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Regular and Chaotic Dynamics

, Volume 23, Issue 5, pp 551–568 | Cite as

The Application of Lagrangian Descriptors to 3D Vector Fields

  • Víctor J. García-Garrido
  • Jezabel Curbelo
  • Ana M. Mancho
  • Stephen Wiggins
  • Carlos R. Mechoso
Article
  • 34 Downloads

Abstract

Since the 1980s, the application of concepts and ideas from dynamical systems theory to analyze phase space structures has provided a fundamental framework to understand long-term evolution of trajectories in many physical systems. In this context, for the study of fluid transport and mixing the development of Lagrangian techniques that can capture the complex and rich dynamics of time-dependent flows has been crucial. Many of these applications have been to atmospheric and oceanic flows in two-dimensional (2D) relevant scenarios. However, the geometrical structures that constitute the phase space structures in time-dependent three-dimensional (3D) flows require further exploration. In this paper we explore the capability of Lagrangian descriptors (LDs), a tool that has been successfully applied to time-dependent 2D vector fields, to reveal phase space geometrical structures in 3D vector fields. In particular, we show how LDs can be used to reveal phase space structures that govern and mediate phase space transport. We especially highlight the identification of normally hyperbolic invariant manifolds (NHIMs) and tori. We do this by applying this methodology to three specific dynamical systems: a 3D extension of the classical linear saddle system, a 3D extension of the classical Duffing system, and a geophysical fluid dynamics f-plane approximation model which is described by analytical wave solutions of the 3D Euler equations. We show that LDs successfully identify and recover the template of invariant manifolds that define the dynamics in phase space for these examples.

Keywords

Lagrangian descriptors phase space structure invariant manifolds invariant tori ergodic decomposition 

MSC2010 numbers

37XX 37D10 37N10 37Mxx 70K43 

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Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  • Víctor J. García-Garrido
    • 1
    • 2
  • Jezabel Curbelo
    • 2
    • 3
  • Ana M. Mancho
    • 2
  • Stephen Wiggins
    • 4
  • Carlos R. Mechoso
    • 5
  1. 1.Departamento de Física y MatemáticasUniversidad de AlcaláAlcalá de HenaresSpain
  2. 2.Instituto de Ciencias MatemáticasCSIC-UAM-UC3M-UCM C/Nicolás Cabrera 15, Campus Cantoblanco UAMMadridSpain
  3. 3.Departamento de MatemáticasFacultad de Ciencias Universidad Autónoma de MadridMadridSpain
  4. 4.School of MathematicsUniversity of BristolBristolUK
  5. 5.Department of Atmospheric and Oceanic SciencesUniversity of California at Los AngelesLos AngelesUSA

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