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

= 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
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