Liénard systems and potential-Hamiltonian decomposition II - Algorithm

Jacques Demongeot (1) , Nicolas Glade (1) , Loïc Forest (2)

1 TIMB
2 TIMC-IMAG - Techniques de l'Ingénierie Médicale et de la Complexité - Informatique, Mathématiques et Applications, Grenoble - UMR 5525

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 PHdecomposition, 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.

Mots clés: Liénard systems; Hodge decomposition; Potential-Hamiltonian decomposition

Domaines: Equations aux dérivées partielles [math.AP]; Systèmes dynamiques [math.DS]

Jacques Demongeot, Nicolas Glade, Loïc Forest. Liénard systems and potential-Hamiltonian decomposition II - Algorithm. Comptes rendus de l'Académie des sciences. Série I, Mathématique, 2007, 344, pp 191-194. ⟨hal-00211817⟩