1.1. INTRODUCTION 5

of sets of initial conditions for which the slow motion

Zε

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 Zε 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 Xε 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