Article (Scientific journals)
On the use of Fourier averages to compute the global isochrons of (quasi)periodic dynamics
Mauroy, Alexandre; Mezić, Igor
2012In Chaos, 22 (3)
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Keywords :
Isochrons; Limit cycle; Koopman operator; Quasi-periodic torus; Eigenfunctions
Abstract :
[en] The concept of isochrons is crucial for the analysis of asymptotically periodic systems. Roughly, isochrons are sets of points that partition the basin of attraction of a limit cycle according to the asymptotic behavior of the trajectories. The computation of global isochrons (in the whole basin of attraction) is however difficult, and the existing methods are inefficient in high-dimensional spaces. In this context, we present a novel (forward integration) algorithm for computing the global isochrons of high-dimensional dynamics, which is based on the notion of Fourier time averages evaluated along the trajectories. Such Fourier averages in fact produce eigenfunctions of the Koopman semigroup associated with the system, and isochrons are obtained as level sets of those eigenfunctions. The method is supported by theoretical results and validated by several examples of increasing complexity, including the 4-dimensional Hodgkin-Huxley model. In addition, the framework is naturally extended to the study of quasiperiodic systems and motivates the definition of generalized isochrons of the torus. This situation is illustrated in the case of two coupled Van der Pol oscillators. © 2012 American Institute of Physics.
Disciplines :
Mathematics
Author, co-author :
Mauroy, Alexandre ;  University of California Santa Barbara > Department of Mechanical Engineering
Mezić, Igor;  University of California Santa Barbara > Department of Mechanical Engineering
Language :
English
Title :
On the use of Fourier averages to compute the global isochrons of (quasi)periodic dynamics
Publication date :
2012
Journal title :
Chaos
ISSN :
1054-1500
eISSN :
1089-7682
Publisher :
American Institute of Physics, Melville, United States - New York
Volume :
22
Issue :
3
Peer reviewed :
Peer Reviewed verified by ORBi
Available on ORBi :
since 02 April 2014

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