
to . The x-axis is the value of k. This plot is a phase space
diagram. This particular pattern being a period doubling
bifurcation. Up until k=0.72 the iterations settle down to one
value. After this point the plot splits into two. This is where
the iterations pass between two values. At k=1.25 the plot splits
again into four values and again at k=1.37, after which the
output is chaotic. If we look further down the graph we can see a
'window' at k=1.75. An area which has only three values. This has
come after a large period of chaos. The same doubling occurs
again splitting the graph into six values. If we were to enlarge
the graph near one of the values at k=1.75 we would find a
picture of the overall graph again. This new picture will also
have a window and we find that our phase plot is fractal in
nature, i.e. no matter how far we zoom in, it is still as
detailed.
Diagram 2.2.2 Plot of
iterations of equation kx2-1. The y-axis shows the values the
iterations settle down to. The x-axis is the value of k.
2.3 Circuit Theory
The circuit described in section 3.1 is a simplified version
of a circuit designed by Robert Newcomb of the University of
Maryland. This version of the circuit is described in an article
by Joseph Neff and Thomas L. Carroll 1. The idea of the two
circuits is to create two chaotic outputs which are synchronised.
If two chaotic systems are constructed which are virtually
identical but separate then they would quickly fall out of phase
with each other because any slight difference between the systems
would be magnified. In order to build a chaotic system whose
parts are synchronous then a system needs to be stable. A stable
system is described as one in which if a small perturbation was
made then its trajectory through phase space changes only a
little from what it might have been if unperturbed. The Russian
mathematician Aleksandr M. Lyapunov realised that one number
could be used to represent the change caused by a perturbation.
If we take a perturbation at a moment in time and then divide it
by the perturbation a small time before and take an average of
this over several values we have a quantity which is called the
Lyapunov multiplier. The Lyapunov multiplier tells us how much on
average a perturbation will change. If the Lyapunov multiplier is
less than one the perturbation will die out. If the multiplier is
greater than one the perturbation will grow and the system is
unstable. All chaotic systems have a Lyapunov multiplier greater
than one and so are always unstable.
In order to create a chaotic systems whose parts were
synchronised Caroll and Louis M. Pecora developed a system
whereby a chaotic system is split into two subsystems. One of
these subsystems can then be duplicated and the first subsystem
can drive both of the duplicate subsystems (see diagram 3.2.1 for
a fuller description).
At the top of the diagram we have a chaotic system. We isolate
the subsystem B-C as a synchronising subsystem. If we then
duplicate this subsystem and use our the subsystem A as our
driving signal for the two duplicate subsystems we will then get
two chaotic outputs which are in step. The two synchronising
subsystems have Lyapunov multipliers less than one so if one of
the systems was perturbed, the output would be out of step but
the perturbation would decrease and the two outputs would soon be
back in step again.
Diagram 2.3.1 Chaotic
systems and Lyapunov Multipliers
2.4 Compass Needle Theory
V. Croquette and C. Poitou 2 suggested a system whereby a
compass needle is placed in a periodic oscillating field. This
system renders large scale chaos, and also period doubling
bifurcation's.
The compass needle set-up suggested will model a bipolar
motor. The normal possible motions of this system are a clockwise
or an anticlockwise rotation. The possible non-linear motions of
this system have not been considered and if they do exist then a
there will be some periodic and also chaotic motions.
The angle of rotation q of the
compass needle is governed by the equation :
 (See
appendix 7.1 for a derivation of this equation)
where I is the moment of inertia of the compass needle, B is
the magnitude of the magnetic field, being driven at the
frequency (w/2p) , and m is the magnetic moment of the compass
needle.
3. Apparatus
3.1 Circuit and explanation
Diagram 3.1.1 Circuit Diagram
In the diagram above we can see the operational amplifiers
used for this experiment. A1, A2, A3, and A5 are op-amps of type
741CN. A4 is of type NE5539N.
The task of the circuit in a very simple form is:- the two
op-amps connected with capacitors, A2 and A3, are oscillators and
so produce a sine wave. This is then amplified exponentially with
time by op-amps A1 and A5. Each time the sine wave gets to a
certain amplitude the high frequency op-amp, A4, resets the
circuit, bringing the amplitude to zero, and the whole process
starts again. The chaos creeps in each time the circuit resets.
