We construct a two-parameter family of moon-shaped billiard tables with boundary made of two circular arcs. These tables fail the defocusing mechanism and other known mechanisms that guarantee ergodicity and hyperbolicity. We analytically study the stability of some periodic orbits and prove there is a class of billiards in this family with elliptic periodic orbits. These moon billiards can be viewed as generalization of annular billiards, which all have Kolmogorov-Arnold-Moser islands. However, the novelty of this paper is that by varying the parameters, we numerically observe a subclass of moon-shaped billiards with a single ergodic component.
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, MA
, 1997
).6.
G.
Benettin
, L.
Galgani
, A.
Giorgilli
, and 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 Mathematics Vol. 1514 (Springer-Verlag, 1992), pp. 62
–82
.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
,” preprint arXiv:1412.0173 (2014
).13.
J.
Chen
, L.
Mohr
, H. K.
Zhang
, and P.
Zhang
, “Ergodicity of the generalized lemon billiards
,” Chaos
23
, 043137
(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
,” in Mathematical Surveys and Monographs
(American Mathematical Society
, Providence, RI
, 2006
), Vol. 127.16.
M. J.
Dias Carneiro
, S.
Oliffson Kamphorst
, and S.
Pintode Carvalho
, “Elliptic islands in strictly convex billiards
,” Ergodic Theory Dyn. Syst.
23
, 799
–812
(2003
).17.
V.
Donnay
, “Using integrability to produce chaos: billiards with positive entropy
,” Commun. Math. Phys.
141
, 225
–257
(1991
).18.
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
).19.
C.
Foltin
, “Billiards with positive topological entropy
,” Nonlinearity
15
, 2053
–2076
(2002
).20.
G.
Gouesbet
, S.
Meunier-Guttin-Cluzel
, and G.
Grhan
, “Periodic orbits in Hamiltonian chaos of the annular billiard
,” Phys. Rev. E
65
, 016212
(2001
).21.
A.
Grigo
, “Billiards and statistical mechanics
,” Ph.D. thesis (Georgia Institute of Technology
, 2009
).22.
A.
Hayli
, “Numerical exploration of a family of strictly convex billiards with boundary of class C2
,” J. Stat. Phys.
83
, 71
–79
(1996
).23.
E.
Heller
and S.
Tomsovic
, “Postmodern quantum mechanics
,” Phys. Today
46
(7
), 38
–46
(1993
).24.
M.
Hénon
and J.
Wisdom
, “The Benettin-Strelcyn oval billiard revisited
,” Physica D
8
, 157
–169
(1983
).25.
M.
Hentschel
and K.
Richter
, “Quantum chaos in optical systems: The annular billiard
,” Phys. Rev. E
66
, 056207
(2002
).26.
R.
Markarian
, “Billiards with Pesin region of measure one
,” Commun. Math. Phys.
118
, 87
–97
(1988
).27.
R.
Markarian
, “Billiards with polynomial decay of correlations
,” Ergodic Theory Dyn. Syst.
24
, 177
–197
(2004
).28.
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
–286
(1982
).29.
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
).30.
Y.
Sinaĭ
, “Dynamical systems with elastic reflections
,” Russ. Math. Surv.
25
, 137
–191
(1970
).31.
H.-J.
Stockmann
, Quantum Chaos: An Introduction
(Cambridge University Press
, Cambridge
, 1999
).32.
M.
Wojtkowski
, “Principles for the design of billiards with nonvanishing Lyapunov exponents
,” Commun. Math. Phys.
105
, 391
–414
(1986
).© 2015 AIP Publishing LLC.
2015
AIP Publishing LLC
You do not currently have access to this content.
