Dynamical Systems
Liénard systems and potential-Hamiltonian decomposition II – algorithm
Comptes Rendus. Mathématique, Volume 344 (2007) no. 3, pp. 191-194.

We show here how to approach with an increasing precision the limit-cycles of Liénard systems, bifurcating from a stable stationary state, by contour lines of Hamiltonian systems derived from a potential-Hamiltonian decomposition of the Liénard flow. We evoke the case (non polynomial) of pure potential systems (n-switches) and pure Hamiltonian systems (2D Lotka–Volterra), and we show that, with the proposed approximation, we can deal with the case of mixed systems (van der Pol or FitzHugh-Nagumo) frequently used for modelling oscillatory systems in biology. We suggest finally that the proposed algorithm, generic for PH-decomposition, can be used for estimating the isochronal fibration in some specific cases near the pure potential or Hamiltonian systems. In a following Note, we will give applications in biology of the potential-Hamiltonian decomposition.

Nous montrons ici comment approcher, avec une précision croissante, les cycles limites des systèmes de Liénard par les courbes de niveau du système hamiltonien obtenu à partir d'une décomposition potentielle-hamiltonienne. Nous présentons des cas non polynomiaux de systèmes purement potentiels (type n-switch) et purement hamiltoniens (type Lotka–Volterra 2D), puis nous donnons des exemples de décomposition potentielle-hamiltonienne pour des systèmes de type van der Pol (mixtes) qui sont d'usage fréquent en modélisation des systèmes biologiques oscillants. Nous suggérons enfin que l'algorithme proposé, générique pour la décomposition potentielle-hamiltonienne, soit utilisé pour estimer la fibration isochrone, dans le cas de systèmes voisins des systèmes purs (potentiels ou hamiltoniens). Dans une Note future, nous décrirons des applications biologiques précises de la décomposition potentielle-hamiltonienne.

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DOI: 10.1016/j.crma.2006.10.013
Demongeot, Jacques 1, 2; Glade, Nicolas 2; Forest, Loic 2

1 Institut Universitaire de France, France
2 TIMC-IMAG UMR CNRS 5525, Faculty of Medicine, University J. Fourier, Grenoble, 38700 La Tronche, France
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Demongeot, Jacques; Glade, Nicolas; Forest, Loic. Liénard systems and potential-Hamiltonian decomposition II – algorithm. Comptes Rendus. Mathématique, Volume 344 (2007) no. 3, pp. 191-194. doi : 10.1016/j.crma.2006.10.013. http://www.numdam.org/articles/10.1016/j.crma.2006.10.013/

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