Painting the phase space of dissipative systems with Lagrangian descriptors

• Published in: Communications in nonlinear science & numerical simulation Vol. 104; p. 106034
• Main Authors: García-Garrido, Víctor J., García-Luengo, Julia
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.01.2022; Elsevier Science Ltd
• Subjects: Attractors; Chaos theory; Dissipative systems; Dynamical systems; Ellipsoids; Energy dissipation; Hamiltonian functions; Lagrangian descriptors; Limit cycles; Nonlinear dynamics; Nonlinear systems; Numerical analysis; Phase space; Phase transitions; Potential energy; Saddle points; Slow manifolds; Strange attractors; Transition ellipsoids; Slow manifolds; Transition ellipsoids; Lagrangian descriptors; Attractors; Limit cycles; Dissipative systems; Phase space
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2021.106034

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 mathematical technique 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. •Phase space structure analysis of dissipative systems with Lagrangian descriptors.•Detection of limit cycles and slow manifolds.•Reconstruction of strange attractors and identification of basin boundaries.•Identification of transition ellipsoids in Hamiltonian systems with dissipation.