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Classical dynamics of a family of billiards with analytic boundaries. (English) Zbl 0622.70011

Summary: The classical dynamics of a billiard which is a quadratic conformal image of the unit disc is investigated. The author gives the stability analysis of major periodic orbits, present the Poincaré maps, demonstrate the mixing properties by following the evolution of a small element in phase space, show the existence of homoclinic points, and calculate the Lyapunov exponent and the Kolmogorov entropy \(h\). It turns out that the system becomes strongly chaotic (positive \(h\)) for sufficiently large deformations of the unit disc. The system shows a generic stochastic transition. The computations suggest that the system is mixing if the boundary is not convex.

MSC:

37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
70K55 Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
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