Static and time-dependent perturbations of the classical elliptical billiard. (English) Zbl 1081.37530

Summary: The elliptical billiard problem defines a two-dimensional integrable discrete dynamical system. Integrability not being a robust property, we study some static and time-dependent perturbations of this problem. For the static case, we observe the transition from integrability to chaos, on some perturbations of the ellipse. Then we study time-dependent perturbations, supposing that the boundary deforms periodically with the time, remaining always an ellipse. We investigate numerically the now four-dimensional phase space, looking mainly at the question of whether or not the velocity of a given trajectory may increase indefinitely.


37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
70H99 Hamiltonian and Lagrangian mechanics
70K20 Stability for nonlinear problems in mechanics
82C05 Classical dynamic and nonequilibrium statistical mechanics (general)
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[1] W. P. Barth, Elliptic moduli curves and Poncelet polygons, Advanced Workshop on Algebraic Geometry, ICTP, Trieste, Italy (1994).
[2] M. V. Berry, Regularity and chaos in classical mechanics, illustrated by three deformations of a circular billiard,Eur. J. Phys. 2:91–102 (1981).
[3] L. A. Bunimovich, On absolutely focusing mirrors, inErgodic Theory and Related Topics, U. Kringel et al., eds. (Springer, Berlin, 1993); L. A. Bunimovich, Conditions on stochasticity of two-dimensional billiards,Chaos 1:187–193 (1991).
[4] E. Canale and R. Markarian, Simulación de billares planos, inAnales IEEE, Segundo Seminario de Informática en el Uruguay (1991), pp. 71–96; and private communication.
[5] S.-J. Chang and R. Friedberg, Elliptical billiards and Poncelet’s theorem,J. Math. Phys. 29(7):1537–1550 (1988). · Zbl 0663.70015
[6] V. J. Donnay, Using integrability to produce chaos: Billiards with positive entropy,Commun. Math. Phys. 141:225–257 (1991). · Zbl 0744.58041
[7] R. Douady, Applications du théorème des tores invariants, Thèse de 3ème Cycle, Université Paris VII (1982).
[8] J. Koiller, R. Markarian, S. Oliffson Kamphorst, and S. Pinto de Carvalho, A geometric framework for time-dependent billiards, inNew Trends in Hamiltonian Systems (World Scientific, Singapore, to appear). · Zbl 0980.37016
[9] J. Koiller, R. Markarian, S. Oliffson Kamphorst, and S. Pinto de Carvalho, Time-dependent Billiards, Preprint (1994);Nonlinearity, to appear. · Zbl 0841.58024
[10] T. Krüger, L. D. Pustyl’nikov, and S. E. Troubetzkoy, Acceleration of bouncing balls in external fields,Nonlinearity 8:397–410 (1995). · Zbl 0826.70013
[11] N. Kutz and E. Zorn, Adiabatische Invarianten und Hannay-Winkel im reichteckigen Billard, Studienarbeit, Technischen Universität Berlin (1988).
[12] S. Laederich and M. Levi, Invariant curves and time dependent potentials,Ergodic Theory Dynam. Syst. 11:365–378 (1991). · Zbl 0742.34050
[13] P. Levallois, Non-integrabilité des billiards définis par certaines perturbations algébriques d’une ellipse et du flot géodesique de certaines perturbations algébriques d’un ellipsoide, Thèse, Université Paris VII (1993).
[14] P. Levallois and M. B. Tabanov, Séparation des séparatrices du billiard elliptique pour une perturbation algébrique et symétrique de l’ellipse,C. R. Acad. Sci. Paris I 316:589–592 (1993). · Zbl 0780.58039
[15] M. Levi, Quasiperiodic motions in superquadratic time-periodic potentials,Commun. Math. Phys. 143:43–83 (1991). · Zbl 0744.34043
[16] A. J. Lichtenberg and M. A. Liebermann,Regular and Stochastic Motion (Springer-Verlag, Berlin, 1983). · Zbl 0506.70016
[17] C. Liverani and M. Wojtkowski, Ergodicity in Hamiltonian systems, SUNY, Stony Brook, Preprint (1992). · Zbl 0824.58033
[18] R. Markarian, New ergodic billiards: Exact results,Nonlinearity 6:819–841 (1993). · Zbl 0788.58034
[19] R. Markarian, S. Oliffson Kamphorst, and S. Pinto de Carvalho, Chaotic properties of the elliptical stadium, preprint (1994);Commun. Math. Phys., to appear. · Zbl 0843.58084
[20] J. Mather, Glancing billiards,Ergodic Theory Dynam. Syst. 2:397–403 (1982). · Zbl 0525.58021
[21] L. D. Pustyl’nikov, Stable and oscillating motions in nonautonomous dynamical systems II.Trans. Moscow Math. Soc. 2:1–101 (1978).
[22] Ya. G. Sinai,Introduction to Ergodic Theory (Princeton University Press, Princeton, New Jersey, 1976). · Zbl 0375.28011
[23] S. Tabachnikov, Billiards, Preprint, University of Arkansas (1994); in ”Panoramas et Synthèses,” SMF, to appear.
[24] M. B. Tabanov, Splitting of separatrices for Birkhoff’s billiard under symmetrical perturbation of an ellipse, inRencontres Franco-Russes de Géometrie (CIRM, France, 1992).
[25] M. Wojtkowski, Principles for the design of billiards with non vanishing Lyapunov exponents,Commun. Math. Phys. 105:391–414 (1986). · Zbl 0602.58029
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