Power Decay of Phase-Difference and Eigenvalue Distributions in a Radial Twist Map

Abstract

A radial twist map whose phase difference has a singularity and decreases monotonically with action is considered. In the region near the vanishing action, almost all orbits become chaotic. It is found that the phase-difference distribution of a single chaotic orbit in that region decays with a power law for the generalized Fermi map and decays exponentially for the separatrix map. It is also found that the distribution of the maximum absolute eigenvalue of their tangent maps decays with a power law. These decay types are related to the rotation number, the transit or waiting time and the orbital instability.