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See detailOn the use of Fourier averages to compute the global isochrons of (quasi)periodic dynamics
Mauroy, Alexandre ULg; Mezić, Igor

in Chaos (2012), 22(3),

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 ... [more ▼]

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. [less ▲]

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See detailFaraday instability on a network
Delon, Giles ULg; Terwagne, Denis ULg; Adami, Nicolas ULg et al

in Chaos (2010)

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See detailThe mayonnaise droplet
Terwagne, Denis ULg; Mack, Nicolas; Dorbolo, Stéphane ULg et al

in Chaos (2009)

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See detailFrom bouncing to boxing
Terwagne, Denis ULg; Gilet, Tristan ULg; Vandewalle, Nicolas ULg et al

in Chaos (2008), 18(4),

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See detailClustering behaviors in networks of integrate-and-fire oscillators
Mauroy, Alexandre ULg; Sepulchre, Rodolphe ULg

in Chaos (2008), 18

Clustering behavior is studied in a model of integrate-and-fire oscillators with excitatory pulse coupling. When considering a population of identical oscillators, the main result is a proof of global ... [more ▼]

Clustering behavior is studied in a model of integrate-and-fire oscillators with excitatory pulse coupling. When considering a population of identical oscillators, the main result is a proof of global convergence to a phase-locked clustered behavior. The robustness of this clustering behavior is then investigated in a population of nonidentical oscillators by studying the transition from total clustering to the absence of clustering as the group coherence decreases. A robust intermediate situation of partial clustering, characterized by few oscillators traveling among nearly phase-locked clusters, is of particular interest. The analysis complements earlier studies of synchronization in a closely related model. [less ▲]

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