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The last thirty years were a period of continuous and intense growth in the subject of dynamical systems. New concepts and techniques and at the same time new areas of applications of the theory were found. The 31st session of the Seminaire de Mathematiques Superieures (SMS) held at the Universite de Montreal in July 1992 was on dynamical systems having as its center theme "Bifurcations and periodic orbits of vector fields". This session of the SMS was a NATO Advanced Study Institute (ASI). This ASI had the purpose of acquainting the participants with some of the most recent developments and of stimulating new research around the chosen center theme. These developments include the major tools of the new resummation techniques with applications, in particular to the proof of the non-accumulation of limit-cycles for real-analytic plane vector fields. One of the aims of the ASI was to bring together methods from real and complex dy namical systems. There is a growing awareness that an interplay between real and complex methods is both useful and necessary for the solution of some of the problems. Complex techniques become powerful tools which yield valuable information when applied to the study of the dynamics of real vector fields. The recent developments show that no rigid frontiers between disciplines exist and that interesting new developments occur when ideas and techniques from diverse disciplines are married. One of the aims of the ASI was to show these multiple interactions at work.
List of contents
Complex Foliations Arising from Polynomial Differential Equations.- Techniques in the Theory of Local Bifurcations: Blow-Up, Normal Forms, Nilpotent Bifurcations, Singular Perturbations.- Six Lectures on Transseries, Analysable Functions and the Constructive Proof of Dulac's Conjecture.- Knots and Orbit Genealogies in Three Dimensional Flows.- Dynamical Systems: Some Computational Problems.- Local Dynamics and Nonlocal Bifurcations.- Singularités d'équations différentielles holomorphes en dimension deux.- Techniques in the Theory of Local Bifurcations: Cyclicity and Desingularization.- Bifurcation Methods in Polynomial Systems.- Algebraic and Geometric Aspects of the Theory of Polynomial Vector Fields.
