1.1. Introduction

Many real systems can be viewed as a combination of slow and fast motions

which leads to complicated double scale equations. Already in the 19th century in

applications to celestial mechanics it was well understood (though without rigorous

justification) that a good approximation of the slow motion can be obtained by

averaging its parameters in fast variables. Later, averaging methods were applied

in signal processing and, rather recently, to model climate–weather interactions (see

[36], [18], [37] and [52]). The classical setup of averaging justified rigorously in [12]

presumes that the fast motion does not depend on the slow one and most of the work

on averaging treats this case only. On the other hand, in real systems both slow and

fast motions depend on each other which leads to the more diﬃcult fully coupled

case which we study here. This setup emerges, in particular, in perturbations of

Hamiltonian systems which leads to fast motions on manifolds of constant energy

and slow motions across them.

In this work we consider a system of differential equations for

Xε

= Xx,y

ε

and

Y

ε

= Yx,y

ε

,

(1.1.1)

dXε(t)

dt

=

εB(Xε(t),

Y

ε(t)),

dY

ε(t)

dt

=

b(Xε(t),

Y

ε(t))

with initial conditions

Xε(0)

= x, Y

ε(0)

= y on the product

Rd

× M where M is a

compact nM-dimensional

C2

Riemannian manifold and B(x, y), b(x, y) are smooth

in x, y families of bounded vector fields on

Rd

and on M, respectively, so that y

serves as a parameter for B and x for b. The solutions of (1.1.1) determine the flow

of diffeomorphisms Φε

t

on

Rd

× M acting by Φε(x,

t

y) = (Xx,y(t),

ε

Yx,y(t)).

ε

Taking

ε = 0 we arrive at the flow Φt = Φ0 t acting by Φt(x, y) = (x, Fxy) t where Fx t is

another family of flows given by Fxy t = Yx,y (t) with Y = Yx,y = Yx,y 0 being the

solution of

(1.1.2)

dY (t)

dt

= b(x, Y (t)), Y (0) = y.

It is natural to view the flow

Φt

as describing an idealised physical system where

parameters x = (x1, ..., xd) are assumed to be constants (integrals) of motion while

the perturbed flow Φε

t

is regarded as describing a real system where evolution of

these parameters is also taken into consideration. Essentially, the proofs of this

paper work also in the slightly more general case when B and b in (1.1.1) together

with their derivatives depend Lipschitz continuously on ε (cf. [55]) but in order to

simplify notations and estimates we do not consider this generalisation here.

Assume that the limit

(1.1.3)

¯(x)

B =

¯

B

y

(x) = lim

T →∞

T

−1

T

0

B(x,

Fxy)dtt

exists and it is the same for ”many” y s. For instance, suppose that µx is an ergodic

invariant measure of the flow Fx t then the limit (1.1.3) exists for µx−almost all y

and is equal to

¯(x)

B =

¯

B

µx

(x) = B(x, y)dµx(y).

If b(x, y) does not, in fact, depend on x then Fx

t

= F

t

and µx = µ are also

independent of x and we arrive at the classical uncoupled setup. Here the Lipschitz

2