Even homoclinic orbits for a non periodic dynamical systems

Abstract

In this paper, we study the existence of even homoclinic solutions
for a nonperiodic dynamical system
$$ \ddot{x}(t)+A\dot{x}(t)+V'(t,x(t))=0,$$ where A is a skew-symmetric constant matrix, t $\in\mathbb{R}$, $x\in\mathbb{R}^N$ and V $\in C^1(\mathbb{R}\times\mathbb{R}^N,\mathbb{R})$,
$V(t,x)=-K(t,x)+W(t,x)$. We assume that $W(t,x)$ does not satisfy the global Ambrosetti-Rabinowitz condition and that the norm of A is sufficiently small.