The Hörmander index of symmetric periodic orbits

Relative Periodic Orbits of Symmetric Lagrangian Systems

Numerical Continuation of Symmetric Periodic Orbits

The bifurcation theory and numerics of periodic orbits of general dynamical systems is well developed, and in recent years there has been rapid progress in the development of a bifurcation theory for symmetric dynamical systems. But there are hardly any results on the numerical computation of those bifurcations yet. In this paper we show how spatiotemporal symmetries of periodic orbits can be e...

Symmetric periodic orbits near a heteroclinic loop

In this paper we consider vector fields in R that are invariant under a suitable symmetry and that posses a “generalized heteroclinic loop” L formed by two singular points (e and e−) and their invariant manifolds: one of dimension 2 (a sphere minus the points e and e−) and one of dimension 1 (the open diameter of the sphere having endpoints e and e−). In particular, we analyze the dynamics of t...

Symmetric Homoclinic Orbits at the Periodic Hamiltonian Hopf Bifurcation

We prove the existence of symmetric homoclinic orbits to a saddle-focus symmetric periodic orbit that appears in a generic family of reversible three degrees of freedom Hamiltonian system due to periodic Hamiltonian Hopf bifurcation, if some coefficient A of the normal form of the fourth order is positive. If this coefficient is negative, then for the opposite side of the bifurcation parameter ...

Action and Index Spectra and Periodic Orbits in Hamiltonian Dynamics

The main theme of this paper is the connection between the existence of infinitely many periodic orbits for a Hamiltonian system and the behavior of its action or index spectrum under iterations. We use the action and index spectra to show that any Hamiltonian diffeomorphism of a closed, rational manifold with zero first Chern class has infinitely many periodic orbits and that, for a general ra...