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).