We present the first natural and visible examples of Hamiltonian systems with divided phase space allowing a rigorous mathematical analysis. The simplest such family (mushrooms) demonstrates a continuous transition from a completely chaotic system (stadium) to a completely integrable one (circle). In the course of this transition, an integrable island appears, grows and finally occupies the entire phase space. We also give the first examples of billiards with a “chaotic sea” (one ergodic component) and an arbitrary (finite or infinite) number of KAM islands and the examples with arbitrary (finite or infinite) number of chaotic (ergodic) components with positive measure coexisting with an arbitrary number of islands. Among other results is the first example of completely understood (rigorously studied) billiards in domains with a fractal boundary.
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December 2001
Research Article|
December 01 2001
Mushrooms and other billiards with divided phase space
Leonid A. Bunimovich
Leonid A. Bunimovich
Southeast Applied Analysis Center, School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
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Chaos 11, 802–808 (2001)
Article history
Received:
August 24 2001
Accepted:
September 25 2001
Citation
Leonid A. Bunimovich; Mushrooms and other billiards with divided phase space. Chaos 1 December 2001; 11 (4): 802–808. https://doi.org/10.1063/1.1418763
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