Abstract The work treats dynamical systems given by ordinary differential equations in the form dXε(t) dt = εB(Xε(t), Y ε (t)) where fast motions Y ε depend on the slow mo- tion Xε (coupled with it) and they are either given by another differential equation dY ε (t) dt = b(Xε(t), Y ε (t)) or perturbations of an appropriate parametric family of Markov processes with freezed slow variables. In the first case we assume that the fast motions are hyperbolic for each freezed slow variable and in the second case we deal with Markov processes such as random evolutions which are combinations of diffusions and continuous time Markov chains. First, we study large deviations of the slow motion Xε from its averaged (in fast variables Y ε ) approximation ¯ ε . The upper large deviation bound justifies the averaging approximation on the time scale of order 1/ε, called the averaging principle, in the sense of convergence in measure (in the first case) or in probability (in the second case) but our real goal is to obtain both the upper and the lower large deviations bounds which together with some Markov property type arguments (in the first case) or with the real Markov property (in the second case) enable us to study (adiabatic) behavior of the slow motion on the much longer exponential in 1/ε time scale, in particular, to describe its fluctuations in a vicinity of an attractor of the averaged motion and its rare (adiabatic) transitions between neighborhoods of such attractors. When the fast motion Y ε does not depend on the slow one we arrive at a simpler averaging setup studied in numerous papers but the above fully coupled case, which better describes real phenomena, leads to much more complicated problems. Received by the editor December 2000 Mathematics Subject Classification. Primary: 34C29 Secondary: 37D20, 60F10, 60J25. Key words and phrases. averaging, hyperbolic attractors, random evolutions,large deviations. The author was partially supported by US–Israel BSF. vi , 2006. 4

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