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

Abstract

This paper introduces a new global dynamics indicator based on the method of Lagrangian Descriptor apt for discriminating ordered and deterministic chaotic motions in multidimensional systems. Its implementation 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 distinguishing different dynamical behaviors, several stability maps of classical systems obtained in the literature with different methods are reproduced. The benchmark examples include the standard map, a 4 dimensional symplectic map, the 2 degrees-offreedom Hénon-Heiles system and a 3 degrees-of-freedom nearly-integrable Hamiltonian with a dense web of resonances supporting diffusion. Contents 1. Introduction 1 2. Lagrangian Descriptors and the ∆LD indicator 3 2.1. Framework of Lagrangian Descriptors 3 2.2. Drivers of the LDs 4 2.3. Regularity of the LD application & the ∆LD indicator 6 3. Applications to flows 9 3.1. The Hénon-Heiles system 9 3.2. The Froeschlé-Guzzo-Lega Hamiltonian 9 4. Applications to mappings 12 4.1. The standard map 12 4.2. A 4-dimensional nearly-integrable mapping 13 5. Summary and conclusive remarks 15 Acknowledgments 16 Appendix A. Choice of the size of the window: temporal & geometrical LDs 16 References 18