Rudnev M., Wiggins S.
Separatrix Splittings Near Resonance in Perturbations of Integrable,
A-Priori Stable
Hamiltonian Systems with Three or More Degrees-of-Freedom
(181K, LaTeX)
ABSTRACT. In this note we consider the problem of the splitting of separatrices
near resonances in perturbations of a-priori stable, integrable
Hamiltonian systems with three or more degrees-of-freedom. For a model
problem that is an $n$ degree-of-freedom generalization of Arnold's
original model we show that
the exponents in the exponentially small (with respect to the
perturbation parameter) upper bound for the measure of the transversality
of the splitting
correspond to the exponents arising in the Nekhoroshev theorem which
describes the evolution of the action variables near a resonant torus.
These exponents are given by $\frac{1}{2(n-m)}$, where $n$ is the number
of degrees-of-freedom and $m$ is the multiplicity of the resonance.
In appropriate coordinates, the problem of separatrix splitting near a
resonant torus
can be viewed as the problem of the splitting of separtrices of a torus
with a certain number of
``fast'' and ``slow'' frequencies (the number of slow frequencies plus one
is the multiplicity of
the resonance). We show that a splitting direction corresponding to a slow
frequency gives rise
to a splitting distance that has an algebraic dependence on the
perturbation parameter, a splitting direction corresponding to a fast
frequency gives rise
to a splitting distance that has an exponentially small dependence on the
perturbation parameter, and if
there is {\em at least one} fast frequency, then the measure of
transversality of the splitting
will be exponentially small with respect to the perturbation
parameter.