Non-linear oscillations of a third-order differential equation

An investigation is conducted into the behavior of the solutions of a third-order non-linear differential equation which is characterized by a non-linearity depending solely upon the Euclidean norm of the associated phase space. The non-linearity represents a central restoring force, which has important applications in modern control theory. For small non-linearities, the existence of a limit cycle is established by a fixed point technique, the approach to the limit cycle is approximated by averaging methods, and the periodic solution is harmonically represented by perturbation. Computer solutions of the differential equation are provided in order to reinforce the analysis. Some related differential equations are discussed including one in which the periodic solution is explicitly prescribed.