Normal Forms, Bifurcations and Finiteness Problems in Differential Equations

Christiane Rousseau, Full Professor, Département de mathématiques et de statistique, Université de Montréal.

A number of recent significant developments in the theory of differential equations are presented in an elementary fashion, many of which are scattered throughout the literature and have not previously appeared in book form, the common denominator being the theory of planar vector fields (real or complex). A second common feature is the study of bifurcations of dynamical systems. Moreover, the book links fields that have developed independently and signposts problems that are likely to become significant in the future.
The following subjects are covered: new tools for local and global properties of systems and families of systems, nonlocal bifurcations, finiteness properties of Pfaffian functions and of differential equations, geometric interpretation of the Stokes phenomena, analytic theory of ordinary differential equations and complex foliations, applications to Hilbert's 16^th problem.

Proceedings of the Nato Advanced Study Institute, held in Montreal, Canada, from 8 to 19 July 2002

Relations between Abelian integrals and limit cycles.- Topics on singularities and bifurcations of vector fields.- Recent advances in the analysis of divergence and singularities.- Local bifurcations of limit cycles, Abel equations and Liénard systems.- Complexity of computations with Pfaffian and Noetherian functions.- Hamiltonian bifurcations and local analytic classification.- Confluence of singular points and Stokes phenomena.- Bifurcations of relaxation oscillations.- Selected topics in differential equations with real and complex time.- Growth rate of the number of periodic points.- Lectures on meromorphic flat connections.- Normal forms, bifurcations and finiteness properties of vector fields.- Aspects of planar polynomial vector fields: global versus local, real versus complex, analytic versus algebraic and geometric.