For our analysis the chaotic output of the circuit is at A1.
Our chaotic attractor is the output at A2 against the output of
A3.
The outputs at various points of the circuit are analysed in
two ways. An oscilloscope was used at first to check the circuit.
After it was known that the circuit was behaving correctly a
computer could be used to take in data which could then be used
for numerical analysis. The computer used a PC26AT A/D board to
take in data. A programme was then used to display this data in
various ways.
To establish whether the circuit was chaotic or not at 12
volts, a number a readings were taken, around thirty cycles of
the attractor, and compared in a very simple fashion. In appendix
7.3 a flow diagram is shown of a programme written by myself
which calculated the Lyapunov multiplier for the circuit. The
output of this programme is shown in section 4.3.
3.2 Reading Equipment
Various equipment was required to read the circuit signals.
In the first instance the signals were read into an
oscilloscope (HAMEG HM203-6). For the computational analysis it
was required to read the signals into the PC26AT digital/analogue
converter card connected to the computer. Due to differences in
reference ground signals of the computer and the circuit there
was a drain from the circuit which collapsed the chaotic
attractor. To remedy this Brookdeal amplifiers were used between
the circuit and the computer as they have a large impedance and
will draw very little current from the circuit. The amplifiers
were set to zero gain, as any gain would give false readings on
the computer.
The PC26AT board used to read in the voltages was set to ± 5 volts bipolar operation. The programs
to read from the board were written in Turbo Pascal.
3.3 Compass Needle Set-up
Diagram 3.3.1 Compass
needle set-up.
If we look at diagram 3.3.1 we see the compass needle held on
an axis which sits between two jewel bearings. The bearings are
housed inside the larger rods. The magnetic field is supplied by
a Helmholtz coil pair (see Appendix 7.2), with the compass needle
sitting equidistant from each coil and on the axis of the coil
pair.
The diagram shows the fibre optic leads. They are arranged in
two pairs, so that light shining on one of the top two would be
transmitted to one of the bottom two fibre optic cables. These
pairs are movable so we can increase the separation between the
pairs from 5 degrees right up to 180 degrees. A light emitting
diode is connected to the end of each of the top two fibre optic
cables. A standard photo diode is connected at the end of the two
bottom cables. The signal from the two photo diodes can then be
interpreted by the computer via the PC26AT A/D converter. As the
compass needle passes between one of the pair of optic cables it
cuts off the beam and the output of the photo diode will be near
to zero giving a reverse pulse. The computer can then time the
distance between this pulse and the next one, which is when the
needle passes the other pair of cables, which will then give the
angular velocity dependent on the angle of separation between the
cables. If we store this information in an array we can then
analyse the data for various frequencies of the driving signal
and obtain a plot of angular velocity against driving frequency.
4. Results
4.1 Electronic Circuit Phase Space
Diagram 4.1.1 Circuit Phase Space at 8 volts
Diagram 4.1.2 Circuit Phase Space at 11 Volts
Diagram 4.1.3 Circuit Phase Space at 11.5 Volts
Diagram 4.1.4 Circuit Phase Space at 12 Volts
4.2 Electronic Circuit Output and
Synchronisation
Diagram 4.2.1 Circuit Output at 8 Volts
Diagram 4.2.2 Output of circuit at 11.5 Volts
Diagram 4.2.3 Output of circuit at 12 Volts (Next two
graphs)
Diagram 4.2.4
Diagram 4.2.5 Output of two synchronised circuits
plotted against each other
4.3 Output of Lyapunov Multiplier
Programme
The listing of all one hundred values outputted by the
programme are too lengthy to list here and irrelevant.
Of important note is the fact that the lowest value outputted
was 4.314. The highest value outputted was 84.45
4.4 Compass Needle observed behaviour.
Due to lack of time, problems building the equipment and
modelling of this system I was unable to obtain data for this
part of the experiment. The compass needle was set-up to model a
bipolar motor and did so successfully. The B field was around 0.1
to 0.3 of a Tesla, which was supplied by 40 Volts running through
the coils. The needle also exhibited non periodic, possibly
chaotic, motion at low frequencies.
