Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-26T00:53:09.106Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-persistence of resonant caustics in perturbed elliptic billiards

Published online by Cambridge University Press:  21 August 2012

SÔNIA PINTO-DE-CARVALHO  and
RAFAEL RAMÍREZ-ROS
SÔNIA PINTO-DE-CARVALHO
Affiliation:
Departamento de Matemática, ICEx, Universidade Federal de Minas Gerais, 30.123–970, Belo Horizonte, Brazil (email: [email protected])
RAFAEL RAMÍREZ-ROS
Affiliation:
Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 6 , December 2013 , pp. 1876 - 1890
Copyright
Copyright © 2012 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Abramowitz, M. and Stegun, I.. Handbook of Mathematical Functions. Dover, New York, 1972.Google Scholar
[2]Baryshnikov, Yu. and Zharnitsky, V.. Sub-Riemannian geometry and periodic orbits in classical billiards. Math. Res. Lett. 13 (2006), 587–598.Google Scholar
[3]Birkhoff, G. D.. Dynamical Systems (American Mathematical Society Colloquium Publications). American Mathematical Society, Providence, RI, 1966, original ed. 1927.Google Scholar
[4]Casas, P. S. and Ramírez-Ros, R.. The frequency map for elliptic billiards. SIAM J. Appl. Dyn. Syst. 10 (2011), 278–324.Google Scholar
[5]Chang, S.-J. and Friedberg, R.. Elliptical billiards and Poncelet’s theorem. J. Math. Phys. 29 (1988), 1537–1550.CrossRefGoogle Scholar
[6]Delshams, A. and Ramírez-Ros, R.. Poincaré–Melnikov–Arnold method for analytic planar maps. Nonlinearity 9 (1996), 1–26.CrossRefGoogle Scholar
[7]Douady, R.. Applications du théorème des tores invariants. Thèse de 3ème Cycle, Univ. Paris VII, 1982.Google Scholar
[8]Dragović, V. and Radnović, M.. Bifurcations of Liouville tori in elliptical billiards. Regul. Chaotic Dyn. 14 (2009), 479–494.Google Scholar
[9]Dragović, V. and Radnović, M.. Poncelet Porisms and Beyond. Birkhäuser, Basel, 2011.CrossRefGoogle Scholar
[10]Gutkin, E. and Katok, A.. Caustics for inner and outer billiards. Comm. Math. Phys. 173 (1995), 101–133.CrossRefGoogle Scholar
[11]Gutkin, E.. Billiard dynamics: a survey with the emphasis on open problems. Regul. Chaotic Dyn. 8 (2003), 1–13.Google Scholar
[12]Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge, 1995.Google Scholar
[13]Knill, O.. On nonconvex caustics of convex billiards. Elem. Math. 53 (1998), 89–106.Google Scholar
[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.CrossRefGoogle Scholar
[15]Lazutkin, V. F.. The existence of caustics for a billiard problem in a convex domain. Math. USSR Izvestija 7 (1973), 185–214.Google Scholar
[16]Mather, J.. Glancing billiards. Ergod. Th. & Dynam. Sys. 2 (1982), 397–403.CrossRefGoogle Scholar
[17]Meiss, J. D.. Symplectic maps, variational principles, and transport. Rev. Mod. Phys. 64 (1992), 795–848.Google Scholar
[18]Poncelet, J. V.. Traité des Propriétés Projectives des Figures. Gauthier-Villars, Paris, 1866.Google Scholar
[19]Ramírez-Ros, R.. Break-up of resonant invariant curves in billiards and dual billiards associated to perturbed circular tables. Physica D 214 (2006), 78–87.Google Scholar
[20]Rothos, V.. Subharmonic bifurcations of localized solutions of a discrete NLS equation. Discrete Contin. Dyn. Syst. (2005), 756–767, supplement volume.Google Scholar
[21]Tabachnikov, S.. Billiards (Panorama et Synthèses). Société Mathématique de France, Paris, 1995.Google Scholar
[22]Whittaker, E. T. and Watson, G. N.. A Course of Modern Analysis. Cambridge University Press, Cambridge, 1927.Google Scholar