Existence of periodic solutions and homoclinic orbits forthird‐order nonlinear differential equations

The existence of periodic solutions for the third‐order differential equation 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 has a quadratic form. Finally, we compare our result to that of Mehri and Niksirat (2001).