Ergodicity of the generalized lemon billiards

Chen, Jingyu; Mohr, Luke; Zhang, Hongkun; Zhang, Peng-Fei

doi:10.1063/1.4850815

Cited by 19 publications

(25 citation statements)

References 26 publications

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“…Alternatively, an asymmetric lemon billiard table is comprised of the intersection of the two disks ( Figure 5, right). We use the designation Q M (R, B, 0) for moons and Q L (R, B, 0) for lemons, replacing the now ambiguous Q(R, B) used for the former class in [16] and the latter in [13].…”

Section: Umbrella Billiardsmentioning

confidence: 99%

“…Conversely, when B approaches 0 and the table approaches the circular case, a host of elliptic islands emerge, becoming long and narrow approaching the integrable case. If R = 1, however, and asymmetric billiards are considered, hyperbolicity [12] and apparent ergodicity [13] often arise, as in Region I in Figure 7.…”

Section: Umbrella Billiardsmentioning

confidence: 99%

“…In [20] the ovals were generalized to a two-parameter family encompassing seven varieties, including special cases of lemon, moon, and a particular example of a class which in this paper we will designate as umbrella billiards, while [4] gives an alternate generalization of [7] to squash billiard tables, on which the elementary defocusing mechanism does not take place. In [25] symmetric lemon billiards were considered, and recently a class of asymmetric lemon-shaped convex billiard tables were constructed in [13], obtained by intersection of two disks in the plane. It was also proved that a subclass of these billiards are indeed hyperbolic using continued fraction techniques [12].…”

Section: Introductionmentioning

confidence: 99%

“…The last two sections give numeric evidence of new ergodic billiards, considering Poincaré surfaces of sections and the transition of periodic points from elliptic to hyperbolic in Section 4, and Lyapunov exponents in Section 5. [13] and [16].) The two-petal flowers in the upper left are analytically known to be ergodic.…”

Section: Introductionmentioning

confidence: 99%

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Ergodicity in Umbrella Billiards

Correia¹,

Cox²,

Zhang³

2017

NHMP

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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.

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Section: Umbrella Billiardsmentioning

confidence: 99%

Section: Umbrella Billiardsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Ergodicity in Umbrella Billiards

Correia¹,

Cox²,

Zhang³

2017

NHMP

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“…Billiards are one of the most used systems to analyse the quantum signatures of classical chaotic motion [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]. Some advantages of the billiards are their extreme simplicity, their straightforward quantization and the possibility to measure many of the relevant quantities in laboratory experiments [18,19,20,21,22,23,24,25].…”

Section: Introductionmentioning

confidence: 99%

Chaos in the Diamond-Shaped Billiard with Rounded Crown

Salazar

Téllez

Jaramillo

et al. 2015

Rev. Acad. Colomb. Cienc. Ex. Fis. Nat.

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We analyse the classical and quantum behaviour of a particle trapped in a diamond-shaped billiard with rounded crown. We defined this billiard as a half stadium connected with a triangular billiard. A parameter ξ smoothly changes the shape of the billiard from an equilateral triangle (ξ = 1) to a diamond with rounded crown (ξ = 0). The parameter ξ controls the transition between the regular and chaotic regimes. The classical behaviour is regular when the control parameter ξ is one; in contrast, the system is chaotic when ξ = 1 even for values of ξ close to one. Several quantities such as Lyapunov exponent and the entropy of the distribution of the incident angle are used to characterize the chaotic behaviour of the classical system. The average information preserved by the classical trajectories increases rapidly as ξ is decreased from 1 and the Lyapunov exponent remains positive for ξ < 1. The Finite Difference Method was implemented in order to solve the quantum counterpart of the billiard. The energy spectrum and eigenstates were numerically computed for different values of ξ < 1. The spacing distribution between adjacent eigenvalues is analysed as a function of ξ, finding a Poisson and a Gaussian Orthogonal Ensemble (GOE) distribution for regular and chaotic regimes respectively. Several scars and bouncing ball states are shown with their corresponding classical periodic orbits. On the other hand, the results found for the quantum billiard are in agreement with the Bohigas-Giannoni-Schmit conjecture and exhibits the standard features of chaotic billiards such as the scarring of the wavefunction.Keywords: quantum chaos, quantum billiards, random matrices, finite difference method.Caos en el billar de forma de diamante y corona redondeada Resumen Se estudia el comportamiento de una partícula en el interior de un billar triangular donde uno de sus lados toma de medio estadio que se llamó billar diamante con corona redondeada o DSRC por su siglas en inglés. Se definió un parámetro ξ que cambia suavemente la forma la frontera partiendo de un billar triangular ξ = 1 a un billar DSRC ξ = 1. Dicho parámetro controla la transición entre el régimen regular y caótico. Clásicamente, el sistema es regular cuando ξ = 1. Por otro lado, el sistema se torna caótico para ξ = 1 incluyendo valores próximos a 1. Se calcula el coeficiente de Lyapunov y la entropía media de la distribución de losángulos de incidencia para caracterizar el comportamiento caótico del sistema. Se observó un rápido crecimiento de la información de las trayectorias hasta saturar la entropía al cambiar levemente la frontera del billar triangular original. A su vez el coeficiente de Lyapunov se mantuvo positivo durante este proceso una vez que ξ se alejaba de 1. Se implementó el método de diferencias finitas FDM para obtener el espectro y los estados propios de la contraparte cuántica del sistema. La distribución de espaciamiento entre primeros vecinos para varios valores de ξ fue construida numéricamente para diferentes valores de ξ encontrando u...

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On Another Edge of Defocusing: Hyperbolicity of Asymmetric Lemon Billiards

Bunimovich

Zhang

2016

Commun. Math. Phys.

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Defocusing mechanism provides a way to construct chaotic (hyperbolic) billiards with focusing components by separating all regular components of the boundary of a billiard table sufficiently far away from each focusing component. If all focusing components of the boundary of the billiard table are circular arcs, then the above separation requirement reduces to that all circles obtained by completion of focusing components are contained in the billiard table. In the present paper we demonstrate that a class of convex tables-asymmetric lemons, whose boundary consists of two circular arcs, generate hyperbolic billiards. This result is quite surprising because the focusing components of the asymmetric lemon table are extremely close to each other, and because these tables are perturbations of the first convex ergodic billiard constructed more than forty years ago.2000 Mathematics Subject Classification. 37D50.

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Ergodicity of the generalized lemon billiards

Cited by 19 publications

References 26 publications

Ergodicity in Umbrella Billiards

Ergodicity in Umbrella Billiards

Chaos in the Diamond-Shaped Billiard with Rounded Crown

On Another Edge of Defocusing: Hyperbolicity of Asymmetric Lemon Billiards

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