Application of the GALI Method to Localization Dynamics in Nonlinear Systems

TL;DR

This paper applies the GALI method to analyze localization and stability of quasiperiodic oscillations in high-dimensional nonlinear Hamiltonian systems and coupled maps, demonstrating its effectiveness in detecting chaos and quasiperiodic structures.

Contribution

It introduces the application of the GALI method to complex nonlinear systems, showing its advantages in identifying dynamical behaviors more efficiently than existing techniques.

Findings

GALI effectively detects chaos and quasiperiodic structures in high-dimensional systems.

The method provides reliable estimates of the dimensionality of quasiperiodic tori.

GALI predicts slow diffusion phenomena more efficiently than previous approaches.

Abstract

We investigate localization phenomena and stability properties of quasiperiodic oscillations in $N$ degree of freedom Hamiltonian systems and $N$ coupled symplectic maps. In particular, we study an example of a parametrically driven Hamiltonian lattice with only quartic coupling terms and a system of $N$ coupled standard maps. We explore their dynamics using the Generalized Alignment Index (GALI), which constitutes a recently developed numerical method for detecting chaotic orbits in many dimensions, estimating the dimensionality of quasiperiodic tori and predicting slow diffusion in a way that is faster and more reliable than many other approaches known to date.