### Abstract

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.

Language | English |
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Number of pages | 23 |

Journal | Chaos: an interdisciplinary journal of nonlinear science |

State | Submitted - 2018 |

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**Global computation of phase-amplitude reduction for limit-cycle dynamics.** / Mauroy, Alexandre; Mezić, Igor.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Global computation of phase-amplitude reduction for limit-cycle dynamics

AU - Mauroy,Alexandre

AU - Mezić,Igor

PY - 2018

Y1 - 2018

N2 - 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.

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.

M3 - Article

JO - Chaos

T2 - Chaos

JF - Chaos

SN - 1054-1500

ER -