Abstract
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≈2π 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.
Highlights
One of the most studied area-preserving maps is the well known standard map (SM hereafter) introduced by Chirikov in [1,2] as a representative model of a multiplet of interacting nonlinear resonances
In this paper we investigate the full dynamics of a generalization of the SM, the so-called Rational Standard Map
Deviations from the expected value of the exponent b ≤ 2 at the values of k = |n| are due to several factors being the most relevant: (i) roundoff errors when v takes very large values, say v > 106, while f (2π u) ∼ O(1); (ii) a finite number N of iterates induces a lack of balance between those trajectories in the ensemble that are trapped by a given accelerator mode and those that escape
Summary
One of the most studied area-preserving maps is the well known standard map (SM hereafter) introduced by Chirikov in [1,2] as a representative model of a multiplet of interacting nonlinear resonances. In order to have a Hamiltonian such that its time-2π map is close to the SM it is necessary to produce an instantaneous jump in y followed by a suitable increase of the value of x This can be achieved using a potential which is the product of V (x) by a 2π periodic δ distribution, to be denoted as δ2π. The dynamics of the SM has been studied in a lot of works for many values of K It is a nearly regular system for K ≪ 1, small chaotic domains are created when increasing K , the transition to large chaos (rotationally invariant curves do not exist) occurs for. In all cases suitable numerical simulations allow to illustrate the different observed phenomena
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