5. Discussion
5.1 Electronic Circuit Phase Space
The onset of chaos was first investigated.
In diagram 4.1.1 the phase space, output of A2 against output
of A3, exhibits a period one attractor. This is at 8 Volts. The
circuit will cycle around this same attractor and the output,
diagram 4.2.1, shows the non chaotic output. A simple sine wave.
Diagram 4.1.2 shows the phase space at 11 volts. The period
one attractor is much the same as at 8 Volts only slightly
larger, compare the Voltages on the X and Y scales.
The 11.5 volts phase space plot is shown in diagram 4.1.3.
This is where we begin to see the onset of chaos. Our attractor
has split into two sections. A smaller spiral and a larger one.
This attractor has a period of about 20 ±
3. This was worked out from continuous readings taken until a set
of data repeated itself. We can see from this diagram the 'jump'
between the two spirals of the attractor. It is not a chaotic
attractor as it does repeat itself, and we can see from the
output at 11.5 volts (diagram 4.2.2) that the output is a sine
wave increasing in amplitude with time, a very easy to model
system.
At 12 volts the circuit exhibits chaos. Diagram 4.1.4 shows
our chaotic attractor for this system. The larger spiral and the
smaller spiral. When the circuit reaches the outside of either
spiral it resets and moves to the centre of the other spiral.
Continuous readings show the circuit following the same phase
space areas but along a slightly different path, sometimes
crossing over previous paths, but never repeating itself.
The output at 12 volts, diagram 4.2.3 and 4.2.4 show our
expected circuit behaviour. The sine wave amplifies, reaches a
certain value, resets to a new value, amplifies again, and
resets. On the graph the path around the two spirals of the
attractor can be seen. The point where the sine wave resets to a
new value is the 'jump' from one spiral of the chaotic attractor
to the other spiral. Looking carefully we can see which is the
smaller spiral as the number of cycles before a reset is less.
5.2 Electronic Circuit Output and
Synchronisation
From the results in section 4.3 of the Lyapunov multiplier we
can see that even at the lowest value after only 100 readings, a
tenth of a cycle of the attractor we have multiplied the
perturbation by over a factor of four.
This shows that the circuit is chaotic.
Diagram 4.2.4 shows the output of the two synchronised
circuits plotted against each other. This graph has a gradient » 1. As we can see the output of the two
circuits is very much the same. Only deviating slightly from a
straight line around the origin.
5.3 Compass Needle observed behaviour.
Although observing the compass needle by eye did show a very
erratic behaviour at low frequencies I am not able to comment any
further on whether the system was chaotic or not.
6. Conclusions
The project as a whole went fairly well. There were many
problems though that rose mainly from loose connections and
faulty equipment.
6.1 Experimental Improvements
The main problem that occurred was the PC26AT board draining
current from the circuit. Although this was overcome the method
employed is not ideal. The Brookdeal amplifiers still drain a
small amount of current from the circuit and affect the attractor
very slightly.
The PC26AT board does read quickly, but for a more thorough
analysis the circuit would need to be slowed down. For this
reason some of the capacitors and resistors on the printed
circuit board are removable and can be changed for others.
The bearings on which the compass needle rotate are adequate
for the analysis but from time to time the needle will stop,
which in theory should not happen, so a better system would be to
use air bearings.
6.2 Comments on Further Work
This project offers many opportunities for further work.
The compass needle project does not require much work to
finish the experimental set-up. All that is required is that the
fibre optic cables are connected to the diodes. This would then
allow experimental reading of the angular velocity of the compass
needle and hopefully a proof of its period doubling bifurcations.
A study of the electronic circuit could be included as a
second year lab experiment. This offers a good introduction into
the field of chaos, which is not understood by the majority of
students. A study of this type would be valuable experience as
the fields to which chaos is being applied are ever expanding. It
also stands to test the students ability to present their gained
knowledge as the methods of chaos are far different to other
projects currently in the second year lab. A possible experiment
would be to study the onset of chaos, phase space e.t.c.., and to
send a coded signal and decipher it using the synchronised
circuits. This requires a small understanding of electronics
which already would be known to the student from previous
courses. The only problem with this type of experiment in the
second year lab is that it does require more background reading
than other experiments. This is mainly due to the fact that chaos
is not covered in any part of the Physics syllabus.
