Global dynamics and diffusion in the rational standard map - Cincotta, Pablo Miguel (2020)
In this paper we study the dynamics of the Rational Standard Map, which is a generalization of the Standard Map. It depends on two parameters, the usual K and a new one, 0 ≤ μ < 1, that breaks the entire character of the perturbing function. By means of analytical and numerical methods it is shown that this system presents significant differences with respect to the classical Standard Map. In particular, for relatively large values of K the integer and semi-integer resonances are stable for some range of μ values. Moreover, for K not small and near suitable values of μ, its dynamics could be assumed to be well represented by a nearly integrable system. On the other hand, periodic solutions or 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 very narrow intervals when 0.9 < μ < 1. Big differences for the domains of existence of rotationally invariant curves (much larger, for μ moderate, or much smaller, for μ close to 1 than for the standard map) appear. While anomalies in the diffusion are observed, for large values of the parameters, the system becomes close to an ergodic one.