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 occupy a nearly full Lebesgue measure set in the phase space. As an important consequence, all orbits are stable in autonomous system with two degrees of freedom, or time-periodic system with one degree of freedom, in the sense that the actions do not change much along the orbits. However, the KAM theory does not guarantee such stability when the system has three or more degrees of freedom for the autonomous case or when it has two or more degrees of freedom for the time-periodic case, simply because the KAM torus can not separate the phase space (or integral manifold) into two disconnected parts. In his celebrated paper [Ar], Arnold constructed an example of nearly integrable Hamiltonian system, where some orbits are unstable. His example is a time periodic system with two degrees of freedom. In Arnold’s example the perturbations are chosen so specifically that all hyperbolic invariant tori preserve in the perturbed system. Hence one can use so called Melnikov method to construct transition chain along which the action has substantial variation. However, in generic case the perturbed systems do not possess such a good property, some resonant gaps between invariant tori break up the transition chain, thus it seems unclear whether one can apply Arnold’s method to find diffusion orbits. Despite of this technical difficulty, Arnold asked whether there is such a phenomenon for a “typical” small perturbation. After near four decades of study some remarkable generalizations of Arnold’s result have been announced ([X1],[DLS1],[Ma5]). A few years ago, Xia [X1] announced that Arnol’d diffusion exists in generic a priori unstable systems, recently Mather announced ([Ma5]) that, under so-called cusp residual condition, Arnold diffusion exists in a priori stable systems with two degrees of freedom in time-periodic case, or with three degrees of freedom in autonomous case. They claim that diffusion orbits 1Department of Mathematics, Nanjing University, Nanjing 210093, China 2Institute of Mathematics, Fudan University, Shanghai 200433, China Typeset by AMS-TEX 1