Normally hyperbolic invariant manifolds (NHIMs) are well-known organizing centers of the dynamics in the phase space of a nonlinear system. Locating such manifolds in systems far from symmetric or integrable, however, has been an outstanding challenge. Here, we develop an automated detection method for codimension-one NHIMs in autonomous dynamical systems. Our method utilizes Stationary Lagrangian Coherent Structures (SLCSs), which are hypersurfaces satisfying one of the necessary conditions of a hyperbolic LCS, and are also quasi-invariant in a well-defined sense. Computing SLCSs provides a quick way to uncover NHIMs with high accuracy. As an illustration, we use SLCSs to locate two-dimensional stable and unstable manifolds of hyperbolic periodic orbits in the classic ABC flow, a three-dimensional solution of the steady Euler equations.
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December 2013
Research Article|
October 16 2013
Detecting invariant manifolds as stationary Lagrangian coherent structures in autonomous dynamical systems
Hiroshi Teramoto;
Hiroshi Teramoto
a)
1
Molecule & Life Nonlinear Sciences Laboratory, Research Institute for Electronic Science, Hokkaido University
, Kita 20 Nishi 10, Kita-ku, Sapporo 001-0020, Japan
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George Haller;
George Haller
2
Institute for Mechanical Systems
, ETH Zürich, CLA J.27, Tannenstrasse 3, 8092 Zürich, Switzerland
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Tamiki Komatsuzaki
Tamiki Komatsuzaki
1
Molecule & Life Nonlinear Sciences Laboratory, Research Institute for Electronic Science, Hokkaido University
, Kita 20 Nishi 10, Kita-ku, Sapporo 001-0020, Japan
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Chaos 23, 043107 (2013)
Article history
Received:
April 08 2013
Accepted:
September 23 2013
Citation
Hiroshi Teramoto, George Haller, Tamiki Komatsuzaki; Detecting invariant manifolds as stationary Lagrangian coherent structures in autonomous dynamical systems. Chaos 1 December 2013; 23 (4): 043107. https://doi.org/10.1063/1.4824314
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