Symmetric homoclinic solutions to the periodic orbits in the Michelson system

Abstract

The Michelson system [D. Michelson, < i> Steady solutions of the Kuramoto–Sivashinsky equation< /i> , Physica D < b> 19< /b> (1986), 89–111] $x'''+x'+0.5x^2=c^2$ for the parameter value $c=1$ is investigated. It was proven in \cite{8} that the system possesses two odd periodic solutions. We shall show that there exist infinitely many homoclinic and heteroclinic connections between them. Moreover, we shall show that the family of homoclinic solutions contains a countable set of odd homoclinic solutions.