A Hamiltonian system with one degree of freedom depending on a slowly periodically varying in time parameter is considered. For every fixed value of the parameter there are separatrices on the phase portrait of the system. When parameter is changing in time, these separatrices are pulsing slowly periodically, and phase points of the system cross them repeatedly. In numeric experiments region swept by pulsing separatrices looks like a region of chaotic motion. However, it is shown in the present paper that if the system possesses some additional symmetry (like a pendulum in a slowly varying gravitational field), then typically in the region in question there are many periodic solutions surrounded by stability islands; total measure of these islands does not vanish and does not tend to 0 as rate of changing of the parameter tends to 0.(c) 1997 American Institute of Physics.
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 ∈ N ∪ {∞, ω} is sufficiently large. A resonance k, ν = 0, where ν = ν(y) ∈ R n+1 is a frequency vector and k ∈ Z n+1 , is called strong if |k| < C . The constant C is determined by H 0 and H 1 and does not depend on ε. Let Q be a domain on the rotator space such that its closure Q is free from strong resonances and let γ be a smooth curve on Q. Then for any small ε > 0 there exists a trajectory whose projection to Q moves in a c| log ε| α ε 1/4neighbourhood 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|.
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