On the determination of periodic solutions of autonomous ordinary differential equations using the homotopy method

This paper deals with the computation of periodic orbits of autonomous ordinary differential equations. A direct numerical method based on the determining equations, as proposed by Seydel (1981), is presented. After discretization, the boundary value problem is solved using an homotopy method. An important point is that the method does not require any initial bifurcation analysis. When the homotopy path does not lead to a solution, a minimization procedure via shooting method may be used to force convergence. The method can be used even when periodic solutions do not exist for the initial set of system parameters. In this case, the system parameters are also manipulated during the minimization procedure in order to provide an oscillatory response. The technique is applied in three different problems and it is shown that very few iterations are usually needed to fmd the periodic solution, no matter whether it is stable or unstable.