Elliptic islands on the elliptical stadium
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We investigate the existence of elliptic islands for a special family of periodic orbits of a two-parameter family of maps corresponding to the billiard problem on the elliptical stadium. The hyperbolic or elliptical character of these orbits is also investigated. Depending on the parameters, we obtain upper bounds of ellipticity for this special family as a lower bound for chaos. On a different region of the parameter space, we can prove that there is no upper bound for the existence of elliptic islands. The main results we use are Birkhoff Normal Form and Moser's Twist Theorem.
12 pages, 6 ps figures
37E40, chaos, 37C05; 37E40, FOS: Physical sciences, Hyperbolic systems with singularities (billiards, etc.), Dynamical Systems (math.DS), elliptic islands, Nonlinear Sciences - Chaotic Dynamics, Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods, 37C05, Strange attractors, chaotic dynamics of systems with hyperbolic behavior, periodic orbits, Dynamical aspects of twist maps, Equilibria and periodic trajectories for nonlinear problems in mechanics, FOS: Mathematics, discrete dynamical systems, billiard, Mathematics - Dynamical Systems, Chaotic Dynamics (nlin.CD)
37E40, chaos, 37C05; 37E40, FOS: Physical sciences, Hyperbolic systems with singularities (billiards, etc.), Dynamical Systems (math.DS), elliptic islands, Nonlinear Sciences - Chaotic Dynamics, Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods, 37C05, Strange attractors, chaotic dynamics of systems with hyperbolic behavior, periodic orbits, Dynamical aspects of twist maps, Equilibria and periodic trajectories for nonlinear problems in mechanics, FOS: Mathematics, discrete dynamical systems, billiard, Mathematics - Dynamical Systems, Chaotic Dynamics (nlin.CD)
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