Computation of minimal periods for ordinary differential equations

Abstract: We consider the problem of finding the shortest possible period for an exactly periodic solution to some given autonomous ordinary differential equation. We show that, given a pair of Lyapunov-like observable functions defined over the state space of the corresponding dynamical system and satisfying a certain pointwise inequality, we can obtain a global lower bound for such periods. We give a method valid for the case of bounding the period of only those solutions which are invariant under a symmetry transformation, as well as bounds for general periodic orbits. If the governing equations are polynomial in the state variables, we can use semidefinite programming to find such auxiliary functions computationally, and thus compute lower bounds which can be rigorously validated using rational arithmetic. We apply our method to the Lorenz and Henon-Heiles systems. For both systems we are able to give validated bounds which are sharp to several decimal places.