Abstract
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.
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