Existence of periodic solutions and homoclinic orbits for third-order nonlinear differential equations

O. Rabiei Motlagh and Z. Afsharnezhad

International Journal of Mathematics and Mathematical Sciences, 2003, vol. 2003, 1-20

Abstract:

The existence of periodic solutions for the third-order differential equation x ¨ ˙ + ω 2 x ˙ = μ F ( x , x ˙ , x ¨ ) is studied. We give some conditions for this equation in order to reduce it to a second-order nonlinear differential equation. We show that the existence of periodic solutions for the second-order equation implies the existence of periodic solutions for the above equation. Then we use the Hopf bifurcation theorem for the second-order equation and obtain many periodic solutions for it. Also we show that the above equation has many homoclinic solutions if F ( x , x ˙ , x ¨ ) has a quadratic form. Finally, we compare our result to that of Mehri and Niksirat (2001).

Date: 2003
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:543981

DOI: 10.1155/S0161171203107089

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