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.
S. Wiggins acknowledges the support of ONR Grant No. N00014-01-1-0769 and EPSRC Grant no. EP/P021123/1. A. M. Mancho acknowledges the support of ONR grant N00014-17-1-3003. V. J. García-Garrido, J.Curbelo and A.M.Mancho thankfully acknowledge the computer resources provided by ICMAT. C.R. Mechoso was supported by the U.S. NSF grant AGS-1245069.
Citation:
Víctor J. García-Garrido, Jezabel Curbelo, Ana M. Mancho, Stephen Wiggins, Carlos R. Mechoso, “The Application of Lagrangian Descriptors to 3D Vector Fields”, Regul. Chaotic Dyn., 23:5 (2018), 551–568
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\paper The Application of Lagrangian Descriptors to 3D Vector Fields
\jour Regul. Chaotic Dyn.
\yr 2018
\vol 23
\issue 5
\pages 551--568
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\crossref{https://doi.org/10.1134/S1560354718050052}
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Linking options:
https://www.mathnet.ru/eng/rcd344
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This publication is cited in the following 18 articles:
Jezabel Curbelo, “Lagrangian descriptors in geophysical flows: a survey”, SeMA, 2025
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Matthaios Katsanikas, Stephen Wiggins, Springer Proceedings in Complexity, Chaos, Fractals and Complexity, 2023, 47
Jezabel Curbelo, Irina I. Rypina, “A Three Dimensional Lagrangian Analysis of the Smoke Plume From the 2019/2020 Australian Wildfire Event”, JGR Atmospheres, 128:21 (2023)
Víctor J. García-Garrido, Stephen Wiggins, “Lagrangian descriptors and the action integral of classical mechanics”, Physica D: Nonlinear Phenomena, 434 (2022), 133206
Matthaios Katsanikas, Makrina Agaoglou, Stephen Wiggins, Ana M. Mancho, “Phase Space Transport in a Symmetric Caldera Potential with Three Index-1 Saddles and No Minima”, Int. J. Bifurcation Chaos, 32:10 (2022)
M. Hillebrand, S. Zimper, A. Ngapasare, M. Katsanikas, S. Wiggins, Ch. Skokos, “Quantifying chaos using Lagrangian descriptors”, Chaos: An Interdisciplinary Journal of Nonlinear Science, 32:12 (2022)
J. Montes, F. Revuelta, F. Borondo, “Lagrangian descriptors and regular motion”, Commun. Nonlinear Sci. Numer. Simul., 102 (2021), 105860
Broncio Aguilar-Sanjuan, Víctor García-Garrido, Vladimír Krajňák, Shibabrat Naik, Stephen Wiggins, “LDDS: Python package for computing and visualizing Lagrangian Descriptors for Dynamical Systems”, JOSS, 6:65 (2021), 3482
C. Niang, A. Maria Mancho, V. Jose Garcia-Garrido, E. Mohino, B. Rodriguez-Fonseca, J. Curbelo, “Transport pathways across the West African Monsoon as revealed by Lagrangian coherent structures”, Sci Rep, 10:1 (2020), 12543
V. J. Garcia-Garrido, Sh. Naik, S. Wiggins, “Tilting and squeezing: phase space geometry of Hamiltonian saddle-node bifurcation and its influence on chemical reaction dynamics”, Int. J. Bifurcation Chaos, 30:4 (2020), 2030008
Sh. Naik, V. J. Garcia-Garrido, S. Wiggins, “Finding NHIM: identifying high dimensional phase space structures in reaction dynamics using Lagrangian descriptors”, Commun. Nonlinear Sci. Numer. Simul., 79 (2019), UNSP 104907
Sh. Naik, S. Wiggins, “Finding normally hyperbolic invariant manifolds in two and three degrees of freedom with Hénon–Heiles-type potential”, Phys. Rev. E, 100:2 (2019), 022204
J. Curbelo, C. R. Mechoso, A. M. Mancho, S. Wiggins, “Lagrangian study of the final warming in the southern stratosphere during 2002: Part II. 3D structure”, Clim. Dyn., 53:3-4 (2019), 1277–1286
F. Balibrea-Iniesta, J. Xie, V. J. Garcia-Garrido, L. Bertino, A. M. Mancho, S. Wiggins, “Lagrangian transport across the upper arctic waters in the Canada basin”, Q. J. R. Meteorol. Soc., 145:718, A (2019), 76–91
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