Thanks

1
Thanks

• I would like to thank all of those who helped me
  through my time as a Masters Student. Their help
  in classes and in projects was greatly
  appreciated.
• I feel that it is necessary to thank my
  supervisor Dr. Randy Kobes, who has helped me not
  only in Thesis projects, but in my decisions
  through my career and for the future. I am not
  sure where I would be without his support.
• This show has been brought to you by the letters
• N - S - E - R - C

2
Chaotic Analysis

• Periodic Orbit Theory

Andrew J Penner Randy Kobes Slaven Peles
3
The System

• Our system was a simple setup of 2 gears, and a
  rod
• This system was set vertically so the effects of
  gravity are ignored.
• One gear is arranged so that it only experiences
  rotational motion
• The second gear is stuck to the surface of the
  first gear, and may move around in a circle
  around the first gear.
• Next a rod of length l is attached to the edge of
  the moving gear. The rod is capable of 360 degree
  motion.
• The point of interest in this system in the
  coordinates at the free tip of the rod. The first
  simplifying assumption is that the angular
  frequency of the moving gear is constant at

4
The Gear and Rod Assembly
5
Mechanics

• What we have from the previous slide is the
  coordinates defined as
• x(R'R)cos(f')-Rcos(f'-f)lcos(q)
• y(R'R)sin(f')-Rsin(f'-f)lsin(q)
• If we consider R'f' Rf,. And take the gear
  ratio to be r R/R'
• this results in a Lagrangian equation of

6
Equations of Motion

• From the Lagrangian, and our first simplifying
  assumption we can easily determine the equation
  of motion for the rod angle,
• To make the system more realistic we consider a
  frictional term
• Then insert it into our equation of motion to get
• Now we have an non-autonomous system, to be of
  more value to our calculations we may consider
  the autonomous version by introducing a
  dimensionless time variable twot. Since our
  assumption has been that the driving frequency is
  constant the phase space for the system consists
  of 3 dimensions ( ) and since f is time
  dependent we expect it to be smooth.
• To simplify matters even more, we may reduce the
  problem to 2 degrees of freedom by using
  Poincare's theorem over the smooth variable f,
  which states that any system may be represented
  in full detail if one only takes a stroboscopic
  view of the attractor. This was done by using a
  common PDE solver and only recording discrete
  evolution steps.

7
Chaos

• There are a few major considerations for a system
  to be considered chaotic. These were developed by
  Devaney
• 1. The trajectory must be transitive
• 2. The topology must be dense
• 3. The trajectory must be non-repeating
• Not all three conditions are necessary, but if
  all three are met you can be assured that chaos
  exists.

8
Stroboscopic Effect

• What we see is a great simplification from the
  projection of the attractor onto the 2D plane.

9
Bifurcation

• An important feature of a chaotic system is the
  dependence that each of the paramters of the
  system must have wrt each other. This dependence
  may be best seen in a bifurcation diagram.
  Furthermore, it is in one of these diagrams that
  the chaos in the system may be located, for
  example consider the logistic diagram. Insert a
  few bifurcation diagram pictures.
• In similar diagrams produced by our system, we
  found several chaotic regimes existsed, the 2 of
  greatest interest was Q0.76, a 1.028, r
  1.088
• and Q 1.2577, a 1.17731, r 1.088.

10
Bifurcation Diagram
11
Focused Bifurcation Diagrams
12
Lyapunov Index

• With the chaotic regime located, a natural next
  question to ask is the degree of chaos given the
  specific parameters. This is measured by the
  Lyapunov index, an exponential value that
  determines how fast a system moves from a
  predictable state to an unpredictable state.
  Traditionally this is measured with averages of a
  pair of trajectories as they evolve, and separate.

13
Lyapunov Exponent

• The Lyapunov index measures the rate of
  divergence between a trajectory with 2 different
  initial conditions

14
Winding Number

• The winding (rotation) number is used to
  determine the convergence of a closed dynamical
  system. Essentially we are observing a path of a
  trajectory through coordinate space. This value
  is rational if the system contains periodic
  paths. For a chaotic system one expects an
  irrational winding number, signifying that the
  orbits on a manifold are not periodic, thus never
  repeating. Like with the case of the Lyapunov
  exponent, the winding number is traditionally
  measured with an average over the system.

