Global dynamics visualisation from Lagrangian Descriptors. Applications to discrete and continuous systems

TL;DR

This paper presents a new global dynamics indicator based on Lagrangian Descriptors that effectively visualizes chaotic and ordered motions in multidimensional systems without requiring tangent vector computations.

Contribution

The paper introduces a novel chaos indicator using Lagrangian Descriptors that simplifies visualization of dynamical behaviors in complex systems, especially near resonances.

Findings

Successfully visualizes chaotic regions in classical systems

Reproduces stability maps for various dynamical systems

Applicable to high-dimensional nearly-integrable Hamiltonian systems

Abstract

This paper introduces a new global dynamics and chaos indicator based on the method of Lagrangian Descriptor apt for discriminating ordered and deterministic chaotic motions in multidimensional systems. The selected implementation of this method requires only the knowledge of orbits on finite time windows and is free of the computation of the tangent vector dynamics (i.e., variational equations are not needed). To demonstrate its ability in visualising different dynamical behaviors, in particular for highlighting chaotic regions, several stability maps of classical systems, obtained with different phase space methods, are reproduced. The benchmark examples are rooted in discrete and continuous nearly-integrable dynamical systems, with prominent features played by resonances. These include the Chirikov standard map, higher dimensional symplectic and volume preserving maps, fundamental models of resonances, and a 3 degrees-of-freedom nearly-integrable Hamiltonian system with a dense web of resonances. The indicator thus appears to be relevant for understanding phase space transport mediated by resonances in nearly-integrable system, as ubiquitous in celestial mechanics or astrodynamics.