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The Application of Lagrangian Descriptors to 3D Vector Fields

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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.

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Authors and Affiliations

  1. Departamento de Física y Matemáticas, Universidad de Alcalá, 28871, Alcalá de Henares, Spain

    Víctor J. García-Garrido

  2. Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM C/Nicolás Cabrera 15, Campus Cantoblanco UAM, 28049, Madrid, Spain

    Víctor J. García-Garrido, Jezabel Curbelo & Ana M. Mancho

  3. Departamento de Matemáticas, Facultad de Ciencias Universidad Autónoma de Madrid, 28049, Madrid, Spain

    Jezabel Curbelo

  4. School of Mathematics, University of Bristol, Bristol, BS8 1TW, UK

    Stephen Wiggins

  5. Department of Atmospheric and Oceanic Sciences, University of California at Los Angeles, Los Angeles, CA, USA

    Carlos R. Mechoso

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  1. Víctor J. García-Garrido
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  2. Jezabel Curbelo
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  3. Ana M. Mancho
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  4. Stephen Wiggins
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Correspondence to Víctor J. García-Garrido.

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García-Garrido, V.J., Curbelo, J., Mancho, A.M. et al. The Application of Lagrangian Descriptors to 3D Vector Fields. Regul. Chaot. Dyn. 23, 551–568 (2018). https://doi.org/10.1134/S1560354718050052

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