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Thanks. I would like to thank all of those who helped me through my time as a Master’s Student. Their help in classes and in projects was greatly appreciated.
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Thanks • I would like to thank all of those who helped me through my time as a Master’s 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
Periodic Orbit Theory Chaotic Analysis Andrew J Penner Randy Kobes Slaven Peles
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 The System
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 ber = R/R' this results in a Lagrangian equation of Mechanics
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 t=wot. 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. Equations of Motion
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. Chaos
What we see is a great simplification from the projection of the attractor onto the 2D plane. Stroboscopic Effect
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 Q=0.76, a = 1.028, r = 1.088 and Q = 1.2577, a = 1.17731, r = 1.088. Bifurcation
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. Lyapunov Index
Lyapunov Exponent • The Lyapunov index measures the rate of divergence between a trajectory with 2 different initial conditions
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. Winding Number
WindingNumber “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
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. Symbolic Dynamics
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.
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. Partitioning
Partitioning • The homoclinic partitioning points as seen on both the Poincare section, and the first return map
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. Symbols
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
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
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
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. Orbit Hunting
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
Larger Orbit Hunting • RNNRRL RNN-RRL RNN, RRL • RNNR-RL RRNN, RL • RRRNLL RRRN-LL RRRN, NRLL
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. Shadowing
Shadowing For kth order shadowing orbits Lp = L1 L2L3…Lk 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
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. Dynamical Averaging
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. Eigenvalues
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
Following the averaging formula We are after a space average of the observable a(x;t) 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; Dynamical Averaging
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
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
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? Future Possibilities
