Isaac Scientific Publishing

New Horizons in Mathematical Physics

Ergodicity in Umbrella Billiards

Download PDF (7141.9 KB) PP. 56 - 67 Pub. Date: September 12, 2017

DOI: 10.22606/nhmp.2017.12004

Author(s)

  • Maria F. Correia
    CIMA-UE, Dept of Mathematics, University of Évora, Rua Romão Ramalho, 59, 7000-671 Évora, Portugal.
  • Christopher Cox*
    Dept of Mathematics, Washington University in St. Louis, One Brookings Drive, St. Louis, MO 63130-4899
  • Hong-Kun Zhang
    Dept of Mathematics and Statistics, University of Massachusetts, 710 North Pleasant Street, Amherst, MA 01003.

Abstract

We investigate a three-parameter family of billiard tables with circular arc boundaries. These umbrella-shaped billiards may be viewed as a generalization of two-parameter moon and asymmetric lemon billiards, in which the latter classes comprise instances where the new parameter is 0. Like those two previously studied classes, for certain parameters umbrella billiards exhibit evidence of chaotic behavior despite failing to meet certain criteria for defocusing or dispersing, the two most well understood mechanisms for generating ergodicity and hyperbolicity. For some parameters corresponding to non-ergodic lemon and moon billiards, small increases in the new parameter transform elliptic 2-periodic points into a cascade of higher order elliptic points. These may either stabilize or dissipate as the new parameter is increased. We characterize the periodic points and present evidence of new ergodic examples.

Keywords

Ergodicity, chaotic billiards, elliptic islands.

References

[1] E. G. Altmann, T. Friedrich, A. E. Motter, H. Kantz, and A. Richter, "Prevalence of marginally unstable periodic orbits in chaotic billiards", Phys Rev E 77, 016205 (2008).

[2] D. Armstead, B. Hunt and E. Ott, "Power-law decay and self-similar distribution in stadium-type billiards", Physica D 193, 96–127 (2004).

[3] M. V. Berry, "Regularity and chaos in classical mechanics, illustrated by three deformations of a circular'billiard'", Eur. J. Phys., 2, 91–102 (1981).

[4] P. Bálint, M. Halász, J. Hernández-Tahuilán, and D. Sanders, "Chaos and stability in a two-parameter family of convex billiard tables", Nonlinearity 24, 1499–1521 (2011).

[5] M. Brack and R. K. Bhaduri, "Semiclassical Physics", Addison-Wesley, Reading, Massachusetts, (1997).

[6] G. Benettin, L. Galgani, A. Giorgilli, J. M. Strelcyn, "Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: A method for computing all of them", Meccanica 15, 9–30 (1980).

[7] G. Benettin and J. M. Strelcyn, "Numerical experiments on the free motion of a point mass moving in a plane convex region: stochastic transition and entropy", Phys. Rev. A 17, 773–785 (1978).

[8] O. Bohigas, D. Boosé, R. Egydio de Carvalho, and V. Marvulle, "Quantum tunneling and chaotic dynamics", Nucl. Phys. A 560, 197–210 (1993).

[9] L. A. Bunimovich, "On ergodic properties of certain billiards", Funct. Anal. Appl. 8, 254–255 (1974).

[10] L. A. Bunimovich, "On absolutely focusing mirror", Ergodic theory and related topics, III (Güstrow, 1990)", Lecturer Notes in Math 1514, Springer-Verlag, 62–82 (1992).

[11] L. Bunimovich and A. Grigo, "Focusing components in typical chaotic billiards should be absolutely focusing", Commun. Math. Phys. 293, 127–143 (2010).

[12] L. A. Bunimovich, H. K. Zhang, and P. Zhang, "On another edge of defocusing: hyperbolicity of asymmetric lemon billiards", arXiv preprint arXiv:1412.0173 (2014).

[13] J. Chen, L. Mohr, H. K. Zhang, and P. Zhang, "Ergodicity of the generalized lemon billiards", Chaos 23, 0431371–12 (2013).

[14] N. Chernov and H.-K. Zhang, "Billiards with polynomial mixing rates", Nonlinearity 18, 1527–1553 (2005).

[15] N. Chernov and R. Markarian, "Chaotic billiards", Mathematical Surveys and Monographs 127, American Mathematical Society, Providence, RI, (2006).

[16] M. F. Correia and H. K. Zhang, "Stability and ergodicity of moon billiards", Chaos 25, 083110 (2015).

[17] M.J. Dias Carneiro, S. Oliffson Kamphorst and S. Pintode Carvalho,"Elliptic islands in strictly convex billiards", Erg. Th. Dynam. Syst. 23, 799–812 (2003).

[18] C. Dettmann, V. Fain, "Linear and nonlinear stability of periodic orbits in annular billiards," arXiv:1609.03652.

[19] V. Donnay, "Using integrability to produce chaos: billiards with positive entropy", Commun. Math. Phys. 141, 225–257 (1991).

[20] H. R. Dullin, P. H. Richter, and A. Wittek, "A two-parameter study of the extent of chaos in a billiard system", Chaos 6, 43–58 (1996).

[21] C. Foltin, "Billiards with positive topological entropy", Nonlinearity 15, 2053–2076 (2002).

[22] G. Gouesbet, S. Meunier-Guttin-Cluzel, and G. Gréhan, "Periodic orbits in Hamiltonian chaos of the annular billiard", Phys. Rev. E 65, 016212 (2001).

[23] A. Grigo, "Billiards and statistical mechanics", Ph.D. Thesis (2009).

[24] A. Hayli, "Numerical exploration of a family of strictly convex billiards with boundary of class C2", J. Stat. Phys. 83, 71–79 (1996).

[25] E. Heller and S. Tomsovic, "Postmodern quantum mechanics", Physics Today 46, 38–46 (1993).

[26] M. Hénon and J. Wisdom, "The Benettin-Strelcyn oval billiard revisited", Physica D 8, 157–169 (1983).

[27] M. Hentschel and K. Richter, "Quantum chaos in optical systems: The annular billiard", Physical Review E 66, 056207 (2002).

[28] R. Markarian, "Billiards with Pesin region of measure one", Commun. Math. Phys. 118, 87–97 (1988).

[29] R. Markarian, "Billiards with polynomial decay of correlations", Ergodic Theory Dynam. Systems 24, 177–197(2004).

[30] N. Saito, H. Hirooka, J. Ford, F. Vivaldi and G. H. Walker, "Numerical study of billiard motion in an annulus bounded by non-concentric circles", Physica D 5, 273–86 (1982).

[31] Z. Sándor, B. érdi, A. Széll, and B. Funk. "The relative Lyapunov indicator: an efficient method of chaos detection". Celest. Mech. Dyn. Astron. 90, 127–?138(2004).

[32] Y. Sinaˇ?, "Dynamical systems with elastic reflections", Russ. Math. Surveys 25, 137–191 (1970).

[33] H.-J. Stockmann, "Quantum Chaos: An Introduction", Cambridge University Press, Cambridge, (1999).

[34] M. Wojtkowski, "Principles for the design of billiards with nonvanishing Lyapunov exponents", Commun. Math. Phys. 105, 391–414 (1986).