Skip to main content Accessibility help
×
Home

Non-persistence of resonant caustics in perturbed elliptic billiards

  • SÔNIA PINTO-DE-CARVALHO (a1) and RAFAEL RAMÍREZ-ROS (a2)

Abstract

A caustic of a billiard table is a curve such that any billiard trajectory, once tangent to the curve, stays tangent after every reflection at the boundary. When the billiard table is an ellipse, any non-singular billiard trajectory has a caustic, which can be either a confocal ellipse or a confocal hyperbola. Resonant caustics—those whose tangent trajectories are closed polygons—are destroyed under generic perturbations of the billiard table. We prove that none of the resonant elliptical caustics persists under a large class of explicit perturbations of the original ellipse. This result follows from a standard Melnikov argument and the analysis of the complex singularities of certain elliptic functions.

Copyright

References

Hide All
[1]Abramowitz, M. and Stegun, I.. Handbook of Mathematical Functions. Dover, New York, 1972.
[2]Baryshnikov, Yu. and Zharnitsky, V.. Sub-Riemannian geometry and periodic orbits in classical billiards. Math. Res. Lett. 13 (2006), 587598.
[3]Birkhoff, G. D.. Dynamical Systems (American Mathematical Society Colloquium Publications). American Mathematical Society, Providence, RI, 1966, original ed. 1927.
[4]Casas, P. S. and Ramírez-Ros, R.. The frequency map for elliptic billiards. SIAM J. Appl. Dyn. Syst. 10 (2011), 278324.
[5]Chang, S.-J. and Friedberg, R.. Elliptical billiards and Poncelet’s theorem. J. Math. Phys. 29 (1988), 15371550.
[6]Delshams, A. and Ramírez-Ros, R.. Poincaré–Melnikov–Arnold method for analytic planar maps. Nonlinearity 9 (1996), 126.
[7]Douady, R.. Applications du théorème des tores invariants. Thèse de 3ème Cycle, Univ. Paris VII, 1982.
[8]Dragović, V. and Radnović, M.. Bifurcations of Liouville tori in elliptical billiards. Regul. Chaotic Dyn. 14 (2009), 479494.
[9]Dragović, V. and Radnović, M.. Poncelet Porisms and Beyond. Birkhäuser, Basel, 2011.
[10]Gutkin, E. and Katok, A.. Caustics for inner and outer billiards. Comm. Math. Phys. 173 (1995), 101133.
[11]Gutkin, E.. Billiard dynamics: a survey with the emphasis on open problems. Regul. Chaotic Dyn. 8 (2003), 113.
[12]Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge, 1995.
[13]Knill, O.. On nonconvex caustics of convex billiards. Elem. Math. 53 (1998), 89106.
[14]Kozlov, V. V. and Treshchëv, D.. Billiards: A Genetic Introduction to the Dynamics of Systems with Impacts (Translations of Mathematical Monographs, 89). American Mathematical Society, Providence, RI, 1991.
[15]Lazutkin, V. F.. The existence of caustics for a billiard problem in a convex domain. Math. USSR Izvestija 7 (1973), 185214.
[16]Mather, J.. Glancing billiards. Ergod. Th. & Dynam. Sys. 2 (1982), 397403.
[17]Meiss, J. D.. Symplectic maps, variational principles, and transport. Rev. Mod. Phys. 64 (1992), 795848.
[18]Poncelet, J. V.. Traité des Propriétés Projectives des Figures. Gauthier-Villars, Paris, 1866.
[19]Ramírez-Ros, R.. Break-up of resonant invariant curves in billiards and dual billiards associated to perturbed circular tables. Physica D 214 (2006), 7887.
[20]Rothos, V.. Subharmonic bifurcations of localized solutions of a discrete NLS equation. Discrete Contin. Dyn. Syst. (2005), 756767, supplement volume.
[21]Tabachnikov, S.. Billiards (Panorama et Synthèses). Société Mathématique de France, Paris, 1995.
[22]Whittaker, E. T. and Watson, G. N.. A Course of Modern Analysis. Cambridge University Press, Cambridge, 1927.

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed