Painting the Phase Space of Dissipative Systems with Lagrangian Descriptors

TL;DR

This paper demonstrates how Lagrangian descriptors can be used to visualize and analyze the geometric structures in phase space of dissipative systems, enhancing understanding of their global dynamics.

Contribution

It applies Lagrangian descriptors to dissipative systems, revealing complex phase space structures and transitions, which is a novel approach in nonlinear dynamics analysis.

Findings

Identifies hyperbolic points, limit cycles, and strange attractors using Lagrangian descriptors.

Detects transition ellipsoids in dissipative Hamiltonian systems.

Provides detailed phase space visualization for complex dynamical behaviors.

Abstract

In this paper we apply the method of Lagrangian descriptors to explore the geometrical structures in phase space that govern the dynamics of dissipative systems. We demonstrate through many classical examples taken from the nonlinear dynamics literature that this tool can provide valuable information and insights to develop a more general and detailed understanding of the global behavior and underlying geometry of these systems. In order to achieve this goal, we analyze systems that display dynamical features such as hyperbolic points with different expansion and contraction rates, limit cycles, slow manifolds and strange attractors. Furthermore, we study how this technique can be used to detect transition ellipsoids that arise in Hamiltonian systems subject to dissipative forces, and which play a crucial role in characterizing trajectories that evolve across an index-1 saddle point of the underlying potential energy surface.