15
Winding Number R

• The Winding Number is a method of tracking a
  trajectory around phase space.
• The black line in the above picture displays a
  winding number of 2/5, since it is rational the
  trajectory is nonchaotic
• The winding number is defined as the asymptotic
  limit over the entire trajectory

16
Symbolic Dynamics

• Symbolic dynamics started by a fellow named
  Hadamard. He was using this method to solve for
  allowed trajectories on spaces of negative
  curvature. Since then there have been several
  applications of this method to physical dynamical
  systems. The idea behind symbolic dynamics is
  that one can trace the path a trajectory takes in
  phase space, by the regions that it passes
  through. This has definite advantages to tracing
  the numerical coordinates of a trajectory
  especially in chaotic dynamics where a true
  representation of a trajectory would require
  infinite precision. The difficult part of this
  method lies in the determination of the
  partitions in phase space.

17
Symbolic Dynamics
The symbol planes for Q 0.76 and Q 1.2577 for
periods up to period 15. The x-axis represents
forward iterations, the y-axis represents
backward iterations.
18
Partitioning

• The breaking of phase space can be made simple by
  the construction of a return map, a map that
  takes a coordinate of phase space and plots it
  against the same coordinate in the next
  evolutionary step. Using this map one could use
  the locat maxima and minima as partitions points,
  howver this method does not work all that well
  for systems with 2 dimensional tendencies, since
  these attractors tend to be double sheeted. If
  one were to use the local maxima and minima
  method, one may end up using more symbols than
  necessary to describe the system. This leads to
  difficulties in later calculations. The method of
  choice was the homoclinic points, where one maps
  the forward iterative map onto the attractor, and
  marks the points of tangency. This method ensures
  a minimal number of symbols used for a system.

19
Partitioning

• The homoclinic partitioning points as seen on
  both the Poincare section, and the first return
  map

20
Symbols

• When calculating the symbolic dynamics of
  possible paths through symbol space the user only
  needs to produce a list of all possible
  permutations and combinations of the symbols
  used, to any period desired. Of course if we
  could leave the list like this, we would
  essentiually be saying that the trajectory of the
  system could go anywhere anytime. Physically we
  expect that this should not be the case, and we
  must consider a set of symbolic limits or
  rules that restrict the orbit in some way.
  These are referred to as fundamental forbidden
  zones. These are regions determined by a return
  map, that may never be entered by any part of a
  trajectory. These are determined in a 1D case by
  a forward iteration on the return map from a
  homoclinic point. For a 2D map one needs to also
  consider the possible reverse iterations (how did
  the trajectory get here points) that form the
  secondary restrictions to a symbolic dynamic
  orbit. Since these orbits are periodic one expect
  repetition in the orbit, and we find that we have
  to test all cyclic permutations of an orbit to
  determine its viability.

21
Symbolic Mathematics

• To determine the rules for allowable orbits one
  must associate a measure to our symbolic system.
• We associate a number in the unit interval to the
  forward and backward sequences. What is common,
  is to associate a sign change depending on the
  value of the symbolic orbit, in our case we chose
  to associate a sign change whenever the symbol N
  appeared in the sequence. The next requirement
  was to create a numerical value for the forward
  sequence.
• Where mi 0,1,2 for si L,R,N if the product of
  the e is one up to that symbol, otherwise mi
  2,1,0 for si L,R,N

22
Symbolic Mathematics

• We perform a similar procedure for the backward
  sequence.
• The value of the symbol is determined by
• Where ni 0,1,2 for si R,N,L if the product of
  the e is one up to that symbol, otherwise ni
  2,1,0 for si R,N,L

23
Ordering

• Now the symbolic measure has been constructed we
  may consider the natural ordering of a symbolic
  set
• This ordering allows us to talk about the winding
  number as a feature of our chaotic system.
• Insert ordering picture

24
Orbit Hunting

• So far the symbolic dynamics has been defines,
  and we have our list of possible orbits. For
  these to be of any futher use, we require the
  ability to actually locate the numerical values
  at least approximately, of these trajectories in
  phase space. Doing this required a few
  techniques, the most important was the use of the
  existance of pseudocycles. Since each pseudocycle
  will always shadow a prime cycle one could simply
  use the smaller orbital positions as guesses for
  larger prime cycles. This had a very high success
  rate. A second method would break up already
  found orbits, and use the peices to make up the
  guesses for larger cycles. This was used as a
  last resort since it is a time consuming process.
  A problem arose with the exponentially increasing
  number of allowed orbits. Methods of combination
  of old orbits became tedious and did not always
  return proper results. This was especially true
  of orbits with very large instability. At this
  point the search method was set up to test all
  possible combinations of points along the
  attractor. Automated methods were much easier,
  but were time consuming, fortunately machines
  like to run night and day for several months.
  Ultimately we came to the conclusion that we
  needed only the more stable orbits, and reduced
  our search a little further. Our search for
  orbits terminated at period 30.

