Open Mushrooms: Stickiness revisited

There is no author summary for this article yet. Authors can add summaries to their articles on ScienceOpen to make them more accessible to a non-specialist audience.

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.

Related collections

Most cited references 21

Record: found
Abstract: found
Article: found

Is Open Access

Long-Time Correlations in the Stochastic Regime

Charles Karney (2005)

The phase space for Hamiltonians of two degrees of freedom is usually divided into stochastic and integrable components. Even when well into the stochastic regime, integrable orbits may surround small stable regions or islands. The effect of these islands on the correlation function for the stochastic trajectories is examined. Depending on the value of the parameter describing the rotation number for the elliptic fixed point at the center of the island, the long-time correlation function may decay as t^-5 or exponentially, but more commonly it decays much more slowly (roughly as t^-1). As a consequence these small islands may have a profound effect on the properties of the stochastic orbits. In particular, there is evidence that the evolution of a distribution of particles is no longer governed by a diffusion equation.