1.1. INTRODUCTION 5 of sets of initial conditions for which the slow motion exhibits substantially different behavior than the averaged one ¯ 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 ¯. 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 V ε,θ x,y 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 F t x 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) = x,y (n) and Y ε (n) = Y ε x,y (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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