Global computation of phase-amplitude reduction for limit-cycle dynamics

Alexandre Mauroy, Igor Mezić

Research output: Contribution to journalArticle

Abstract

Recent years have witnessed increasing interest to phase-amplitude reduction of limit-cycle dynamics. Adding an amplitude coordinate to the phase coordinate allows to take into account the dynamics transversal to the limit cycle and thereby overcomes the main limitations of classic phase reduction (strong convergence to the limit cycle and weak inputs). While previous studies mostly focus on local quantities such as infinitesimal responses, a major and limiting challenge of phase-amplitude reduction is to compute amplitude coordinates globally, in the basin of attraction of the limit cycle.
In this paper, we propose a method to compute the full set of phase-amplitude coordinates in the large. Our method is based on the so-called Koopman (composition) operator and aims at computing the eigenfunctions of the operator through Laplace averages (in combination with the harmonic balance method). This yields a forward integration method that is not limited to two-dimensional systems. We illustrate the method by computing the so-called isostables of limit cycles in two and three-dimensional state spaces, as well as their responses to strong external inputs.
LanguageEnglish
Number of pages23
JournalChaos: an interdisciplinary journal of nonlinear science
Volume28
Issue number7
StatePublished - Jul 2018

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Limit Cycle
cycles
Eigenvalues and eigenfunctions
Laplace transformation
Harmonic Balance
Computing
Basin of Attraction
Composition Operator
Two-dimensional Systems
Laplace
Strong Convergence
attraction
Eigenfunctions
Chemical analysis
State Space
eigenvectors
Limiting
harmonics
operators
Three-dimensional

Cite this

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abstract = "Recent years have witnessed increasing interest to phase-amplitude reduction of limit-cycle dynamics. Adding an amplitude coordinate to the phase coordinate allows to take into account the dynamics transversal to the limit cycle and thereby overcomes the main limitations of classic phase reduction (strong convergence to the limit cycle and weak inputs). While previous studies mostly focus on local quantities such as infinitesimal responses, a major and limiting challenge of phase-amplitude reduction is to compute amplitude coordinates globally, in the basin of attraction of the limit cycle.In this paper, we propose a method to compute the full set of phase-amplitude coordinates in the large. Our method is based on the so-called Koopman (composition) operator and aims at computing the eigenfunctions of the operator through Laplace averages (in combination with the harmonic balance method). This yields a forward integration method that is not limited to two-dimensional systems. We illustrate the method by computing the so-called isostables of limit cycles in two and three-dimensional state spaces, as well as their responses to strong external inputs.",
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Global computation of phase-amplitude reduction for limit-cycle dynamics. / Mauroy, Alexandre; Mezić, Igor.

In: Chaos: an interdisciplinary journal of nonlinear science, Vol. 28, No. 7, 07.2018.

Research output: Contribution to journalArticle

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AB - Recent years have witnessed increasing interest to phase-amplitude reduction of limit-cycle dynamics. Adding an amplitude coordinate to the phase coordinate allows to take into account the dynamics transversal to the limit cycle and thereby overcomes the main limitations of classic phase reduction (strong convergence to the limit cycle and weak inputs). While previous studies mostly focus on local quantities such as infinitesimal responses, a major and limiting challenge of phase-amplitude reduction is to compute amplitude coordinates globally, in the basin of attraction of the limit cycle.In this paper, we propose a method to compute the full set of phase-amplitude coordinates in the large. Our method is based on the so-called Koopman (composition) operator and aims at computing the eigenfunctions of the operator through Laplace averages (in combination with the harmonic balance method). This yields a forward integration method that is not limited to two-dimensional systems. We illustrate the method by computing the so-called isostables of limit cycles in two and three-dimensional state spaces, as well as their responses to strong external inputs.

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