Lagrangian descriptors for two dimensional, area preserving, autonomous and nonautonomous maps

Carlos Lopesino, Francisco Balibrea, Stephen R Wiggins*, Ana M Mancho

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

22 Citations (Scopus)
335 Downloads (Pure)

Abstract

In this paper we generalize the method of Lagrangian descriptors to two dimensional, area preserving, autonomous and nonautonomous discrete time dynamical systems. We consider four generic model problems - a hyperbolic saddle point for a linear, area-preserving autonomous map, a hyperbolic saddle point for a nonlinear, area-preserving autonomous map, a hyperbolic saddle point for linear, area-preserving nonautonomous map, and a hyperbolic saddle point for nonlinear, area-preserving nonautonomous map. The discrete time setting allows us to evaluate the expression for the Lagrangian descriptors explicitly for a certain class of norms. This enables us to provide a rigorous setting for the notion that the "singular sets" of the Lagrangian descriptors correspond to the stable and unstable manifolds of hyperbolic invariant sets, as well as to understand how this depends upon the particular norms that are used. Finally we analyze, from the computational point of view, the performance of this tool for general nonlinear maps, by computing the "chaotic saddle" for autonomous and nonautonomous versions of the Hénon map.

Original languageEnglish
Pages (from-to)40-51
Number of pages12
JournalCommunications in Nonlinear Science and Numerical Simulation
Volume27
Issue number1-3
Early online date6 Mar 2015
DOIs
Publication statusPublished - 1 Oct 2015

Bibliographical note

Date of Acceptance: 02/03/2015

Keywords

  • Autonomous map
  • Chaotic saddle
  • Lagrangian descriptor
  • Nonautonomous map

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