Liénard systems and potential-Hamiltonian decomposition - Applications in biology

Loïc Forest (1) , Nicolas Glade (1) , Jacques Demongeot (1)

1 TIMB

In separated 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 ordinary differential equations belonging to n-switches and Liénard system 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 behaviours of both the potential and Hamiltonian parts.

Mots clés: Liénard system; Hodge decomposition; potential-Hamiltonian decomposition; dynamical systems

Domaines: Systèmes dynamiques [math.DS]; Adaptation et Systèmes auto-organisés [nlin.AO]; Autre [q-bio.OT]

Loïc Forest, Nicolas Glade, Jacques Demongeot. Liénard systems and potential-Hamiltonian decomposition - Applications in biology. Comptes Rendus Biologies, 2007, 330 (2), pp.97-106. ⟨10.1016/j.crvi.2006.12.001⟩. ⟨hal-00346975⟩