The Frequency Map for Billiards inside Ellipsoids

Casas, Pablo S.; Ramírez-Ros, Rafael

doi:10.1137/10079389x

Cited by 12 publications

(20 citation statements)

References 41 publications

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“…10(a). The stochastic layer around the next convergent ν = 42 137 is only accessed very briefly. Finally the level-hierarchy is left via the stochastic layer around ν = 49 160 and by passing through ν = 15 49 and ν = 3 10 (not shown).…”

Section: Discussionmentioning

confidence: 99%

Three-dimensional billiards: Visualization of regular structures and trapping of chaotic trajectories

et al. 2018

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The dynamics in three-dimensional (3D) billiards leads, using a Poincaré section, to a four-dimensional map, which is challenging to visualize. By means of the recently introduced 3D phase-space slices, an intuitive representation of the organization of the mixed phase space with regular and chaotic dynamics is obtained. Of particular interest for applications are constraints to classical transport between different regions of phase space which manifest in the statistics of Poincaré recurrence times. For a 3D paraboloid billiard we observe a slow power-law decay caused by long-trapped trajectories, which we analyze in phase space and in frequency space. Consistent with previous results for 4D maps, we find that (i) trapping takes place close to regular structures outside the Arnold web, (ii) trapping is not due to a generalized island-around-island hierarchy, and (iii) the dynamics of sticky orbits is governed by resonance channels which extend far into the chaotic sea. We find clear signatures of partial transport barriers. Moreover, we visualize the geometry of stochastic layers in resonance channels explored by sticky orbits.

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Section: Discussionmentioning

confidence: 99%

Three-dimensional billiards: Visualization of regular structures and trapping of chaotic trajectories

et al. 2018

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“…We recall some concepts related to periodic trajectories of billiards inside ellipses. These results can be found, for instance, in [9,24]. To begin with, we introduce the function ρ : Λ → R given by the quotient of elliptic integrals…”

Section: The Planar Casementioning

confidence: 93%

“…It turns out that given any ellipsoid of the form (1) and any proper coordinate subspace of R n , there exist infinitely many sets of n − 1 distinct nonsingular caustics such that their tangent trajectories are periodic with even period, say m 0 , and any of their impact points becomes its reflection with respect to that coordinate subspace after m 0 /2 bounces. We will not prove this claim, since the proof requires some convoluted ideas developed in [24,25].…”

Section: Conjecturementioning

confidence: 99%

On Cayley conditions for billiards inside ellipsoids

Ramírez-Ros

2014

Nonlinearity

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All the segments (or their continuations) of a billiard trajectory inside an ellipsoid of R n are tangent to n − 1 quadrics of the pencil of confocal quadrics determined by the ellipsoid. The quadrics associated to periodic billiard trajectories verify certain algebraic conditions. Cayley found them in the planar case. Dragović and Radnović generalized them to any dimension. We rewrite the original matrix formulation of these generalized Cayley conditions as a simpler polynomial one. We find several remarkable algebraic relations between caustic parameters and ellipsoidal parameters that give rise to nonsingular periodic trajectories.

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“…The integralξ 1 does not depend on ǫ. The integralξ 1 is bounded using ideas from the proof of Lemma 23 in [45]. The integralξ 1 is bounded using (C.3).…”

Section: Appendix a Proof Of Propositionmentioning

confidence: 99%

Exponentially Small Asymptotic Formulas for the Length Spectrum in Some Billiard Tables

Martín

Ramírez-Ros

Tamarit-Sariol

2016

Experimental Mathematics

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Let q ≥ 3 be a period. There are at least two (1, q)-periodic trajectories inside any smooth strictly convex billiard table, and all of them have the same length when the table is an ellipse or a circle. We quantify the chaotic dynamics of axisymmetric billiard tables close to their borders by studying the asymptotic behavior of the differences of the lengths of their axisymmetric (1, q)-periodic trajectories as q → +∞. Based on numerical experiments, we conjecture that, if the billiard table is a generic axisymmetric analytic strictly convex curve, then these differences behave asymptotically like an exponentially small factor q −3 e −rq times either a constant or an oscillating function, and the exponent r is half of the radius of convergence of the Borel transform of the well-known asymptotic series for the lengths of the (1, q)-periodic trajectories. Our experiments are restricted to some perturbed ellipses and circles, which allows us to compare the numerical results with some analytical predictions obtained by Melnikov methods and also to detect some non-generic behaviors due to the presence of extra symmetries. Our computations require a multiple-precision arithmetic and have been programmed in PARI/GP.

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The Frequency Map for Billiards inside Ellipsoids

Cited by 12 publications

References 41 publications

Three-dimensional billiards: Visualization of regular structures and trapping of chaotic trajectories

Three-dimensional billiards: Visualization of regular structures and trapping of chaotic trajectories

On Cayley conditions for billiards inside ellipsoids

Exponentially Small Asymptotic Formulas for the Length Spectrum in Some Billiard Tables

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