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International Journal of Mathematics and Mathematical Sciences
Volume 2003, Issue 4, Pages 209-228
http://dx.doi.org/10.1155/S0161171203107089

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

Department of Mathematics, Faculty of Mathematical Sciences, Ferdowsi University, Mashhad, Iran

Received 23 July 2001; Revised 18 February 2002

Copyright © 2003 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The existence of periodic solutions for the third-order differential equation x¨˙+ω2x˙=μF(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).