How sticky is the chaos/order boundary?

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Abstract

In dynamical systems with divided phase space, the vicinity of the boundary between regular and chaotic regions is often “sticky,” that is, trapping orbits from the chaotic region for long times. Here, we investigate the stickiness in the simplest mushroom billiard, which has a smooth such boundary, but surprisingly subtle behaviour. As a measure of stickiness, we investigate P(t), the probability of remaining in the mushroom cap for at least time t given uniform initial conditions in the chaotic part of the cap. The stickiness is sensitively dependent on the radius of the stem r via the Diophantine properties of ρ = (2/π) arccos r. Almost all ρ give rise to families of marginally unstable periodic orbits (MUPOs) where P(t) ∼ C/t, dominating the stickiness of the boundary. Here we consider the case where ρ is MUPO-free and has continued fraction expansion with bounded partial quotients. We show that t^2 P(t) is bounded, varying infinitely often between values whose ratio is at least 32/27. When ρ has an eventually periodic continued fraction expansion, that is, a quadratic irrational, t^2 P(t) converges to a log-periodic function. In general, we expect less regular behaviour, with upper and lower exponents lying between 1 and 2. The results may shed light on the parameter dependence of boundary stickiness in annular billiards and generic area preserving maps.
Original languageEnglish
Title of host publicationDynamical Systems, Ergodic Theory, and Probability: in Memory of Kolya Chernov
Subtitle of host publicationConference Dedicated to the Memory of Nikolai Chernov Dynamical Systems, Ergodic Theory, and Probability May 18–20, 2015 University of Alabama at Birmingham, Birmingham, Alabama
EditorsAlexander M Bloch, Leonid A Bunimovich, Paul H Jung, Lex G Oversteegen, Yakov G Sinai
PublisherAmerican Mathematical Society
Pages111-128
Number of pages18
ISBN (Electronic)9781470442248
ISBN (Print)9781470427733
DOIs
Publication statusPublished - 30 Sep 2017

Publication series

NameContemporary Mathematics
PublisherAmerican Mathematical Society
Volume698
ISSN (Print)0271-4132

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