A lower bound for chaos on the elliptical stadium

Canale, Eduardo; Markarian, Roberto; Kamphorst, Sylvie Oliffson; Carvlho, Sônia Pinto de

doi:10.1016/s0167-2789(97)00232-7

Cited by 17 publications

(22 citation statements)

References 2 publications

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“…It has been confirmed by analysis and numerical computation [7,26,27,28,33,34] that this billiard is fully chaotic (ergodic) for a sizeable but strictly limited region in the parameter space, defined by the stable two-bounce horizontal periodic orbit on one and the pantografic orbits on the other side. Our investigations of the ESB and TEB billiards confirm the suggestion by Del Magno [29] that in spite of apparently similar stadium-like shapes, these two billiards have essentially different dynamical properties.…”

Section: Introductionmentioning

confidence: 86%

“…We are stressing the fact that notable regions of full chaos have been discovered in billiards with elliptical arcs and piecewise flat boundaries, indicating that such billiards deserve further attention [26,27,28,29]. In our previous work we analyzed several types of billiards with noncircular arcs (parabolic, hyperbolic, elliptical and generalized power-law), exhibiting mixed dynamics [30,31,32].…”

Section: Introductionmentioning

confidence: 93%

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Chaotic properties of the truncated elliptical billiards

Lopac¹,

Šimić²

2011

Communications in Nonlinear Science and Numerical Simulation

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Chaotic properties of symmetrical two-dimensional stadium-like billiards with elliptical arcs are studied numerically and analytically. For the two-parameter truncated elliptical billiard the existence and linear stability of several lowest-order periodic orbits are investigated in the full parameter space. Poincaré plots are computed and used for evaluation of the degree of chaoticity with the box-counting method. The limit of the fully chaotic behavior is identified with circular arcs. Above this limit, for flattened elliptical arcs, mixed dynamics with numerous stable elliptic islands is present, similarly as in the elliptical stadium billiards. Below this limit the full chaos extends over the whole region of elongated shapes and the existing orbits are either unstable or neutral. This is conspicuously different from the behavior in the elliptical stadium billiards, where the chaotic region is strictly bounded from both sides. To examine the mechanism of this difference, a generalization to a novel three-parameter family of boundary shapes is proposed and suggested for further evaluation.

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Section: Introductionmentioning

confidence: 86%

Section: Introductionmentioning

confidence: 93%

Chaotic properties of the truncated elliptical billiards

Lopac¹,

Šimić²

2011

Communications in Nonlinear Science and Numerical Simulation

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“…Billiards with δ < 1 − γ This subfamily of elliptical stadia has elongated semiellipses. The corresponding billiards have been investigated in [3,4,5,6], with the principal aim to establish the limiting shapes beyond which the billiard is fully chaotic. Parameters a and h used there to describe the boundary are connected with our parameters δ and γ through relations…”

Section: Classical Dynamics Of the Elliptical Stadium Billiardmentioning

confidence: 99%

“…The results reported in [4,6] suggest that in the parameter space there exists a lower limit above which the billiard is chaotic, as consequence of the existence of the stable pantographic orbits. In In [5], an earlier conjecture of Donnay [3], stating that the lower limit of chaos is set by relations 1 < a < 4 − 2 √ 2 and h > 2a 2 √ a 2 − 1, was investigated.…”

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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“…Quantum versions of both kind of models have also been considered [11][12][13][14][15]. Note that these one-dimensional classical systems allow direct comparison of theoretical results with experimental ones [16,17] and that the formalism used in their characterization can immediately be extended to the billiards [18,19] as well to time-dependent potential well problems [20][21][22][23][24].…”

Section: Introductionmentioning

confidence: 99%

Scaling properties of the Fermi-Ulam accelerator model

Silva¹,

Ladeira²,

Leonel³

et al. 2006

Braz. J. Phys.

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The chaotic low energy region (chaotic sea) of the Fermi-Ulam accelerator model is discussed within a scaling framework near the integrable to non-integrable transition. Scaling results for the average quantities (velocity, roughness, energy etc.) of the simplified version of the model are reviewed and it is shown that, for small oscillation amplitude of the moving wall, they can be described by scaling functions with the same characteristic exponents. New numerical results for the complete model are presented. The chaotic sea is also characterized by its Lyapunov exponents.

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A lower bound for chaos on the elliptical stadium

Cited by 17 publications

References 2 publications

Chaotic properties of the truncated elliptical billiards

Chaotic properties of the truncated elliptical billiards

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

Scaling properties of the Fermi-Ulam accelerator model

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