Dynamical robustness of discrete conservative systems: Harper and generalized standard maps

TL;DR

This paper investigates the statistical behavior of discrete conservative maps, including the Harper map and a generalized standard map, revealing universal patterns in their phase space distributions related to stability and chaos.

Contribution

It demonstrates the universality of statistical behaviors across different conservative maps and introduces a generalized standard map with complex phase space structures.

Findings

Harper map shares the same statistical universality class as standard and web maps.

Probability distributions transition from Gaussian to q-Gaussian as parameters vary.

Generalized standard map exhibits complex mixture behaviors due to sticky regions.

Abstract

In recent years, statistical characterization of the discrete conservative dynamical systems (more precisely, paradigmatic examples of area-preserving maps such as the standard and the web maps) has been analyzed extensively and shown that, for larger parameter values for which the Lyapunov exponents are largely positive over the entire phase space, the probability distribution is a Gaussian, consistent with Boltzmann-Gibbs (BG) statistics. On the other hand, for smaller parameter values for which the Lyapunov exponents are virtually zero over the entire phase space, we verify this distribution appears to approach a $q$-Gaussian (with $q \simeq 1.935$), consistent with $q$-statistics. Interestingly, if the parameter values are in between these two extremes, then the probability distributions happen to exhibit a linear combination of these two behaviours. Here, we numerically show that the Harper map is also in the same universality class of the maps discussed so far. This constitutes one more evidence on the robustness of this behavior whenever the phase space consists of stable orbits. Then, we propose a generalization of the standard map for which the phase space includes many sticky regions, changing the previously observed simple linear combination behavior to a more complex combination.