Decaying Oscillatory Perturbations of Hamiltonian Systems in the Plane

Smoothed Dynamics of Highly Oscillatory Hamiltonian Systems

We consider the numerical treatment of Hamiltonian systems that contain a potential which grows large when the system deviates from the equilibrium value of the potential. Such systems arise, e.g., in molecular dynamics simulations and the spatial discretization of Ha-miltonian partial diierential equations. Since the presence of highly oscillatory terms in the solutions forces any explicit int...

Dissipative Perturbations of 3d Hamiltonian Systems

In this article we present some results concerning natural dissipative perturbations of 3d Hamiltonian systems. Given a Hamiltonian system ẋ = PdH, and a Casimir function S, we construct a symmetric covariant tensor g, so that the modified (so-called “metriplectic”) system ẋ = PdH + gdS satisfies the following conditions: dH is a null vector for g, and dS(gdS) ≤ 0. Along solutions to a dynamica...

On Hamiltonian Perturbations of Hyperbolic Systems of Conservation Laws I: Quasi-Triviality of Bi-Hamiltonian Perturbations

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 bi-Hamiltonian perturbation can be eliminated in all orders of the perturbative expansion by a change of coordinates on the infinite jet space depending rationally on ...

Periodic Solutions of Pendulum-Like Hamiltonian Systems in the Plane

By the use of a generalized version of the Poincaré–Birkhoff fixed point theorem, we prove the existence of at least two periodic solutions for a class of Hamiltonian systems in the plane, having in mind the forced pendulum equation as a particular case. Our approach is closely related to the one used by Franks in [15], but the proof remains at a more elementary level. 2010 Mathematics Subject ...

Arnold diffusion in perturbations of analytic integrable Hamiltonian systems