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In many complex systems one can distinguish "fast" and "slow"
processes with radically di?erent velocities. In mathematical
models based on di?er- tialequations,
suchtwo-scalesystemscanbedescribedbyintroducingexpl- itly a small
parameter?on the left-hand side ofstate equationsfor the "fast"
variables, and these equationsare referredto assingularly
perturbed. Surpr- ingly, this kind of equation attracted attention
relatively recently (the idea of distinguishing "fast" and "slow"
movements is, apparently, much older). Robert O'Malley, in comments
to his book, attributes the originof the whole
historyofsingularperturbationsto the celebratedpaperofPrandtl 79].
This was an extremely short note, the text of his talk at the Third
International Mathematical Congress in 1904: the young author
believed that it had to be literally identical with his ten-minute
long oral presentation. In spite of its length, it had a tremendous
impact on the subsequent development. Many famous mathematicians
contributed to the discipline, having numerous and important
applications. We mention here only the name of A. N. Tikhonov,
whodevelopedattheendofthe1940sinhisdoctoralthesisabeautifultheory
for non-linear systems where the fast variables can almost reach
their eq- librium states while the slow variables still remain near
their initial values: the aerodynamics of a winged object like a
plane or the "Katiusha" rocket may serve an example of such a
system. It is generally accepted that the probabilistic modeling of
real-world p- cesses is more adequate than the deterministic
modeling.
Two-scale systems described by singularly perturbed SDEs have been the subject of ample literature. However, this new monograph develops subjects that were rarely addressed and could be given the collective description "Stochastic Tikhonov-Levinson theory and its applications." The book provides a mathematical apparatus designed to analyze the dynamic behaviour of a randomly perturbed system with fast and slow variables. In contrast to the deterministic Tikhonov-Levinson theory, the basic model is described in a more realistic way by stochastic differential equations. This leads to a number of new theoretical questions but simultaneously allows us to treat in a unified way a surprisingly wide spectrum of applications like fast modulations, approximate filtering, and stochastic approximation.
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