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Mushrooms and other billiards with divided phase space
1Southeast Applied Analysis Center, School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332.
Chaos (Woodbury, N.Y.)
|June 5, 2003
Summary
Researchers reveal novel Hamiltonian systems with divided phase space, offering insights into the transition from chaotic to integrable systems. These findings include new examples of billiards with complex structures and fractal boundaries.
Area of Science:
- Mathematical Physics
- Dynamical Systems Theory
- Statistical Mechanics
Background:
- Hamiltonian systems with divided phase space are crucial for understanding complex dynamics.
- Previous studies lacked natural, mathematically tractable examples for rigorous analysis.
Purpose of the Study:
- To introduce the first natural and rigorously analyzable examples of Hamiltonian systems with divided phase space.
- To explore the transition from chaotic to integrable dynamics in these systems.
- To present novel examples of billiards with complex phase space structures.
Main Methods:
- Analysis of "mushroom" family of Hamiltonian systems exhibiting continuous transitions.
- Introduction of billiards with "chaotic sea" and multiple KAM (Kolmogorov-Arnold-Moser) islands.
- Study of billiards with fractal boundaries.
Main Results:
- Demonstration of a continuous transition from fully chaotic (stadium) to fully integrable (circle) systems.
- Observation of integrable islands appearing, growing, and eventually dominating the phase space.
- First examples of systems with multiple coexisting ergodic components and KAM islands.
- First rigorously studied billiards in domains with fractal boundaries.
Conclusions:
- The presented systems provide a rigorous framework for studying divided phase spaces.
- These examples bridge the gap between completely chaotic and completely integrable systems.
- The findings offer new perspectives on the structure of phase space in dynamical systems.