Existence of a Smooth Hamiltonian Circle Action near Parabolic Orbits and Cuspidal Tori

Bifurcations of Normally Parabolic Tori in Hamiltonian Systems

Existence of Diffusion Orbits in a priori Unstable Hamiltonian Systems

Under open and dense conditions we show that Arnold diffusion orbits exist in a priori unstable and time-periodic Hamiltonian systems with two degrees of freedom. 1, Introduction and Results By the KAM (Kolmogorov, Arnold and Moser) theory we know that there are many invariant tori in nearly integrable Hamiltonian systems with arbitrary n degrees of freedom. These tori are of n dimension and oc...

Hamiltonian Systems Near Relative Periodic Orbits

We give explicit differential equations for a symmetric Hamiltonian vector field near a relative periodic orbit. These decompose the dynamics into periodically forced motion in a Poincaré section transversal to the relative periodic orbit, which in turn forces motion along the group orbit. The structure of the differential equations inherited from the symplectic structure and symmetry propertie...

Parabolic resonances in near integrable Hamiltonian systems

When an integrable Hamiltonian system, possessing an m-resonant lower dimensional normally parabolic torus is perturbed, a parabolic m-resonance occurs. If, in addition, the isoenergetic nondegeneracy condition for the integrable system fails, the near integrable Hamiltonian exhibits a at parabolic m-resonance. It is established that most kinds of parabolic resonances are persistent in n (n 3) ...

On the Generic Existence of Periodic Orbits in Hamiltonian Dynamics

We prove several generic existence results for infinitely many periodic orbits of Hamiltonian diffeomorphisms or Reeb flows. For instance, we show that a Hamiltonian diffeomorphism of a complex projective space or Grassmannian generically has infinitely many periodic orbits. We also consider symplectomorphisms of the two-torus with irrational flux. We show that such a symplectomorphism necessar...