Rudnev M., Wiggins S.
Existence of exponentially small separatrix splitting
and homoclinic connections
between whiskered tori in weakly hyperbolic near-integrable Hamiltonian
systems. Revised version.
(305K, LaTeX)
ABSTRACT. In this paper we 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 {\em "Arnold diffusion"} in case when the number of degrees of freedom $n\geq 3$. Namely, we consider a
widely studied model of a pendulum weakly coupled with $n-1$ rotors with the degeneracies in the unperturbed
Hamiltonian, corresponding to different time-scales, existing in the problem.
Using an alloy of the iterative and direct methods developed within the last years we
give exponentially small upper bounds for the splitting measure of transversality for the
case of an even, analytic perturbation, thus improving the estimate of Gallavotti [1994], which he
calls quasiflat, and generalizing the analogous recent estimate of Delshams et al. [1996] 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, we 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 in fact more stable in the sense that they have larger
Nekhoroshev exponent. Nevertheless, we emphasize that getting uniform estimates for an arbitrary $n\geq3$ 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,
we 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 us to bound the splitting distance from below.
Furthermore, having such a perturbation, for the case of three degrees of freedom, we use a simple number-theoretical
argument to find the asymptotics of the Fourier series with exponentially small coefficients involved. This enables us
to compute the numerous homoclinic orbits for the whiskered tori of 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, thus proving the existence of an infinite number of heteroclinic connections between tori with close diophantine frequencies.
We 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 our example shows. These obstacles can be also possibly avoided in the same fashion for an arbitrary
$n\geq3$ if one treats the case when the frequencies of the rotors are near a resonance of multiplicity $n-3$ or $n-2$.