### 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.

< 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.

#### Keywords

Differential equations; symmetric homoclinic orbits; rigorous numerical analysis

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