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 = Φt 0 acting by Φt(x, y) = (x, F t x y) where F t x is another family of flows given by F t x y = Yx,y(t) with Y = Yx,y = Y 0 x,y 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) = ¯ y (x) = lim T →∞ T −1 T 0 B(x, F t x y)dt exists and it is the same for ”many” y s. For instance, suppose that µx is an ergodic invariant measure of the flow F t x then the limit (1.1.3) exists for µx−almost all y and is equal to ¯(x) = ¯ 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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