Chaotic properties of the truncated elliptical billiards

Lopac, Vjera; Šimić, Ana

doi:10.1016/j.cnsns.2010.03.007

Cited by 3 publications

(5 citation statements)

References 54 publications

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“…As in previous works on billiards formed from piecewise-smooth components [1,3,7,[13][14][15][16][17][18], for a large set of parameter values, we find coexistence of stability islands and chaotic components in phase space. However, we also find numerically that billiards which are sufficiently close to the limiting skewed stadia appear to have no remaining stability islands-the phase space is completely filled by a single chaotic, ergodic component.…”

Section: Introductionsupporting

confidence: 82%

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Chaos and stability in a two-parameter family of convex billiard tables

et al. 2011

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Abstract. We study, by numerical simulations and semi-rigorous arguments, a twoparameter family of convex, two-dimensional billiard tables, generalizing the one-parameter class of oval billiards of Benettin-Strelcyn [BS78]. We observe interesting dynamical phenomena when the billiard tables are continuously deformed from the integrable circular billiard to different versions of completely-chaotic stadia. In particular, we conjecture that a new class of ergodic billiard tables is obtained in certain regions of the two-dimensional parameter space, when the billiards are close to skewed stadia. We provide heuristic arguments supporting this conjecture, and give numerical confirmation using the powerful method of Lyapunov-weighted dynamics.

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Section: Introductionsupporting

confidence: 82%

“…The numerical methods used are briefly described in the appendix B. Numerical results for other billiard models formed by piecewisesmooth curves can be found in [1,3,7,[13][14][15][16][17][18].…”

Section: Parameter Dependence Of Dynamicsmentioning

confidence: 99%

Chaos and stability in a two-parameter family of convex billiard tables

et al. 2011

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“…The method used to estimate the occupation fraction is the box-counting analysis [35,36]. The phase space is divided in a grid and the fraction of boxes that contain at least one point of a given orbit is a good approximation of the area this orbit fills.…”

Section: Parameter Spacementioning

confidence: 99%

Dynamical properties of the soft-wall elliptical billiard

et al. 2016

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Physical systems such as optical traps and microwave cavities are realistically modeled by billiards with soft walls. In order to investigate the influence of the wall softness on the billiard dynamics, we study numerically a smooth two-dimensional potential well that has the elliptical (hard-wall) billiard as a limiting case. Considering two parameters, the eccentricity of the elliptical equipotential curves and the wall hardness, which defines the steepness of the well, we show that (1) whereas the hard-wall limit is integrable and thus completely regular, the soft wall elliptical billiard exhibits chaos, (2) the chaotic fraction of the phase space depends nonmonotonically on the hardness of the wall, and (3) the effect of the hardness on the dynamics depends strongly on the eccentricity of the billiard. We further show that the limaçon billiard can exhibit enhanced chaos induced by wall softness, which suggests that our findings generalize to quasi-integrable systems.

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“…Many aspects of classical and quantum chaos have been widely studied by means of billiards with different shapes [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]. We offer a brief review about some commonly used methods to characterize the classical and quantum motion of particles inside billiards.…”

Section: Introductionmentioning

confidence: 99%

“…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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Chaotic properties of the truncated elliptical billiards

Cited by 3 publications

References 54 publications

Chaos and stability in a two-parameter family of convex billiard tables

Chaos and stability in a two-parameter family of convex billiard tables

Dynamical properties of the soft-wall elliptical billiard

Chaos in the Diamond-Shaped Billiard with Rounded Crown

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