Probing rare physical trajectories with Lyapunov weighted dynamics

TL;DR

This paper introduces a novel numerical method to efficiently identify rare trajectories with unusual chaoticity in high-dimensional systems, demonstrated on nonlinear chains and with potential applications in celestial mechanics and turbulence.

Contribution

The paper presents a new Lyapunov weighted dynamics algorithm enabling the study of rare trajectories in high-dimensional systems, overcoming previous limitations of low-dimensional methods.

Findings

Successfully identified soliton solutions in the Fermi-Pasta-Ulam chain

Detected chaotic-breathers with the new method

Demonstrated efficiency in high-dimensional systems

Abstract

The transition from order to chaos has been a major subject of research since the work of Poincare, as it is relevant in areas ranging from the foundations of statistical physics to the stability of the solar system. Along this transition, atypical structures like the first chaotic regions to appear, or the last regular islands to survive, play a crucial role in many physical situations. For instance, resonances and separatrices determine the fate of planetary systems, and localised objects like solitons and breathers provide mechanisms of energy transport in nonlinear systems such as Bose-Einstein condensates and biological molecules. Unfortunately, despite the fundamental progress made in the last years, most of the numerical methods to locate these 'rare' trajectories are confined to low-dimensional or toy models, while the realms of statistical physics, chemical reactions, or astronomy are still hard to reach. Here we implement an efficient method that allows one to work in higher dimensions by selecting trajectories with unusual chaoticity. As an example, we study the Fermi-Pasta-Ulam nonlinear chain in equilibrium and show that the algorithm rapidly singles out the soliton solutions when searching for trajectories with low level of chaoticity, and chaotic-breathers in the opposite situation. We expect the scheme to have natural applications in celestial mechanics and turbulence, where it can readily be combined with existing numerical methods