Existence of homoclinic solutions for a class of second order systems

In this paper we study the existence of nontrivial homoclinic solutions for a second order differential equation of the form q̈+Aq̇−L(t)q+Wq(t,q)=0. Here, we assume that AA is a antisymmetric constant matrix, L(t)L(t) is a continuous positive definite symmetric matrix valued function depending periodically on tt and assume that potential WW is asymptotically quadratic as |q|→0|q|→0 and |q|→∞|q|→∞.