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Nonlinear Dynamics

, Volume 96, Issue 1, pp 685–702 | Cite as

Trajectory-free approximation of phase space structures using the trajectory divergence rate

  • Gary K. NaveJr.Email author
  • Peter J. Nolan
  • Shane D. Ross
Original Paper

Abstract

This paper introduces the trajectory divergence rate, a scalar field which locally gives the instantaneous attraction or repulsion of adjacent trajectories. This scalar field may be used to find highly attracting or repelling invariant manifolds, such as slow manifolds, to rapidly approximate hyperbolic Lagrangian coherent structures, or to provide the local stability of invariant manifolds. This work presents the derivation of the trajectory divergence rate and the related trajectory divergence ratio for two-dimensional systems, investigates their properties, shows their application to several example systems, and presents their extension to higher dimensions.

Keywords

Vector fields Phase space structure Computational geometry Normally hyperbolic invariant manifolds 

Notes

Acknowledgements

This work was supported by National Science Foundations Grants Division of Atmospheric and Geospace Sciences (Grant No. 1520825) Division of Civil, Mechanical and Manufacturing Innovation (Grant No. 1537349) and Division of Mathematical Sciences (Grant No. 1821145) and by the Biological Transport (BioTrans) Interdisciplinary Graduate Education Program at Virginia Tech. We thank Pierre Lermusiaux, P.J. Haley, and the MIT-MSEAS team for providing the MSEAS model data.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Engineering Mechanics ProgramVirginia TechBlacksburgUSA

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