Periodic orbits in Hamiltonian systems with involutory symmetries

TL;DR

This paper investigates the existence and structure of periodic solutions near symmetric equilibria in Hamiltonian systems with involutory symmetries, revealing conditions for symmetric and non-symmetric periodic families.

Contribution

It provides new results on the existence and number of periodic solutions in Hamiltonian systems with involutory symmetries, including symmetric and non-symmetric families.

Findings

Existence of a two-parameter family of symmetric periodic solutions with symmetry R.

Presence of 2 to 12 non-symmetric periodic solution families near the equilibrium.

Systems with both symmetries exhibit similar periodic solution structures.

Abstract

We study the existence of families of periodic solutions in a neighbourhood of a symmetric equilibrium point in two classes of Hamiltonian systems with involutory symmetries. In both classes, involutions reverse the sign of the Hamiltonian function. In the first class we study a Hamiltonian system with a reversing involution R acting symplectically. We first recover a result of Buzzi and Lamb showing that the equilibrium point is contained in a three dimensional conical subspace which consists of a two parameter family of periodic solutions with symmetry R and there may or may not exist two families of non-symmetric periodic solutions, depending on the coefficients of the Hamiltonian. In the second problem we study an equivariant Hamiltonian system with a symmetry S that acts anti-symplectically. Generically, there is no S-symmetric solution in a neighbourhood of the equilibrium point. Moreover, we prove the existence of at least 2 and at most 12 families of non-symmetric periodic solutions. We conclude with a brief study of systems with both forms of symmetry, showing they have very similar structure to the system with symmetry R.