Homoclinic solutions for a forced generalized Liénard system

Abstract

In this article, we find a special class of homoclinic solutions which tend to 0 as t→ ± ∞, for a forced generalized Liénard system ẍ + f1(x)ẋ + f2(x)ẋ2 + g(x) = p(t). Since it is not a small perturbation of a Hamiltonian system, we cannot employ the well-known Melnikov method to determine the existence of homoclinic solutions. We use a sequence of periodically forced systems to approximate the considered system, and find their periodic solutions. We prove that the sequence of those periodic solutions has an accumulation which gives a homoclinic solution of the forced Liénard type system. © 2012 Zhang.