An extremely simple example for demonstrating chaotic dynamics in conservative systems numerically is that of Birkhoff’s billiard [3.1], i.e. the frictionless motion of a particle on a plane billiard table bounded by a closed curve [3.2]–[3.7]. The limiting cases of strictly regular (integrable) and strictly irregular (ergodic’ or ‘mixed’) systems can be illustrated, as well as the typical case, which shows a complicated mixture of regular and irregular behavior. The onset of chaos follows the so-called Poincaré scenario, i.e. the consecutive destruction of invariant tori for increasing deviation from integrability as described by the KAM-theory and the Poincaré-Birkhoff theorem discussed in Chap. 2.
KeywordsPhase Space Periodic Orbit Boundary Curve Invariant Torus Regular Orbit
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- [3.7]H. J. Korsch, B. Mirbach, and H.-J. Jodl, Chaos und Determinismus in der klassischen Dynamik: Billard—Systeme als Modell, Praxis d. Naturwiss. (Phys.) 36(7) (1987) 2Google Scholar
- [3.9]M. V. Berry, Regular and irregular motion, in: S. Jorna, editor, Topics in Nonlinear Dynamics, page 16. Am. Inst. Phys. Conf. Proc. Vol. 46 (1978) (reprinted in R. S. MacKay and J. D. Meiss, Hamiltonian Dynamical Systems, Adam Hilger, Bristol 1987.Google Scholar
- [3.12]J. M. Greene, R. S. MacKay, F. Vivaldi, and M. J. Feigenbaum, Universal behaviour in families of area-preserving maps, Physica D3 (1981) 468 (reprinted in: R. S. MacKay and J. D. Meiss, Hamiltonian Dynamical Systems, Adam Hilger, Bristol 1987.Google Scholar
- [3.17]M. Robnik, Regular and chaotic billiard dynamics in magnetic fields, in: S. Sakar, editor, Nonlinear Phenomena and Chaos, page 303, Hilger, Bristol 1986.Google Scholar