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.
