Open Mushrooms: Stickiness revisited

TL;DR

This paper studies mushroom billiards, identifying conditions for reduced stickiness by destroying MUPOs using continued fractions, and quantifies how the survival probability decays over time in open systems.

Contribution

It introduces a generalized mushroom model, characterizes MUPOs via continued fractions, and derives exact decay expressions for open mushroom billiards.

Findings

A zero measure set of parameters eliminates MUPOs, reducing stickiness.

Exact formulas for survival probability decay are obtained for specific mushroom shapes.

The system's stickiness can be controlled by tuning the control parameter .

Abstract

We investigate mushroom billiards, a class of dynamical systems with sharply divided phase space. For typical values of the control parameter of the system $\rho$, 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 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 a zero measure set of control parameter values $\rho\in(0,1)$ for which all MUPOs are destroyed and therefore the system is less sticky. The open mushroom (billiard with a hole) is then considered in order to quantify the stickiness exhibited and exact leading order expressions for the algebraic decay of the survival probability function $P(t)$ are calculated for mushrooms with triangular and rectangular stems.