Abstract
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.
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References
W. P. Barth, Elliptic moduli curves and Poncelet polygons, Advanced Workshop on Algebraic Geometry, ICTP, Trieste, Italy (1994).
M. V. Berry, Regularity and chaos in classical mechanics, illustrated by three deformations of a circular billiard,Eur. J. Phys. 2:91–102 (1981).
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).
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.
S.-J. Chang and R. Friedberg, Elliptical billiards and Poncelet's theorem,J. Math. Phys. 29(7):1537–1550 (1988).
V. J. Donnay, Using integrability to produce chaos: Billiards with positive entropy,Commun. Math. Phys. 141:225–257 (1991).
R. Douady, Applications du théorème des tores invariants, Thèse de 3ème Cycle, Université Paris VII (1982).
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).
J. Koiller, R. Markarian, S. Oliffson Kamphorst, and S. Pinto de Carvalho, Time-dependent Billiards, Preprint (1994);Nonlinearity, to appear.
T. Krüger, L. D. Pustyl'nikov, and S. E. Troubetzkoy, Acceleration of bouncing balls in external fields,Nonlinearity 8:397–410 (1995).
N. Kutz and E. Zorn, Adiabatische Invarianten und Hannay-Winkel im reichteckigen Billard, Studienarbeit, Technischen Universität Berlin (1988).
S. Laederich and M. Levi, Invariant curves and time dependent potentials,Ergodic Theory Dynam. Syst. 11:365–378 (1991).
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).
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).
M. Levi, Quasiperiodic motions in superquadratic time-periodic potentials,Commun. Math. Phys. 143:43–83 (1991).
A. J. Lichtenberg and M. A. Liebermann,Regular and Stochastic Motion (Springer-Verlag, Berlin, 1983).
C. Liverani and M. Wojtkowski, Ergodicity in Hamiltonian systems, SUNY, Stony Brook, Preprint (1992).
R. Markarian, New ergodic billiards: Exact results,Nonlinearity 6:819–841 (1993).
R. Markarian, S. Oliffson Kamphorst, and S. Pinto de Carvalho, Chaotic properties of the elliptical stadium, preprint (1994);Commun. Math. Phys., to appear.
J. Mather, Glancing billiards,Ergodic Theory Dynam. Syst. 2:397–403 (1982).
L. D. Pustyl'nikov, Stable and oscillating motions in nonautonomous dynamical systems II.Trans. Moscow Math. Soc. 2:1–101 (1978).
Ya. G. Sinai,Introduction to Ergodic Theory (Princeton University Press, Princeton, New Jersey, 1976).
S. Tabachnikov, Billiards, Preprint, University of Arkansas (1994); in “Panoramas et Synthèses,” SMF, to appear.
M. B. Tabanov, Splitting of separatrices for Birkhoff's billiard under symmetrical perturbation of an ellipse, inRencontres Franco-Russes de Géometrie (CIRM, France, 1992).
M. Wojtkowski, Principles for the design of billiards with non vanishing Lyapunov exponents,Commun. Math. Phys. 105:391–414 (1986).
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Koiller, J., Markarian, R., Oliffson Kamphorst, S. et al. Static and time-dependent perturbations of the classical elliptical billiard. J Stat Phys 83, 127–143 (1996). https://doi.org/10.1007/BF02183642
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DOI: https://doi.org/10.1007/BF02183642