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

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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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  1. Institute of Applied Mathematics and Mechanics, Donetsk, 83114, Ukraine

    A. O. Ignatyev

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  1. A. O. Ignatyev
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Correspondence to A. O. Ignatyev.

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Russian Text © The Author(s), 2020, published in Matematicheskie Zametki, 2020, Vol. 107, No. 3, pp. 391-399.

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Ignatyev, A.O. On the Existence of Homoclinic Orbits in Nonautonomous Second-Order Differential Equations. Math Notes 107, 435–441 (2020). https://doi.org/10.1134/S0001434620030074

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  • DOI: https://doi.org/10.1134/S0001434620030074

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