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

or the second-order differential equation ẍ + f(t) ẋ + 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), ẋ (t) satisfying the conditions limt_→±∞x(t) = 0 and limt_→±∞ẋ (t) = 0. The specific case in which all the solutions of this differential equation are homoclinic is considered. [ABSTRACT FROM AUTHOR]