The existence of Shilnikov homoclinic orbits in the Michelson system: A computer assisted proof
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 (+/- c root 2, 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,root 2-, 0, 0) possessing a one-dimensional unstable manifold with the equilibrium (c root 2-, 0, 0) possessing a one-dimensional stable manifold. The method used in the proof can be applied to other reversible systems.
shilnikov homoclinic orbit computer assisted proof brouwer degree