(Georgia Institute of Technology, 1999)
Chow, Shui-Nee; Li, Yong; Yi, Yingfei
Generalizing the degenerate KAM theorem under the Rüssmann non-degeneracy
and the isoenergetic KAM theorem, we employ a quasi-linear iterative scheme to study
the persistence and frequency preservation of invariant tori on a smooth sub-manifold
for a real analytic, nearly integrable Hamiltonian system. Under a nondegenerate
condition of Rüssmann type on the sub-manifold, we shall show the following: a) the majority of the unperturbed tori on the sub-manifold will persist; b) the perturbed toral frequencies can be partially preserved according to the maximal degeneracy of the Hessian of the unperturbed system and be fully preserved if the Hessian is nondegenerate; c) the Hamiltonian admits normal forms near the perturbed tori of arbitrarily prescribed high order. Under a sub-isoenergetic nondegenerate condition on an energy surface, we shall show that the majority of unperturbed tori give rise to invariant tori of the perturbed system of the same energy which preserve the ratio of certain components of the respective frequencies.
(Georgia Institute of Technology, 1999)
Chow, Shui-Nee; Liu, Weishi; Yi, Yingfei
We derive a general center manifolds theory for a class of compact invariant sets of flows generated by a smooth vector fields in R^n. By applying the Hadamard graph transform technique, it is shown that, associated to certain dynamical characteristics of the linearized flow along the invariant set, there exists an invariant manifold (called a center manifold) of the invariant set which contains every locally bounded solution (in particular, contains the invariant set) and is persistent under small perturbations.
(Georgia Institute of Technology, 1997)
Chow, Shui-Nee; Liu, Weishi; Yi, Yingfei
We study dynamics of flows generated from smooth vector fields in R^n in the vicinity of an invariant and closed smooth manifold Y. By applying the Hadamard graph transform technique, we show that there exists an
invariant manifold (called a center manifold of Y) based on the information of
the linearization along Y, which contains every locally bounded solution and
is persistent under small perturbations.