Arnold diffusion far from strong resonances in multidimensional a priori unstable Hamiltonian systems

摘要

We prove the existence of Arnold diffusion in a typical a priori unstable Hamiltonian system outside a small neighbourhood of strong resonances. More precisely, we consider a near-integrable Hamiltonian system with Hamiltonian H=H _0+εH _1+O(ε ~2), where the unperturbed Hamiltonian H _0 is essentially the product of a one-dimensional pendulum and n-dimensional rotator. Coordinates y=(y _1,y _n) on the rotator space are first integrals in the unperturbed system and become slow variables after perturbation. The main result is as follows. Suppose that the time-periodic perturbation H _1 is C ~r-generic, r∈?∪{∞,ω} is sufficiently large. A resonance 〈k,v〉=0, where ν=ν(y)∈? ~(n+1) is a frequency vector and k∈Z ~(n+1), is called strong if |k|0 there exists a trajectory whose projection to Q moves in a c|log ε| ~αε ~(1/4)-neighbourhood of γ(α≥n ~2+2n/r-n-1 is any constant and c=c(α)>0) with average velocity along γ of order ε/| log ε|.