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Existence of exponentially small separatrix splittings and homoclinic connections between whiskered tori in weakly hyperbolic near-integrable Hamiltonian systems. (English) Zbl 0941.37021

Summary: The authors study a generalization of Arnold’s original example in which he discussed the existence of a mechanism for instability caused by the splitting of the homoclinic manifolds of the weakly hyperbolic tori, that has subsequently been referred to as “Arnold diffusion” in case when the number of degrees of freedom \(n\geq 3\). Namely, they consider a widely studied model of a pendulum weakly coupled with \(n - 1\) rotors with the degeneracies in the unperturbed Hamiltonian, corresponding to different timescales, existing in the problem.
Using an alloy of the iterative and direct methods developed within the last years the authors give exponentially small upper bounds for the splitting measure of transversality for the case of an even analytic perturbation, thus improving the estimate of G. Gallavotti [Rev. Math. Phys. 6, 343-411 (1994; Zbl 0798.58036)], which he calls quasiflat, and generalizing the analogous recent estimate of A. Delshams, V. Gelfreich, A. Jorba and T. Seara [Commun. Math. Phys. 189, 35-71 (1997; Zbl 0897.34042)] for the rapidly quasiperiodically forced pendulum to a much larger class of Hamiltonian systems.
In particular, the exponentially small upper bound for the transversality measure of the splitting applies when the Hamiltonian has extra degeneracies, namely when the frequencies on a torus become near-resonant. In fact, they show that in such a case the quantity in question becomes smaller, which is the incarnation of the general fact that resonant regions in the action space are more stable in the sense that they have larger Nekhoroshev exponent. Nevertheless, it is emphasized that getting uniform estimates for an arbitrary \(n\geq 3\) is very hard unless one makes some additional assumptions on the approximation properties of the frequency vector.
Although recent developments show that the first order of canonical perturbation theory, given by Melnikov integrals, generally cannot be accepted as the leading-order answer for the splitting distance for the case of more than two degrees of freedom, including the rapidly quasiperiodically forced pendulum problem, the authors suggest an analytic perturbation, the majority of whose Fourier components are strictly non-zero, for which Melnikov integrals can be vindicated as the leading-order approximation for the components of the splitting distance in different directions if the frequencies on the invariant tori satisfy certain arithmetic conditions. This allows to bound the splitting distance from below.
Furthermore, having such a perturbation, for the case of three degrees of freedom, they use a simple number-theoretical argument to find the asymptotics of the Fourier series with exponentially small coefficients involved. This enables to compute the numerous homoclinic orbits for the whiskered tori of a asymptotically full measure, and by proving the domineering contribution of the first order of perturbation theory for the transversality measure, to suggest a leading-order answer for this quantity, which implies the existence of an infinite number of heteroclinic connections between tori with close diophantine frequencies.
The authors elucidate the numerous arithmetic issues that obstruct getting a compact leading-order approximation for the splitting size, most of which can be overcome in the case of three degrees of freedom, as their example shows. These obstacles can be also possibly avoided in the same fashion for an arbitrary \(n\geq 3\) if one treats the cse when the frequencies of the rotors are near a resonance of multiplicity \(n-3\) or \(n-2\).

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
37C55 Periodic and quasi-periodic flows and diffeomorphisms
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
70H05 Hamilton’s equations
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References:

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