Symmetric Heteroclinic Connections in the Michelson System: A Computer Assisted Proof

Abstract

In this paper we present a new technique of proving the existence of an infinite number of symmet- ric heteroclinic and homoclinic solutions. This technique combines the covering relations method introduced by Zgliczynski (Topol. Methods Nonlinear Anal., 8 (1996), pp. 169-177; Nonlinearity ,1 0 (1997), pp. 243-252) with symmetry properties of a dynamical system. As an example we present a computer assisted proof of the existence of an infinite number of heteroclinic connections be- tween equilibrium points in the Kuramoto-Sivashinsky ODE (D. Michelson, Phys. D, 19 (1986), pp. 89-111). Moreover, we present the proof of the existence of an infinite number of heteroclinic connections between periodic orbits and equilibrium points. 1. Introduction. The aim of this paper is to present a new method for proving the ex- istence of symmetric homoclinic or heteroclinic solutions in systems possessing the reversing symmetry property. In (5) (and references given there) a method of proving the existence of time reversing symmetric homoclinic and heteroclinic solutions for dynamical systems is presented and it is called the fixed set iteration method. It applies to dynamical systems with continuous and discrete time. The basic idea of such a method is to search for the points u which are invariant under the symmetry and whose trajectories converge to an equilibrium point or a periodic orbit. This allows us to conclude that the trajectories of the points u must be homoclinic or heteroclinic. Galias and Zgliczynski (3) presented the method for proving the existence of homoclinic and heteroclinic solutions for maps R 2 → R 2 . This result was applied to the planar circular restricted three body problem (1, 16), where the existence of an infinite number of homoclinic and heteroclinic connections between periodic orbits was shown. In this paper we demonstrate how to combine these two methods for proving the existence of symmetric homoclinic or heteroclinic orbits in systems possessing the reversing symmetry property. Moreover, we present some generalization of the Galias-Zgliczynski method. We show how to prove the existence of heteroclinic orbits between objects possessing unequal dimensions—for example, the equilibrium points and periodic orbits.