Branching of periodic orbits in reversible Hamiltonian systems

Abstract

Abstract. This paper deals with the dynamics of time-reversible Hamiltonian vector fields with 2 and 3 degrees of freedom around an elliptic equilibrium point in presence of symplectic involutions. The main results discuss the existence of one-parameter families of reversible periodic solutions terminating at the equilibrium. The main techniques used are Birkhoff and Belitskii normal forms combined with the LiapunovSchmidt reduction.

Normal forms of $ω$-Hamiltonian vector fields with symmetries

“…There are several methods to understand the local qualitative behavior of a vector field and one of them is the normal form theory. Some authors, as in [12,16,17,21,22], use this theory to study limit cycles, families of periodic orbits, relative equilibria, relative periodic solutions, etc.…”

Normal form theory for reversible equivariant vector fields

“…Examples are equivariant vector fields, symplectic, volume preserving, reversible and combinations thereof. Many authors have used normal form theory in distinct contexts to study limit cycles, family of periodic orbits, relative equilibria and relative periodic solutions (see, for example [8,13,14,16,17]). The resulting normal forms in each of these contexts can then be expressed in terms of the group invariants.…”

Centers with equal period functions

“…In [1,11] the converse problem was considered, i. e. under which conditions a center is reversible in a generalized sense. In [4,12,13,14] reversible centers were considered in relation to bifurcation problems.…”