Stickiness in mushroom billiards

Eduardo G. Altmann; Adilson E. Motter; Holger Kantz

doi:10.1063/1.1979211

Stickiness in mushroom billiards

Eduardo G. Altmann^*, Adilson E. Motter, Holger Kantz

^*Corresponding author for this work

Physics and Astronomy

Research output: Contribution to journal › Article › peer-review

62 Scopus citations

Abstract

We investigate the dynamical properties of chaotic trajectories in mushroom billiards. These billiards present a well-defined simple border between a single regular region and a single chaotic component. We find that the stickiness of chaotic trajectories near the border of the regular region occurs through an infinite number of marginally unstable periodic orbits. These orbits have zero measure, thus not affecting the ergodicity of the chaotic region. Notwithstanding, they govern the main dynamical properties of the system. In particular, we show that the marginally unstable periodic orbits explain the periodicity and the power-law behavior with exponent γ=2 observed in the distribution of recurrence times.

Original language	English (US)
Article number	033105
Journal	Chaos
Volume	15
Issue number	3
DOIs	https://doi.org/10.1063/1.1979211
State	Published - Sep 2005

Funding

E.G.A. is supported by CAPES (Brazil) and DAAD (Germany). A.E.M. is supported by the U.S. Department of Energy under Contract No. W-7405-ENG-36.

ASJC Scopus subject areas

General Physics and Astronomy
Applied Mathematics
Statistical and Nonlinear Physics
Mathematical Physics

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10.1063/1.1979211

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title = "Stickiness in mushroom billiards",

abstract = "We investigate the dynamical properties of chaotic trajectories in mushroom billiards. These billiards present a well-defined simple border between a single regular region and a single chaotic component. We find that the stickiness of chaotic trajectories near the border of the regular region occurs through an infinite number of marginally unstable periodic orbits. These orbits have zero measure, thus not affecting the ergodicity of the chaotic region. Notwithstanding, they govern the main dynamical properties of the system. In particular, we show that the marginally unstable periodic orbits explain the periodicity and the power-law behavior with exponent γ=2 observed in the distribution of recurrence times.",

author = "Altmann, {Eduardo G.} and Motter, {Adilson E.} and Holger Kantz",

note = "Funding Information: E.G.A. is supported by CAPES (Brazil) and DAAD (Germany). A.E.M. is supported by the U.S. Department of Energy under Contract No. W-7405-ENG-36.",

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AU - Altmann, Eduardo G.

AU - Motter, Adilson E.

AU - Kantz, Holger

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PY - 2005/9

Y1 - 2005/9

N2 - We investigate the dynamical properties of chaotic trajectories in mushroom billiards. These billiards present a well-defined simple border between a single regular region and a single chaotic component. We find that the stickiness of chaotic trajectories near the border of the regular region occurs through an infinite number of marginally unstable periodic orbits. These orbits have zero measure, thus not affecting the ergodicity of the chaotic region. Notwithstanding, they govern the main dynamical properties of the system. In particular, we show that the marginally unstable periodic orbits explain the periodicity and the power-law behavior with exponent γ=2 observed in the distribution of recurrence times.

AB - We investigate the dynamical properties of chaotic trajectories in mushroom billiards. These billiards present a well-defined simple border between a single regular region and a single chaotic component. We find that the stickiness of chaotic trajectories near the border of the regular region occurs through an infinite number of marginally unstable periodic orbits. These orbits have zero measure, thus not affecting the ergodicity of the chaotic region. Notwithstanding, they govern the main dynamical properties of the system. In particular, we show that the marginally unstable periodic orbits explain the periodicity and the power-law behavior with exponent γ=2 observed in the distribution of recurrence times.

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Stickiness in mushroom billiards

Abstract

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