Creating transverse homoclinic connections in planar billiards. (English. Russian original) Zbl 1120.37040

Donnay, V. J.

J. Math. Sci., New York 128, No. 2, 2747-2753 (2005); translation from Zap. Nauchn. Semin. POMI 300, 122-134 (2003).

Summary: Given a planar billiard system containing stable and unstable manifolds that intersect nontransversely, we show how to make a local perturbation to the boundary that causes the intersection to become transverse. We apply these ideas to billiards inside an ellipse.

MSC:

37J45	Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
37D50	Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
70K44	Homoclinic and heteroclinic trajectories for nonlinear problems in mechanics
37D10	Invariant manifold theory for dynamical systems

Keywords:

planar billiard system; stable and unstable manifolds; local perturbation

References:

[1]	M. Bialy, ”Convex billiards and a theorem by E. Hopf,” Math. Z., 214, No.1, 147–154 (1993). · Zbl 0790.58023 · doi:10.1007/BF02572397
[2]	I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinai, Ergodic Theory, Springer-Verlag, New York (1982).
[3]	A. Delshams and R. Ramirez-Ros, ”Poincare-Melnikov-Arnold method for analytic planar maps,” Nonlinearity, 9, No.1, 1–26 (1996). · Zbl 0887.58029 · doi:10.1088/0951-7715/9/1/001
[4]	A. Delshams and R. Ramirez-Ros, ”On Birkhoff’s conjecture about convex billiards,” in: Proceedings of the 2nd Catalan Days on Applied Mathematics (Odeillo, 1995), Presses Univ. Perpignan, Perpignan (1995), pp. 85–94. · Zbl 0911.58021
[5]	V. J. Donnay, ”Using integrability to produce chaos: billiards with positive entropy,” Comm. Math. Phys., 141, 225–257 (1991). · Zbl 0744.58041 · doi:10.1007/BF02101504
[6]	V. J. Donnay, ”Transverse homoclinic connections for geodesic flows,” in: Hamiltonian Dynamical Systems (Cincinnati, 1992), Springer, New York (1995), pp. 115–125. · Zbl 0835.58027
[7]	M. M. Dvorin and V. F. Lazutkin, ”Existence of an infinite number of elliptic and hyperbolic periodic trajectories for convex billiards,” Funkts. Anal. Prilozhen., 7, No.2, 20–27 (1973). · Zbl 0298.58006
[8]	V. Gelfreich, ”A century of separatrices splitting in Hamiltonian dynamical systems: perturbation theory, exponential smallness,” in: XIII International Congress on Mathematical Physics, Int. Press, Boston-London (2000), pp. 73–86. · Zbl 1026.37053
[9]	V. Gelfreich and V. F. Lazutkin, ”Splitting of separatrices: perturbation theory and exponential smallness,” Uspekhi Mat. Nauk, 56, No.3, 79–142 (2001). · Zbl 1071.37039
[10]	N. Innami, ”Geometry of geodesics for convex billiards and circular billiards,” Nihonkai Math. J., 13, No.1, 73–120 (2002). · Zbl 1035.37027
[11]	G. I. Kim, ”Elliptic Birkhoff’s billiards with C 2-generic global perturbations,” Bull. Korean Math. Soc., 36, No.1, 147–159 (1999). · Zbl 0934.34036
[12]	G. Knieper and H. Weiss, ”A surface with positive curvature and positive topological entropy,” J. Differential Geom., 39, No.2, 229–249 (1994). · Zbl 0809.53043
[13]	H. E. Lomeli, ”Perturbations of elliptic billiards,” Phys. D, 99, No.1, 59–80 (1996). · Zbl 0890.58081 · doi:10.1016/S0167-2789(96)00061-9
[14]	G. Paternain, ”Real analytic convex surfaces with positive topological entropy and rigid body dynamics,” Manuscripta Math., 78, No.4, 397–402 (1993). · Zbl 0791.58073 · doi:10.1007/BF02599321
[15]	D. Petroll, Existenz und Transversalitat von Homoklinen und Heteroklinen Orbits beim Geodatischen Fluss, Thesis, Universitat Freiburg (1996).
[16]	Ya. G. Sinai, Introduction to Ergodic Theory, Princeton Univ. Press, Princeton (1976).
[17]	M. B. Tabanov, ”Separatrices splitting for Birkhoff’s billiard in symmetric convex domain, closed to an ellipse,” Chaos, 4, No.4, 595–606 (1994). · Zbl 1055.37551 · doi:10.1063/1.166037
[18]	M. Wojtkowski, ”Principles for the design of billiards with nonvanishing Lyapunov exponent,” Comm. Math. Phys., 105, 319–414 (1986). · Zbl 0602.58029 · doi:10.1007/BF01205934
[19]	M. Wojtkowski, ”Two applications of Jacobi fields to the billiard ball problem,” J. Differential Geom., 40, No.1, 155–164 (1994). · Zbl 0812.58067