1.1. INTRODUCTION 5
of sets of initial conditions for which the slow motion

exhibits substantially
different behavior than the averaged one
¯
Z but also enable us to go further and to
investigate much longer exponential in 1/ε time behavior of
Zε.
Namely, we will
be able to study exits of the slow motion from a neighborhood of an attractor of
the averaged one and transitions of between basins of attractors of
¯.
Z Such
evolution, which becomes visible only on much longer than 1/ε time scales, is
usually called adiabatic in the framework of averaging. In the simpler case when
the fast motion does not depend on the slow one such results were discussed in
[49]. Still, even in this uncoupled situation descriptions of transitions of the slow
motion between attractors of the averaged one were not justified rigorously both
in the Markov processes case of [29] and in the dynamical systems case of [49].
Extending these technique to three scale equations may exhibit stochastic resonance
type phenomena producing a nearly periodic motion of the slowest motion which
is described in Section 1.10 below. These problems seem to be important in the
study of climate–weather interactions and they were discussed in [18] and [37]
in the framework of a model describing transitions between steady climatic states
with weather evolving as a fast chaotic system and climate playing the role of
the slow motion. Such ”very long” time description of the slow motion is usually
impossible in the traditional averaging setup which deals with perturbations of
integrable Hamiltonian systems. In the fully coupled situation we cannot work just
with one hyperbolic flow but have to consider continuously changing fast motions
which requires a special technique. In particular, the full flow Φε
t
on
Rd
×M defined
above and viewed as a small perturbation of the partially hyperbolic system
Φt
plays
an important role in our considerations. It is somewhat surprising that the ”very
long time” behavior of the slow motion which requires certain ”Markov property
type” arguments still can be described in the fully coupled setup which involves
continuously changing fast hyperbolic motions. It turns out that the perturbed
system still possesses semi-invariant expanding cones and foliations and a certain
volume lemma type result on expanding leaves plays an important role in our
argument for transition from small time were perturbation techniques still works
to the long and ”very long” time estimates.
It is plausible that moderate deviations type results can be proved for Vx,yε,θ
when 1/2 θ 1 and that the distribution of Vx,y
ε,1/2
(·) in y converges to the
distribution of a Gaussian diffusion process in Rd. Still, this requires somewhat
different methods and it will not be discussed here. In this regard we can mention
limit theorems obtained in [14] for a system of two heavy and light particles which
leads to an averaging setup for a billiard flow. For the simpler case when b does
not depend on x, i.e. when all flows Fx t are the same, the moderate deviations and
Gaussian approximations results were obtained previously in [50]. Related results
in this uncoupled situation concerning Hasselmann’s nonlinear (strong) diffusion
approximation of the slow motion were obtained in [56].
We consider also the discrete time case where (1.1.1) is replaced by difference
equations for sequences
Xε(n)
= Xx,y(n)
ε
and Y
ε(n)
= Yx,y
ε
(n), n = 0, 1, ... so that
Xε(n
+ 1)
Xε(n)
=
εB(Xε(n),
Y
ε(n)),
(1.1.10)
Y
ε(n
+ 1) = FXε(n)Y
ε(n), Xε(0)
= x, Y
ε(0)
= y
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