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Geometric singular perturbation theory. (English) Zbl 0840.58040
Johnson, Russell (ed.), Dynamical systems. Lectures given at the 2nd session of the Centro Internazionale Matematico Estivo (CIME) held in Montecatini Terme, Italy, June 13-22, 1994. Berlin: Springer-Verlag. Lect. Notes Math. 1609, 44-118 (1995).
These are lectures given at the second session of the Centro Internazionale Matematico Estivo (CIME) in 1994. The goal is an exposition of the geometric approach to singular perturbation problems. Singularly perturbed equations gain their special structure from the presence of differing time scales. The fundamental tool in their analysis, from perspective taken in the lectures, is the set of theorems due to Fenichel. The first step is then to explain these theorems and their significance. At the same time, new proofs of Fenichel’s three main results are outlined. The contents of the lectures: introduction, invariant manifold theorems, Fenichel normal form, tracking with differential forms, exchange lemma, generalizations and future directions [58 ref.].
MSC:
37J40Perturbations, normal forms, small divisors, KAM theory, Arnol’d diffusion
37-01Instructional exposition (Dynamical systems and ergodic theory)
37-99Dynamic systems and ergodic theory (MSC2000)