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 i...

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Bibliographic Details
Published in:Mathematical Notes 2020-03, Vol.107 (3-4), p.435-441
Main Author: Ignatyev, A. O.
Format: Article
Language:English
Subjects:
Differential equations
Liapunov functions
Mathematics
Mathematics and Statistics
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Summary: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.
ISSN:0001-4346
1067-9073
1573-8876
DOI:10.1134/S0001434620030074