Open mushrooms: stickiness revisited

CP Dettmann, O Georgiou

Research output: Contribution to journalArticle (Academic Journal)

23 Citations (Scopus)


We investigate mushroom billiards, a class of dynamical systems with sharply divided phase space. For typical values of the control parameter of the system ρ, an infinite number of marginally unstable periodic orbits (MUPOs) exist making the system sticky in the sense that unstable orbits approach regular regions in phase space and thus exhibit quasi-regular behaviour for long periods of time. The problem of finding these MUPOs is expressed as the well-known problem of finding optimal rational approximations of a real number, subject to some system-specific constraints. By introducing a generalized mushroom and using properties of continued fractions, we describe for the first time a zero measure set of control parameter values ρ (0, 1) for which all MUPOs are destroyed and therefore the system is less sticky, leading to a different power law exponent for the Poincaré recurrence time distribution statistics. The open mushroom (billiard with a hole) is then considered in order to quantify the stickiness exhibited due to MUPOs and exact leading order expressions for the algebraic decay of the survival probability function are calculated for mushrooms with triangular and rectangular stems. Numerical simulations are also performed which confirm our predictions for both sticky and less sticky mushrooms.
Translated title of the contributionOpen mushrooms: Stickiness revisited
Original languageEnglish
Article number195102
Number of pages30
JournalJournal of Physics A: Mathematical and Theoretical
Issue number19
Publication statusPublished - May 2011

Bibliographical note

Publisher: IOP

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