Bounding Periodic Orbits in Second Order Systems

Abstract

This paper provides an upper bound for the number of periodic orbits in planar systems. The research results in, [7], and, [8], allows one to produce a bound on the number of periodic orbits/limit cycles. Introducing the concept of Maximal Grade and Maximal Number of Periodic Orbits, a simple algebraic calculation leads to an upper bound on the number of periodic trajectories for general second order systems. In particular, it also applies to polynomial ODE’s. As far as the author is aware, such a powerful result is not available in the literature. Instead, the methods in this paper provide a tool to determine an upper bound on the periodic orbits/limit cycles for a wide range of dynamical systems.