Perturbation theory

Abstract

These lecture notes from a six-hour seminar aim to illustrate some of the recently developed techniques in the perturbation theory for ordinary and partial differential equations. This is done by describing a number of typical examples, preference being given to those useful for applications in mechanics or chemistry and with emphasis on problems on which the author has done research himself. The first half of the article deals with several singular perturbation phenomena, some old, some new, most of them non-linear. The second part is concerned with multi-scale expansions for certain nonlinear systems of ordinary differential equations, depending in a regular way on a parameter, near a point where a stationary solution bifurcates. The long last section is a summary of an as yet unpublished paper by the author dealing with bifurcation of solutions at a point where the real parts of two pairs of conjugate complex eigenvalues of the linearized problem vanish simultaneously. The article limits itself to the formal and heuristic aspects of perturbation theory. No convergence proofs or error estimates are included.

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