[Systèmes de Liénard et décomposition potentielle-hamiltonienne II – algorithme]
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.
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.
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@article{CRMATH_2007__344_3_191_0,
author = {Demongeot, Jacques and Glade, Nicolas and Forest, Loic},
title = {Li\'enard systems and {potential-Hamiltonian} decomposition {II} {\textendash} algorithm},
journal = {Comptes Rendus. Math\'ematique},
pages = {191--194},
publisher = {Elsevier},
volume = {344},
number = {3},
year = {2007},
doi = {10.1016/j.crma.2006.10.013},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2006.10.013/}
}
TY - JOUR AU - Demongeot, Jacques AU - Glade, Nicolas AU - Forest, Loic TI - Liénard systems and potential-Hamiltonian decomposition II – algorithm JO - Comptes Rendus. Mathématique PY - 2007 SP - 191 EP - 194 VL - 344 IS - 3 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2006.10.013/ DO - 10.1016/j.crma.2006.10.013 LA - en ID - CRMATH_2007__344_3_191_0 ER -
%0 Journal Article %A Demongeot, Jacques %A Glade, Nicolas %A Forest, Loic %T Liénard systems and potential-Hamiltonian decomposition II – algorithm %J Comptes Rendus. Mathématique %D 2007 %P 191-194 %V 344 %N 3 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2006.10.013/ %R 10.1016/j.crma.2006.10.013 %G en %F CRMATH_2007__344_3_191_0
Demongeot, Jacques; Glade, Nicolas; Forest, Loic. Liénard systems and potential-Hamiltonian decomposition II – algorithm. Comptes Rendus. Mathématique, Tome 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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