Estimating Regions of Existence of Unstable Periodic Orbits Using Computer-Based Techniques

Abstract

In this paper, we present a systematic way of proving the existence of periodic solutions to a dynamical system using numerical techniques. In particular, we consider an n-dimensional ordinary differential equation, periodic in time, and state necessary conditions to prove the existence of both stable and unstable periodic solutions with the use of ordinary differential equation solvers (implemented by computer) and a knowledge of their error bounds. The method used is based on a simple shooting method and is an alternative to the Kantorovich theorem. A particularly difficult problem is considered in which the Lipschitz constant is large along unstable solutions. This leads to a modified scheme incorporating a multipoint shooting method to prove the existence of periodic solutions. The multipoint shooting method is then applied to a model in epidemiology to estimate a region of existence of an unstable periodic solution.