The Existence of Shilnikov Homoclinic Orbits in the Michelson System: A Computer Assisted Proof

Abstract

In this paper we present a new topological tool which allows us to prove the existence of Shilnikov homoclinic or heteroclinic solutions. We present an application of this method to the Michelson system $y'''+y'+0.5y^2=c^2$ [16]. We prove that there exists a countable set of parameter values $c$ for which a pair of the Shilnikov homoclinic orbits to the equilibrium points $(\pm c\sqrt2,0,0)$ appear. This result was conjectured by Michelson [16]. We also show that there exists a countable set of parameter values for which there exists a heteroclinic orbit connecting the equilibrium $(-c\sqrt2,0,0)$ possessing a one-dimensional unstable manifold with the equilibrium $(c\sqrt2,0,0)$ possessing a one-dimensional stable manifold. The method used in the proof can be applied to other reversible systems. To verify the assumptions of the main topological theorem for the Michelson system, we use rigorous computations based on interval arithmetic.