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 (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 from its averaged (in fast variables Y ε) approximation
¯
X 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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