Liénard systems and potential-Hamiltonian decomposition I - Methodology

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

Following the Hodge decomposition of regular vector fields we can decompose the second member of any Liénard system into 2 (non-unique) polynomials, the first corresponding to potential and the second to Hamiltonian dynamics. This polynomial Hodge decomposition is called potential-Hamiltonian, denoted PH-decomposition, and we give it for any polynomial differential system of dimension 2. We will give in a future Note an algorithm expliciting the PH-decomposition in the neighborhood of particular orbits, like a limit-cycle for Liénard systems, the method being applicable for any polynomial differential system of dimension 2.

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

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

Jacques Demongeot, Nicolas Glade, Loïc Forest. Liénard systems and potential-Hamiltonian decomposition I - Methodology. Comptes rendus de l'Académie des sciences. Série I, Mathématique, 2007, 344, pp 121-126. ⟨hal-00211816⟩