Sticky and Non-sticky open Mushrooms

Abstract

A billiard is a dynamical system in which a particle alternates between motion in a straight line and specular reflections from a boundary. The mushroom billiard forms a class of dynamical systems with sharply divided phase in two dimensions. Its mixed phase space is composed of a single completely regular (integrable) component and a single chaotic and ergodic component. For typical values of the control parameter of the system, an infinite number of marginally unstable periodic orbits (MUPOs) exist making the system sticky in the sense that unstable periodic orbits approach regular regions in phase space and thus exhibit 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 number, subject to some system-specific constraints. We introduce a measure zero set of control parameter values for which all MUPOs are destroyed and therefore the system is non-sticky. The open mushroom (billiard with a hole) is considered and the asymptotic survival probability function P (t) is calculated for both cases.