Chaotic behavior in lemon-shaped billiards with elliptical and hyperbolic boundary arcs

Lopac, Vjera; Mrkonjić, Ivana; Radić, Danko

doi:10.1103/physreve.64.016214

Cited by 21 publications

(21 citation statements)

References 59 publications

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“…Similar catalogue of periodic orbits have also been observed on lemon-shaped billiards with parabolic boundary arcs 17 and with elliptical hyperbolic boundaries arcs. 18 The table approaches the circular table as we continue decreasing B. This trend is clear from Figs.…”

Section: -10supporting

confidence: 51%

Ergodicity of the generalized lemon billiards

Chen

Mohr

Zhang

et al. 2013

Chaos: An Interdisciplinary Journal of Nonlinear Science

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In this paper, we study a two-parameter family of convex billiard tables, by taking the intersection of two round disks (with different radii) in the plane. These tables give a generalization of the one-parameter family of lemon-shaped billiards. Initially, there is only one ergodic table among all lemon tables. In our generalized family, we observe numerically the prevalence of ergodicity among the some perturbations of that table. Moreover, numerical estimates of the mixing rate of the billiard dynamics on some ergodic tables are also provided.

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Section: -10supporting

confidence: 51%

Ergodicity of the generalized lemon billiards

Chen

Mohr

Zhang

et al. 2013

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…The family of lemon billiards was introduced by Heller and Tomsovic in 1993 [2], and has been studied in a number of works [8][9][10][11][12], most recently by Lozej [3] and Bunimovich et al [13], and in our recent work [1]. The lemon billiard boundary is defined by the intersection of two circles of equal unit radius with the distance between their center 2B being less than their diameters and B ∈ (0, 1), and is given by the following implicit equations in Cartesian coordinates…”

Section: The Definition Of the Lemon Billiards And Their Classical Dynamical Propertiesmentioning

confidence: 99%

Classical and Quantum Mixed-Type Lemon Billiards without Stickiness

Lozej

Lukman

Robnik

2021

Nonlinear Phenomena in Complex Systems

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The boundary of the lemon billiards is defined by the intersection of two circles of equal unit radius with the distance 2B between their centers, as introduced by Heller and Tomsovic in Phys. Today 46 38 (1993). This paper is a continuation of our recent paper on classical and quantum ergodic lemon billiard (B = 0:5) with strong stickiness effects published in Phys. Rev. E 103 012204 (2021). Here we study the classical and quantum lemon billiards, for the cases B = 0:42; 0:55; 0:6, which are mixed-type billiards without stickiness regions and thus serve as ideal examples of systems with simple divided phase space. The classical phase portraits show the structure of one large chaotic sea with uniform chaoticity (no stickiness regions) surrounding a large regular island with almost no further substructure, being entirely covered by invariant tori. The boundary between the chaotic sea and the regular island is smooth, except for a few points. The classical transport time is estimated to be very short (just a few collisions), therefore the localization of the chaotic eigenstates is rather weak. The quantum states are characterized by the following universal properties of mixed-type systems without stickiness in the chaotic regions: (i) Using the Poincare-Husimi (PH) functions the eigenstates are separated to the regular ones and chaotic ones. The regular eigenenergies obey the Poissonian statistics, while the chaotic ones exhibit the Brody distribution with various values of the level repulsion exponent β, its value depending on the strength of the localization of the chaotic eigenstates. Consequently, the total spectrum is well described by the Berry-Robnik-Brody (BRB) distribution. (ii) The entropy localization measure A (also the normalized inverse participation ratio) has a bimodal universal distribution, where the narrow peak at small A encompasses the regular eigenstates, theoretically well understood, while the peak at larger A comprises the chaotic eigenstates, and is well described by the beta distribution. (iii) Thus the BRB energy level spacing distribution captures two effects: the divided phase space dictated by the classical Berry-Robnik parameter ρ2 measuring the relative size of the largest chaotic region, in agreement with the Berry-Robnik picture, and the localization of chaotic PH functions characterized by the level repulsion (Brody) parameter β. (iv) Examination of the PH functions shows that they are supported either on the classical invariant tori in the regular islands or on the chaotic sea, where they are only weakly localized. With increasing energy the localization of chaotic states decreases, as the PH functions tend towards uniform spreading over the classical chaotic region, and correspondingly β tends to 1.

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“…In our chosen system of units the mass and the velocity of the particle are m = 1 and V = 1, respectively, hence both X and V x lie in the interval [-1,1]. For billiards with noncircular boundary segments [8,25] such variables are computationally more convenient than those containing the arc length variable suitable for the billiard boundaries with circular arcs [26]. Since X and V x are canonically conjugated variables and our billiard is a Hamiltonian system which reduces to the collision-to-collision symplectic twist map, the phase space and the corresponding Poincaré sections are area preserving.…”

Section: Classical Dynamics Of the Elliptical Stadium Billiardmentioning

confidence: 99%

Chaotic dynamics of the elliptical stadium billiard in the full parameter space

Lopac

Mrkonjić

Pavin

et al. 2006

Physica D: Nonlinear Phenomena

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Dynamical properties of the elliptical stadium billiard, which is a generalization of the stadium billiard and a special case of the recently introduced mushroom billiards, are investigated analytically and numerically. In dependence on two shape parameters δ and γ, this system reveals a rich interplay of integrable, mixed and fully chaotic behavior. Poincaré sections, the box counting method and the stability analysis determine the structure of the parameter space and the borders between regions with different behavior. Results confirm the existence of a large fully chaotic region surrounding the straight line δ = 1 − γ corresponding to the Bunimovich circular stadium billiard. Bifurcations due to the hour-glass and multidiamond orbits are described. For the quantal elliptical stadium billiard, statistical properties of the level spacing fluctuations are examined and compared with classical results.

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Chaotic behavior in lemon-shaped billiards with elliptical and hyperbolic boundary arcs

Cited by 21 publications

References 59 publications

Ergodicity of the generalized lemon billiards

Ergodicity of the generalized lemon billiards

Classical and Quantum Mixed-Type Lemon Billiards without Stickiness

Chaotic dynamics of the elliptical stadium billiard in the full parameter space

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