Dynamical problems and phase transitions

Issued as Financial status report, Technical reports [nos. 1-12], and Final report, Project B-06-68

Persistence of Invariant Tori on Submanifolds in Hamiltonian Systems

AMS (MOS). Mathematics Subject Classification. 58F05, 58F27, 58F30.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.The first author is partially supported by NSFC grant 19971042, National 973 key Project: Nonlinearity in China and the outstanding youth project of Ministry of Education of China. The second author is partially supported by NSF grant DMS9803581. This work is partially done when the second and third authors were visiting the National University of Singapore

Synchronization, stability and normal hyperbolicity

Synchronization is studied in the framework of invariant manifold theory. Normal hyperbolicity and its persistence are applied to give general results on synchronization and its stability. Simple numerics illustrate the importance of the stability issue