Hyperbolic structures in hamiltonian systems

Persistence of Hyperbolic Tori in Hamiltonian Systems

We generalize the well-known result of Graff and Zehnder on the persistence of hyperbolic invariant tori in Hamiltonian systems by considering non-Floquet, frequency varying normal forms and allowing the degeneracy of the unperturbed frequencies. The preservation of part or full frequency components associated to the degree of non-degeneracy is considered. As applications, we consider the persi...

Generalized Hamiltonian Structures for Ermakov Systems

We construct Poisson structures for Ermakov systems, using the Er-makov invariant as the Hamiltonian. Two classes of Poisson structures are obtained, one of them degenerate, in which case we derive the Casimir functions. In some situations, the existence of Casimir functions can give rise to superintegrable Ermakov systems. Finally, we characterize the cases where linearization of the equations...

Pre-hamiltonian Structures for Integrable Nonlinear Systems

Pre-Hamiltonian matrix operators in total derivatives are considered; they are defined by the property that their images are subalgebras of the Lie algebra of evolutionary vector fields. This construction is close to the Lie algebroids over infinite jet spaces. We assign a class of these operators and the brackets induced in their pre-images to integrable KdV-type hierarchies of symmetry flows ...

On Hamiltonian perturbations of hyperbolic systems of conservation laws

We study the general structure of formal perturbative solutions to the Hamiltonian perturbations of spatially one-dimensional systems of hyperbolic PDEs vt + [φ(v)]x = 0. Under certain genericity assumptions it is proved that any bihamiltonian perturbation can be eliminated in all orders of the perturbative expansion by a change of coordinates on the infinite jet space depending rationally on t...

Remarks on the definition of hyperbolic tori of Hamiltonian systems