On the existence of periodic solution of perturbed generalized Li\'enard equations

Islam Boussaada (1) , A. Raouf Chouikha (2)

1 LMRS - Laboratoire de Mathématiques Raphaël Salem
2 LAGA - Laboratoire Analyse, Géométrie et Applications

Under conditions of Levinson-Smith type, we prove the existence of a $\tau$-periodic solution for the perturbed generalized Li\'enard equation u''+\phi(u,u')u'+\psi(u)=\epsilon\omega(\frac{t}{\tau},u,u') with periodic forcing term. Also we deduce sufficient condition for existence of a periodic solution for the equation u''+\sum_{k=0}^{2s+1} p_k(u){u'}^k=\epsilon\omega(\frac{t}{\tau},u,u'). Our method can be applied also to the equation u''+[u^2+(u+u')^2-1]u'+u=\epsilon\omega(\frac{t}{\tau},u,u'). The results obtained are illustrated with numerical examples.

Domaines: Automatique / Robotique; Systèmes dynamiques [math.DS]

Islam Boussaada, A. Raouf Chouikha. On the existence of periodic solution of perturbed generalized Li\'enard equations. Electronic Journal of Differential Equations, 2006, http://ejde.math.txstate.edu/Volumes/2006/140/abstr.html. ⟨hal-01123648⟩