A theoretical framework for Lagrangian descriptors

Carlos Lopesino, Francisco Balibrea, Victor Garcia-Garrido, Stephen Wiggins, Ana M Mancho

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

29 Citations (Scopus)
239 Downloads (Pure)

Abstract

This paper provides a theoretical background for Lagrangian Descriptors. The goal of achieving rigourous proofs that justify the ability of LDs to detect invariant manifolds is simplified by introducing an alternative definition for LDs. The definition is stated for n-dimensional systems with general time dependence, however we rigorously prove that this method reveals the stable and unstable manifolds of hyperbolic points in four particular cases: a hyperbolic saddle point for linear autonomous systems, a hyperbolic saddle point for nonlinear autonomous systems, a hyperbolic saddle point for linear nonautonomous systems and a hyperbolic saddle point for nonlinear nonautonomous systems. We also discuss further rigorous results which show the ability of LDs to highlight additional invariants sets, such as n-tori. These results are just a simple extension of the ergodic partition theory which we illustrate by applying this methodology to well-known examples, such as the planar field of the harmonic oscillator and the 3D ABC flow. Finally, we
provide a thorough discussion on the requirement of the objectivity (frame-invariance) property for tools designed to reveal phase space structures and their implications for Lagrangian descriptors.
Original languageEnglish
Number of pages25
JournalInternational Journal of Bifurcation and Chaos
Volume27
Issue number1
Early online date1 Jan 2017
DOIs
Publication statusPublished - Jan 2017

Keywords

  • Langrangian descriptors
  • Hyperbolic trajectories
  • stable and unstable manifolds
  • n-tori
  • Invariant sets

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