Zhihong Jeff XIA

Department of Mathematics
Northwestern University
Evanston, IL 60208, USA
xia@math.nwu.edu

Arnold Diffusion, Variational Method and Arnold-Mather Mechanism

Abstract: We use variational method to study the problem of Arnold diffusion and instabilities in near-integrable Hamiltonian systems. With generic assumptions, verifiable by Melnikov integrals in a priori unstable situation and perhaps other methods in a priori stable situation, we construct transition chain of arbitrary length, crossing gaps of any sizes between invariant tori. We also obtain shadowing properties in the transition chain, non-integrability of the system and many other interesting properties. Our method is strictly variational and applies to cases where the normally hyperbolic (weakly) set is 2-D (for diffeos, 3-D for flows). Its generalization to higher dimensional normally hyperbolic set is currently being studied.