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Birkhäuser - Birkhäuser Mathematics | Bifurcations of Planar Vector Fields and Hilbert's Sixteenth Problem

Bifurcations of Planar Vector Fields and Hilbert's Sixteenth Problem

Series: Modern Birkhäuser Classics, Vol. 164

Roussarie, Robert

Originally published as volume 164 in the Progress in Mathematics series

1998, XVII, 204 p. 51 illus.

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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. 

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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 recently developed methods. The book, reflecting the state of the art, can also be used for teaching special courses.
(Mathematical Reviews)

 

Content Level » Research

Keywords » bifurcation diagrams - desingularization theory - differential equations - dynamical systems - limit cycles - limit periodic sets - singular limit sets

Related subjects » Birkhäuser Mathematics

Table of contents 

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.

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