Bifurcations and Periodic Orbits of Vector Fields

Abstract

The main topic of this book is the theory of bifurcations of vector fields, i.e. the study of families of vector fields depending on one or several parameters and the changes (bifurcations) in the topological character of the objects studied as parameters vary. In particular, one of the phenomena studied is the bifurcation of periodic orbits from a singular point or a polycycle. The following topics are discussed in the book: Divergent series and resummation techniques with applications, in particular to the proofs of the finiteness conjecture of Dulac saying that polynomial vector fields on R2 cannot possess an infinity of limit cycles. The proofs work in the more general context of real analytic vector fields on the plane. Techniques in the study of unfoldings of singularities of vector fields (blowing up, normal forms, desingularization of vector fields). Local dynamics and nonlocal bifurcations. Knots and orbit genealogies in three-dimensional flows. Bifurcations and applications: computational studies of vector fields. Holomorphic differential equations in dimension two. Studies of real and complex polynomial systems and of the complex foliations arising from polynomial differential equations. Applications of computer algebra to dynamical systems. 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.