Probing the local dynamics of periodic orbits by the generalized alignment index (GALI) method

TL;DR

This paper extends the GALI chaos detection method to analyze local dynamics near periodic orbits, revealing distinct behaviors for stable and unstable orbits in Hamiltonian flows and symplectic maps.

Contribution

It introduces a novel application of GALI to study local periodic orbit dynamics, providing theoretical and numerical insights into stability and chaos indicators.

Findings

GALIs tend to zero with specific power laws for stable orbits in flows

GALIs fluctuate around non-zero values for stable orbits in maps

GALIs decay exponentially for unstable periodic orbits

Abstract

As originally formulated, the Generalized Alignment Index (GALI) method of chaos detection has so far been applied to distinguish quasiperiodic from chaotic motion in conservative nonlinear dynamical systems. In this paper we extend its realm of applicability by using it to investigate the local dynamics of periodic orbits. We show theoretically and verify numerically that for stable periodic orbits the GALIs tend to zero following particular power laws for Hamiltonian flows, while they fluctuate around non-zero values for symplectic maps. By comparison, the GALIs of unstable periodic orbits tend exponentially to zero, both for flows and maps. We also apply the GALIs for investigating the dynamics in the neighborhood of periodic orbits, and show that for chaotic solutions influenced by the homoclinic tangle of unstable periodic orbits, the GALIs can exhibit a remarkable oscillatory behavior during which their amplitudes change by many orders of magnitude. Finally, we use the GALI method to elucidate further the connection between the dynamics of Hamiltonian flows and symplectic maps. In particular, we show that, using for the computation of GALIs the components of deviation vectors orthogonal to the direction of motion, the indices of stable periodic orbits behave for flows as they do for maps.