Sample trajectory for a=1.0 , b=8.0
This project was originally started as an exam project in the Classical and quantum chaos course taught by Predrag Cvitanović et al.
It was inspired by Kai T. Hansen and Predrag Cvitanović Symbolic Dynamics and Markov Partitions for the Stadium Billiard (still a preprint?) which introduced a biinfinite symbol dynamics for the Bunimovich stadium billiards, which has been established as a standard example of a non-escaping 2D billiards system with convex bouncing surfaces. The present state of the revised article is available as a ps.gz file.
A parallel project on Hard Bunimovich stadium and washboard diffusion (available in a PostScript version), was carried out by Jonas Lundbek Hansen
Bunimovich Stadium - central section has length 2a.
The Bunimovich Stadium is a system in which a particle moves within a boundary defined by opposite semi-circles a distance 2a apart - connected with two line pieces.
Kai Hansen proposed a study of a version of the well known Stadium model where bouncing on the stadium-edge was replaced by a movement in a smooth potential. The potential is defined such that a particle with energy 1 will be confined to the same region as the particle in the normal stadium.
Let z(x,y) denote the distance from the central line of the center rectangular part of the stadium (and thus the distance from the center in the semi-circles). We introduce a family of Soft potentials
V(x,y) = z(x,y)^b
A Hamiltonian particle moving in this potential will approach the dynamics of the normal stadium as b goes towards infinity. Thus it can be studied, how the system of periodic orbits and general chaology changes as one moves from the strongly chaotic Hard Wall stadium towards the non-chaotic systems for b<=2.
The hard Stadium has one parameter -- the stadium length. Since both the circle billiard (a=0) and the pair of infinite walls (a = infinity) are integrable (and thus non-chaotic) the lyap.exp. can be expected to have a maximum for some finite a. For 'non-extreme' values of a the system is hard chaotic/ergodic (no islands of stability, 'energy-surface' well connected). Apart from the lyapunov exponent, magnitudes such pressure on sections of the the container wall could be of interest.
A variation of the hard stadium is the washboard system: A infinite set of stadii are stacked on the flat sides and the line-pieces are removed. Thus one gets a infinite corridor (width 2a) whose sides are composed of open semi-circles. In the single stadium this means that whenever a straight wall is meet - the 'ball' simply passes into another stadium. Different transport/diffusion aspects of this system could be of interest.
(by J.L.Hansen) Lyapunov exponent as function of a (based on simple average divergence of neighboring trajectories. Chaoticity is maximal for a=0.5. The dotted line marks an estimate based on Periodic Orbit Theory using the Cvitanovic/Hansen Symbol Dynamics. (To be described below)
The biinfinite symbol dynamics used throughout this project are introduced by
Predrag Cvitanović and
Kai T. Hansen.
The basic idea is simple:
For each 'stay' in a semi circle : C = (+/-)(#(bounces on the wall) - 1)
For each 'stay' in the rectangular region : R = (+/-)(#(bounces on the wall))
C is positive if the x-axis (y=0) was crossed an odd number of times since last exit from the (other) semi-circle -- otherwise negative.
R is negative if the trajectory (after leaving a semi-circle) crosses the x-axis before making a wall bounce (in the rectangular region)
The stadium has a C(2v) symmetry (symmetric in x,y axis plus 180 deg rotation). Thus the quater-stadium is the 'fundamental domain' and a periodic orbit here will correspond to half a periodic orbit in the full stadium. In this project the orbits are considered in the fundamental domain but displayed in the full stadium - since this is more intuitive.
Below is shown how the different values of C and R share the Poincare section at the border between the semi-circle and the rectangular region. q corresponds to the y-component of the position, sinfi to sine of the velocity angle (proportional to the y-component of the velocity).
(click to enlarge) Symbol dynamics in the Poincare section of the hard stadium for a=1.0. Corner feature corresponds to more than 1 bounce in the circle. (the concentric "half ovals"'s center corresponds to the Whispering gallery orbit with infinitely many bounces. The horizontal Stripes on the second picture correspond to ...-2,-1,0,1,2 bounces in the rectangular region. (white ~ R=0, bluish colors ~ R<0 etc. )
(click to enlarge) Symbol dynamics in the poincare section of the hard stadium for a=10.0. Notice that the "corner feature" is unchanged except for the signs.
The original exam project from Sept 1995 : Dynamics of the Soft Wall Bunimovich stadium is available in a Post script version . There is a minor error in this project : The pictures of the symbol dynamics divisions of the Poincare section has a false vertical feature in the middle.
The present version of the project remedies the to weaknesses mentioned above. Furthermore it goes beyond the original project in the following respects :
(click to enlarge) a=1.0 b=120 (|C| drawn instead of C)
(click to enlarge) a=2.0 b=50 (|C| drawn instead of C)
(click to enlarge) a=0.2 b=18 (|C| drawn instead of C)
(click to enlarge) a=1.0 b=6 (|C| drawn instead of C)
As usual q along x-axis, sinfi along y-axis.
The above graph shows how the |C|>0 feature contracts from both sides and looses loops as b decreases.
Instead of trying to derive a Symbol dynamics for each value of b from the manifold-structure, One could try to take the some set of periodic orbits for the hard stadium (they can be found with some effort): A periodic orbit of the hard stadium (ie. a fix point of the mapping of the Poincare section unto itself) is likely to lie close to a corresponding periodic orbit in Soft Stadiums with very high b. The latter is likely to lie close to some corresponding periodic orbit for some slightly lower b etc.To test the perspectives of the strategy above we have tested how far down one can follow different types of orbits.
With a better optimized fix-point search algorithm the periodic orbits
may be traceable further down (presently implemented: a 2D Newton-type search -
based on a finite difference estimate of the Jacobian of the map.)
But even with some improvements the above method could properly not be used below b=20. (too many orbits would be lost on the way - and there is no immediate way to estimate how many new orbits might become possible without having a 'hard big brother'!)
One of the other problems of Using Hard Stadium symbol dynamics becomes clear when looking at the two last periodic orbits : (0,1) and (0,1)(0,0) are easily distinguishable orbits in the hard stadium, and their respective set of derived soft orbits for b>0 are likewise easily distinguished, but if the 'naive' Symbol dynamics of the soft stadium (based on local extrema as 'bounces'), the (0,1)(0,0) descendents should be given the symbol (0,1)(0,1) since they all have a very weak maximum/bounce in the flat parsing of the rectangle!! When (0,1)(0,1) is different from 2 repeats of (0,1) the notation cant be used in periodic orbit theory.
I will present an outline of the present state of the project in a talk on May 27. 11.15 in Aud A.
In addition to the material presented here some LIVE simulations will be presented on the overhead projector