25
Orbit Hunting

• For short orbits, the first return map was
  essential for finding the allowed orbits, the
  left is the first return map, the right is the
  second return map both for Q 0.76

26
Larger Orbit Hunting

• RNNRRL ? RNN-RRL ? RNN, RRL
• ? RNNR-RL ? RRNN, RL
• RRRNLL ? RRRN-LL ? RRRN, NRLL

27
Shadowing

• The symbolic dynamics described so far involves
  what is referred to as prime orbits or prime
  cycles. Another set of orbits are referred to as
  pseudo-orbits, or pseudo cycles. These are a set
  of orbits made by joining 2 or more prime cycles.
  Physically they look very similar, and
  mathimatically they are also very similar. A
  complete symbolic dynamic description requires
  both sets of values. For example consider the
  period 4 orbit RNRL, and the 2 period 2 orbits,
  RN and RL. One could combine the 2 period 2
  orbits to make RN-RL a pseudocycle whith similar
  properties as the prime cycle RNRL.

28
Shadowing
For kth order shadowing orbits Lp L1 L2
L3Lk np n1 n2 n3 nk Tp T1 T2 T3
Tk Ap A1 A2 A3 Ak Where A is
any observable that may be extracted from the
system
29
Dynamical Averaging

• Now we fix our dynamical system by adding in some
  statistical mechanics. Because nothing really
  makes something sound complicated like
  statistical mechanics. We have around 10000
  orbits, each are periodic, and some slightly more
  stable than others. But the question is What do
  we want with them, or how do they relate to
  chaotic dynamics? Well, in the next several
  slides of very complicated code, we will discuss
  line by line how each of these orbits contributed
  to the analysis of the chaos in the system and
  four alternate commands that could have been used.

30
Eigenvalues

• The most important peice of information that is
  drawn from the periodic orbits is the stability
  eigenvalue. Each of these orbits contained a
  point in phase space which was used to solve the
  Jacobian matrix. The Jacobian matrix for each
  point in the trajectory was then multiplied, and
  the eigenvalues of the resulting matrix were
  determined. One expects the eigenvalues to sum to
  a negative value, due to the divergence of the
  system. By Haken's theorem, we expect that one of
  the eigenvalues will be 1. There in we obtain two
  eigenvalues, one greater than zero, and one less
  than zero. The eigenvalue that is less than zero
  corresponds to a stable contractive system, the
  eigenvalue that is greater than zero corresponds
  to an expanding unstable system. The positive
  eigenvalue is the eigenvalue of interest. These
  play a very important role in the dynamical
  averaging of a system.

31
Eigenvalues

• Nonlinear flow equations are difficult to
  evaluate analytically and thus we use
  approximations to linearize the flow

When calculating the eigenvalues one just
multiplies the matrix J
Now find the eigenvalues Li of Ji
32
Dynamical Averaging

• Following the averaging formula
• We are after a space average of the observable
  a(xt)
• Using geometric arguments over phase space we
  find that for infinitesimal strip of phase space
  Mi is approximately equal to the unstable
  eigenvalue Li of the orbit occupying that space.
  This allows the spatial average to be reduced to
  a discrete sum over all the orbits, further
  considering that we want a full averaged system,
  we also expect to average over time,
• which in the case of periodic orbits is a sum
  over each element in the orbit. Ultimately after
  considering all the averages we have a double
  sum

33
Three Observables

• We have 3 observables, the escape rate, the
  Lyapunov exponent, and the winding number
• When considering these values and the periodic
  orbit theory, all we have to do is replace Ap
  with the following values
• Where Rp is the winding ratio of the pth periodic
  orbit

34
Escape Rate g
35
Winding Number R
36
Lyapunov Exponent
37
Summary of Results

• We found the expectation that the escape rate is
  zero was met by both systems
• The Winding Number met the brute force value
  within decimal places
• The Lyapunov exponent also met the brute force
  value within decimal places

38
Future Possibilities

• Better analysis may be performed or better
  results by use of smoothing
• Applications of this research are open to any
  chaotic system that has one or two dimensions
• Researching symbolic dynamics for higher
  dimensional systems is the only restriction, the
  methods for dynamical averaging are open to
  higher dimension?

39
References