Some third-order ordinary differential equations

The paper deals with periodic orbits in three systems of ordinary differential equations. Two of the systems, the Falkner-Skan equations and the Nose equations, do not possess fixed points, and yet interesting dynamics can be found. Here, periodic orbits emerge in bifurcations from heteroclinic cycles, connecting fixed points at infinity. We present existence results for such periodic orbits and discuss their properties using careful asymptotic arguments. In the final part results about the Nose equations are used to explain the dynamics in a dissipative perturbation, related to a system of dynamo equations.