Global dynamics and diffusion in the rational standard map

Abstract

In this paper we study the dynamics of the Rational Standard Map, which is a generalization of theStandard Map. It depends on two parameters, the usual K and a new one, 0 ≤ μ < 1, that breaksthe entire character of the perturbing function. By means of analytical and numerical methods it isshown that this system presents significant differences with respect to the classical Standard Map. Inparticular, for relatively large values of K the integer and semi-integer resonances are stable for somerange of μ values. Moreover, for K not small and near suitable values of μ , its dynamics could beassumed to be well represented by a nearly integrable system. On the other hand, periodic solutionsor accelerator modes also show differences between this map and the standard one. For instance,in case of K ≈ 2 π accelerator modes exist for μ less than some critical value but also within verynarrow intervals when 0 . 9 < μ < 1. Big differences for the domains of existence of rotationallyinvariant curves (much larger, for μ moderate, or much smaller, for μ close to 1 than for the standardmap) appear. While anomalies in the diffusion are observed, for large values of the parameters, thesystem becomes close to an ergodic one.