7. Appendices
7.1 Derivation of the Compass Needle
Equation
The equation of motion for the compass needle is derived thus:
t is the torque. Theta is the angle
between the dipole and the magnetic field.
The torque is also equal to the first derivative of the
angular acceleration times the moment of inertia, I.
We can now substitute the second derivative of theta for
angular velocity and the time dependent form of the magnetic
field.
Take the moment of inertia over to the other side.
The algebraic equation below gives us the last step, where
theta is substituted for A and omega times t is substituted for B
7.2 Helmholtz Coils
For our compass needle experiment we require a uniform B field
over the area through which the needle rotates. To acquire such a
field a Helmholtz coil pair is used. If two identical coils has N
loops carrying current I in the same sense, and their separation
is the same as their radius, the magnetic field along their
common axis between the two coils is: -
where r is the distance from the midpoint between the coils, m 0 is the permeability of free space, N is
the number of turns on the coils, I is the current through the
coils and a is the coil separation or radius.
Diagram 7.2.1 shows the field strength at a distance, r, from
the central point between the two coils, shown here as the two
groups of vertical lines. As can be seen the field is nearly
constant between the two coils and drops away as we get outside
them. This set-up obtains the required linear field over the area
specified.
Diagram 7.2.1 Field strength with respect to separation
7.3 Programme for calculating Lyapunov
multiplier for circuit.
The idea of the following programme was to calculate the
Lyapunov multiplier for the circuit. The method employed was to
take a certain point of the output, find the next point where the
output was the same and then calculating the Lyapunov multiplier
by following both paths and averaging the difference.
The data file used contained 32000 points of data from the
output of the circuit. Corresponding to roughly 30 cycles of the
attractor.
This was then accessed as N(x). N being the file treated as an
array and x being the data point number in that file.
On the flow diagram below we see the starting value N(y) from
which to work on. The programme was run one hundred times using
every 100th value for N(y), working out the Lyapunov multiplier
and then outputting it to a file.
On the second step, is N(y) roughly equal to N(z)), the exact
criterion for the Boolean expression was that 0.01 >
(N(y)-N(z)) > -0.01 and also that -1´
10-6 > (N(y)-N(z-1)) > 1´ 10-6.
This meant that the previous value of N(z) was very close to N(y)
and that the next value only introduced a small perturbation.
The programme would then work out the differences between the
next 100 pairs of values and calculate the Lyapunov multiplier by
taking the average.
Flow diagram showing the programme steps.
8. References and Bibliography
-
- 1. Joseph Neff and Thomas L. Caroll
The Amateur
Scientist, Circuits That Get Chaos in Sync
Scientific American pp 101-103, August 1993
2. V. Croquette and C. Poitou
Cascade of period doubling bifurcation’s
and large scale stochasticity in the
motions of a compass
J. Physique Lettres 42 (1981) L-537 L-539
3. William L. Ditto and Louis M. Pecora
Mastering Chaos
Scientific American pp 62-68, August 1993
4. D. F. Escande and F. Doveil
Renormalization Method for the Onset of
Stochasticity in a Hamiltonian
System
Physics Letters, pp 307- 310, volume 83A,
number 7
5. John H. Moore, Christopher C. Davies and
Michael A. Coplan
Magnetic Field Control
Building Scientific Apparatus pp 323-324
Addison-Wesley Publishing company, 1983
6. James Gleick
Chaos
Sphere Books Ltd, 1991
7. Vernon D. Barger and Martin G. Olsson
Classical Electricity and Magnetism
Allyn and Bacon Inc. 1987
8. R.Z. Sagdeev, D. A. Usikov and G. M.
Zaslavsky
Non-linear Physics, From the Pendulum to
Turbulence and Chaos
Contemporary concepts in Physics Volume 4
Harwood Academic Publishers, 1988
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