Global dynamics of coupled standard maps

TL;DR

This paper extends the use of the GALI method to analyze the dynamics of a 4D symplectic map composed of coupled standard maps, efficiently distinguishing between regular and chaotic motions.

Contribution

It introduces the application of GALI indices to multi-dimensional maps, demonstrating faster and reliable identification of dynamical behaviors in coupled standard maps.

Findings

GALI indices effectively distinguish regular and chaotic motion.

GALI$_3$ and GALI$_4$ are faster than previous indices.

High accuracy in computing percentages of regular and chaotic trajectories.

Abstract

Understanding the dynamics of multi--dimensional conservative dynamical systems (Hamiltonian flows or symplectic maps) is a fundamental issue of non-linear science. The Generalized ALignment Index (GALI), which was recently introduced and applied successfully for the distinction between regular and chaotic motion in Hamiltonian systems \cite{sk:6}, is an ideal tool for this purpose. In the present paper we make a first step towards the dynamical study of multi--dimensional maps, by obtaining some interesting results for a 4--dimensional (4D) symplectic map consisting of N=2 coupled standard maps \cite{Kan:1}. In particular, using the new GALI$_3$ and GALI$_4$ indices, we compute the percentages of regular and chaotic motion of the map equally reliably but much faster than previously used indices, like GALI$_2$ (known in the literature as SALI).