International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 4, Pages 209-228
doi: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 O. Rabiei Motlagh and Z. Afsharnezhad. 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).