Comptes Rendus
Dynamical Systems
Liénard systems and potential-Hamiltonian decomposition III – applications
Comptes Rendus. Mathématique, Volume 344 (2007) no. 4, pp. 253-258.

In the two previous Notes, we described the mathematical aspects of the potential-Hamiltonian (PH) decomposition, in particular for n-switches and Liénard systems. In the present Note, we give some examples of biological regulatory systems susceptible to be decomposed. We show that they can be modeled in terms of 2D-ODE belonging to n-switches and Liénard systems families. Although simplified, these models can be decomposed in a set of equations combining a potential and a Hamiltonian part. We discuss about the advantage of such a PH-decomposition for understanding the mechanisms involved in their regulatory abilities. We suggest a generalized algorithm to deal with differential systems having a second part of rational fraction type (frequently used in metabolic systems). Finally, we comment what can be interpreted as a precise signification in biological systems from the dynamical behaviors of both the potential and Hamiltonian parts.

Dans les deux Notes précédentes, nous avons décrit la décomposition potentielle-Hamiltonienne pour des systèmes de type n-switch ou Liénard. Leurs équations sont bien adaptées à la modélisation des systèmes dynamiques en biologie. Nous donnons ici des exemples de systèmes de régulation biologique pouvant être écrits sous la forme d'équations de Liénard et également sous forme de systèmes n-switch. Nous discutons ensuite de l'intérêt de connaître les contributions potentielles et Hamiltoniennes de ces systèmes dans la compréhension de leurs mécanismes. Pour terminer, nous suggérons un algorithme prenant en compte des systèmes différentiels à second membre de type fraction rationnelle rencontrés dans les modèles métaboliques, pour lesquels les parties potentielle et Hamiltonienne ont des significations biologiques précises. On explique comment utiliser en pratique cette décomposition au voisinage de leurs attracteurs.

Published online:
DOI: 10.1016/j.crma.2006.11.014

Nicolas Glade 1; Loic Forest 1; Jacques Demongeot 1, 2

1 TIMC-IMAG UMR CNRS 5525, University J. Fourier Grenoble, Faculty of Medicine, 38700 La Tronche, France
2 Institut Universitaire de France
     author = {Nicolas Glade and Loic Forest and Jacques Demongeot},
     title = {Li\'enard systems and {potential-Hamiltonian} decomposition {III} {\textendash} applications},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {253--258},
     publisher = {Elsevier},
     volume = {344},
     number = {4},
     year = {2007},
     doi = {10.1016/j.crma.2006.11.014},
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PB  - Elsevier
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%A Loic Forest
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Nicolas Glade; Loic Forest; Jacques Demongeot. Liénard systems and potential-Hamiltonian decomposition III – applications. Comptes Rendus. Mathématique, Volume 344 (2007) no. 4, pp. 253-258. doi : 10.1016/j.crma.2006.11.014.

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