Existence of Periodic Solutions and Bounded Invariant Sets for Non-Autonomous Third-Order Differential Equantion

O. Rabiei Motlagh and Z. Afsharnezhad


    In this paper we consider the third-order non-autonomous differential equation
$\dot\ddot{x}+\omega^2\dot{x}= \Muy F(x,\dot{x},ddot{x},t,\epsilon)                 (1)
where the C^r (r > 2) map F is periodic in t and $\muy, \epsilon$  are real parameters. We give some conditions for Eq. (1) in order to reduce it to a second-order differential equation. Then we show that the periodic (homoclinic ) solutions of the second-order equation imply the periodic (homoclinic) solutions of Eq. (1). Then we use the Hopf bifurcation theorem for the second-order equation and obtain periodic solutions and non-trivial bounded invariant sets for it. The effect of the quadratic terms on Eq.  (1) is also studied.