Recent Advances in Open Billiards with Some Open Problems

Dettmann, Carl P.

doi:10.1142/9789814340700_0011

Cited by 15 publications

(16 citation statements)

References 76 publications

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“…In this respect, care should be taken as it is known that discretization can introduce 'fake' eigenvalues (not related to the dynamics) even for linear hyperbolic toral automorphisms, [42], and similar phenomena could potentially appear in open systems. From a physical viewpoint, we expect that some of the methods introduced could be adapted to approximate open systems in different practical situations, such as open billiards [17]. Moreover, note that the formalism is equally valid in situations where some piecewise constant observable is introduced [14] or where an infinite countable partition is used as in intermittent maps [43].…”

Section: Discussionmentioning

confidence: 99%

Follow the fugitive: an application of the method of images to open systems

Cristadoro¹,

Knight²,

Esposti³

2013

J. Phys. A: Math. Theor.

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Borrowing and extending the method of images we introduce a theoretical framework that greatly simplifies analytical and numerical investigations of the escape rate in open systems. As an example, we explicitly derive the exact size-and position-dependent escape rate in a Markov case for holes of finite size. Moreover, a general relation between the transfer operators of the closed and corresponding open systems, together with the generating function of the probability of return to the hole is derived. This relation is then used to compute the small hole asymptotic behavior, in terms of readily calculable quantities. As an example we derive logarithmic corrections in the second order term. Being valid for Markov systems, our framework can find application in many areas of the physical sciences such as information theory, network theory, quantum Weyl law and via Ulam's method can be used as an approximation method in general dynamical systems.

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Section: Discussionmentioning

confidence: 99%

Follow the fugitive: an application of the method of images to open systems

Cristadoro¹,

Knight²,

Esposti³

2013

J. Phys. A: Math. Theor.

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“…Notice that Proposition 2.2 offers the converse of this statement. 4 This also yields the first part of point (1) of Theorem A.…”

Section: Interpretation Of B F and Induced Invariantsmentioning

confidence: 54%

“…Given some dynamical system on a topological space and an open subset (called hole in the following), it is natural to study the associated surviving set, that is, the collection of all points which never enter this subset. In this framework, the theory of open dynamical systems is, for instance, concerned with escape rates, conditionally invariant measures and other closely related concepts, see for example [1][2][3][4][5][6][7][8] for more information and further references. Recently, there has been an increased interest in understanding families of suitably parametrized interval holes of one-dimensional maps whose surviving sets fulfill certain properties, see for instance [9][10][11][12][13][14][15].…”

Section: Introductionmentioning

confidence: 99%

The bifurcation set as a topological invariant for one-dimensional dynamics

Fuhrmann¹,

Gröger²,

Passeggi³

2019

Preprint

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For a continuous map on the unit interval or circle, we define the bifurcation set to be the collection of those interval holes whose surviving set is sensitive to arbitrarily small changes of their position. By assuming a global perspective and focusing on the geometric and topological properties of this collection rather than the surviving sets of individual holes, we obtain a novel topological invariant for one-dimensional dynamics.We provide a detailed description of this invariant in the realm of transitive maps and observe that it carries fundamental dynamical information. In particular, for transitive non-minimal piecewise monotone maps, the bifurcation set encodes the topological entropy and strongly depends on the behavior of the critical points.

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“…These systems, which have been the focus of mathematical research 2,[11][12][13] , provide insight into physical phenomena such as encountered in celestial mechanics 14,15 , statistical mechanics 6,16,17 , tokamak physics 18,19 , as well as being important models for quantum chaos [20][21][22] . Billiard systems connect experimental physics and mathematics through experiments employing both two and three dimensional geometries that may be either open 2 or closed.…”

Section: Introductionmentioning

confidence: 99%

The Iris billiard: Critical geometries for global chaos

Page

Antoine

Dettmann

et al. 2020

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We introduce the Iris Billiard, consisting of a point particle enclosed by a unit circle enclosing a central scattering ellipse of fixed elongation (defined as the ratio of the semi-major to the semi-minor axes). When the ellipse degenerates to a circle, the system is integrable, otherwise it displays mixed dynamics. Poincaré sections are displayed for different elongations. Recurrence plots are then applied to the long-term chaotic dynamics of trajectories launched from the unstable period-2 orbit along the semi-major axis i.e., one that initially alternately collides with the ellipse and the circle. We obtain numerical evidence of a set of critical elongations at which the system transitions to global chaos. The transition is characterized by an endogenous escape event, E , which is the first time a trajectory launched from the unstable period-2 orbit misses the ellipse. The angle of escape, θ esc and distance of closest approach, d min of the escape event are studied, and are shown to be exquisitely sensitive to the elongation. The survival probability that E has not occurred after n collisions is shown to follow an exponential distribution.

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Recent Advances in Open Billiards with Some Open Problems

Cited by 15 publications

References 76 publications

Follow the fugitive: an application of the method of images to open systems

Follow the fugitive: an application of the method of images to open systems

The bifurcation set as a topological invariant for one-dimensional dynamics

The Iris billiard: Critical geometries for global chaos

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