Eur. Phys. J. Special Topics 223, 2705–2722 (2014)
https://doi.org/10.1140/epjst/e2014-02287-6

Review

Continuation of periodic orbits in symmetric Hamiltonian and conservative systems

¹ Departamento de Matemática Aplicada II e Instituto de Matemáticas de la Universidad de Sevilla, Camino de los Descubrimientos s/n, Sevilla 41092, Spain
² Departamento de Ciencias Físicas, Matemáticas y de la Computación, Universidad Cardenal Herrera, San Bartolomé 55, 46115 Alfara del Patriarca – Valencia, Spain

^a e-mail: jgv@us.es

Received: 11 August 2014
Revised: 17 October 2014
Published online: 10 December 2014

Abstract

We present and review results on the continuation and bifurcation of periodic solutions in conservative, reversible and Hamiltonian systems in the presence of symmetries. In particular we show how two-point boundary value problem continuation software can be used to compute families of periodic solutions of symmetric Hamiltonian systems. The technique is introduced with a very simple model example (the mathematical pendulum), justified with a theoretical continuation result and then applied to two non trivial examples: the non integrable spring pendulum and the continuation of the figure eight solution of the three body problem.