On the Existence of Homoclinic Orbits in Nonautonomous Second-Order Differential Equations

For the second-order differential equation $\ddot x+f(t)\dot x+g(t)x=0$, the method of Lyapunov functions is used to obtain sufficient conditions for the existence of homoclinic trajectories, i.e., solutions $x(t)$, $\dot x(t)$ satisfying the conditions $\lim_{t\to\pm\infty}x(t)=0$ and $\lim_{t\to\pm\infty}\dot x(t)=0$. The specific case in which all the solutions of this differential equation are homoclinic is considered.