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 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.

The Shannon entropy: An efficient indicator of dynamical stability

“…where τ = 2π η −1 t and δ 2π is the 2π -periodic delta function 95 defined through its Fourier expansion. The above set of equations 96 corresponds to the flow of the Hamiltonian (see [26] for details 97 concerning the numerical equivalence between the map and the 98 Hamiltonian flow) 99…”

“…Chaotic diffusion in high-dimensional Hamiltonian systems in both limits of weak and strong chaos has been largely investigated (see for instance the discussion given in [20] and references therein) while for studies in low dimensional systems we refer to [21][22][23][24][25][26].…”

“…On the other hand, in [25,26,30] it was shown analytically and numerically that the Shannon entropy is also a very powerful tool to measure correlations among the successive values of the phases involved in highly chaotic, almost ergodic, low dimensional maps as the whisker mapping and its generalization to cope with diffusion in Arnold model [21], and the standard map as well as the rational standard map, both for large values of the perturbation parameters.…”

“…The fit was done in 63 ln(Var(I f )) = ln(D) + b ln t, in a similar fashion as in [26] and [20] 64 where both coefficients were derived in different systems. When-65 ever b ≈ 1, D would lead to the expected diffusion coefficient 66 provided that correlations among the phases are negligible.…”

Anomalous diffusion in single and coupled standard maps with extensive chaotic phase spaces

“…In addition, in [29], the authors investigated the presence of phase correlations using the Shannon entropy to distinguish between strong correlations in the time evolution associated with anomalous diffusion, while in [30] mechanisms for controlling the escape of orbits in the SM were studied. A generalized SM, the so-called rational SM, was considered in [31] (see references therein regarding the model's introduction), where an extensive study and comparison between the different dynamical and diffusion trends appearing in the SM were performed.…”