[Une méthode fréquentielle pour le calcul de stabilité des solutions périodiques des systèmes dynamiques]
Dans cette Note, nous présentons une méthode numérique fréquentielle pour déterminer la stabilité des solutions périodiques d'un système dynamique. La méthode, basée sur la théorie de Floquet et le développement en série de Fourier (méthode de Hill), consiste à extraire les valeurs propres physiques de l'ensemble des valeurs propres numériques du système perturbé étendu dans le domaine fréquentiel. En combinant alors la méthode de l'équilibrage harmonique et la méthode asymptotique numérique avec la précédente méthode de Hill, on obtient un outils de continuation purement fréquentiel où le calcul de la stabilité des solutions suivies est quasiment immédiat. Afin de valider la méthode, nous appliquons la méthode de continuation à un oscillateur de Duffing forcé.
In this Note, we present a harmonic-based numerical method to determine the local stability of periodic solutions of dynamical systems. Based on the Floquet theory and the Fourier series expansion (Hill method), we propose a simple strategy to sort the relevant physical eigenvalues among the expanded numerical spectrum of the linear periodic system governing the perturbed solution. By mixing the harmonic-balance method and asymptotic numerical method continuation technique with the developed Hill method, we obtain a purely-frequency based continuation tool able to compute the stability of the continued periodic solutions in a reduced computation time. To validate the general methodology, we investigate the dynamical behavior of the forced Duffing oscillator with the developed continuation technique.
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Mots-clés : Systèmes dynamiques, Stabilité, Méthode de Hill, Méthode de continuation, Méthode de l'équilibrage harmonique
Arnaud Lazarus 1 ; Olivier Thomas 1
@article{CRMECA_2010__338_9_510_0,
author = {Arnaud Lazarus and Olivier Thomas},
title = {A harmonic-based method for computing the stability of periodic solutions of dynamical systems},
journal = {Comptes Rendus. M\'ecanique},
pages = {510--517},
publisher = {Elsevier},
volume = {338},
number = {9},
year = {2010},
doi = {10.1016/j.crme.2010.07.020},
language = {en},
}
TY - JOUR AU - Arnaud Lazarus AU - Olivier Thomas TI - A harmonic-based method for computing the stability of periodic solutions of dynamical systems JO - Comptes Rendus. Mécanique PY - 2010 SP - 510 EP - 517 VL - 338 IS - 9 PB - Elsevier DO - 10.1016/j.crme.2010.07.020 LA - en ID - CRMECA_2010__338_9_510_0 ER -
Arnaud Lazarus; Olivier Thomas. A harmonic-based method for computing the stability of periodic solutions of dynamical systems. Comptes Rendus. Mécanique, Volume 338 (2010) no. 9, pp. 510-517. doi : 10.1016/j.crme.2010.07.020. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2010.07.020/
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