Bifurcations of Planar Vector Fields and Hilbert's Sixteenth Problem

In a coherent, exhaustive and progressive way, this book presents the tools for studying local bifurcations of limit cycles in families of planar vector fields. A systematic introduction is given to such methods as division of an analytic family of functions in its ideal of coefficients, and asymptotic expansion of non-differentiable return maps and desingularisation. The exposition moves from classical analytic geometric methods applied to regular limit periodic sets to more recent tools for singular limit sets.

The methods can be applied to theoretical problems such as Hilbert's 16th problem, but also for the purpose of establishing bifurcation diagrams of specific families as well as explicit computations.

The book as a whole is a well-balanced exposition that can be recommended to all those who want to gain a thorough understanding and proficiency in the recently developed methods. The book, reflecting the current state of the art, can also be used for teaching special courses.
(Mathematical Reviews)

Presents tools for studying local bifurcations of limit cycles in families of planar vector fields Provides systematic introduction to classical analytic geometric methods applied to regular limit periodic sets and more recent tools for singular limit sets Well-balanced basis for teaching special courses

Autorentext

Robert Roussarie was Professor of Mathematics at the University of Burgundy, France.

Zusammenfassung

"A well-balanced exposition that can be recommended to all those who want to gain a thorough understanding and proficiency in recently developed methods. The book, reflecting the state of the art, can also be used for teaching special courses."

--Mathematical Reviews

Inhalt
Preface.- 1 Families of Two-dimensional Vector Fields.- 2 Limit Periodic Sets.- 3 The 0-Parameter Case.- 4 Bifurcations of Regular Limit Periodic Sets.- 5 Bifurcations of Elementary Graphics.- 6 Desingularization Theory and Bifurcation of Non-elementary Limit Periodic Sets.- Bibliography.- Index.