Bifurcation of Planar Vector Fields and Hilbert’s Sixteenth Problem

verfasst von: Robert Roussarie

Verlag: Birkhäuser Basel

Buchreihe : Progress in Mathematics

Enthalten in: Professional Book Archive

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Über dieses Buch

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

Inhaltsverzeichnis

Frontmatter

Chapter 1. Families of Two-dimensional Vector Fields

Abstract

In this section we will consider individual vector fields. They can be considered as 0-parameter families. We assume these vector fields to be of class at least C¹. This will be sufficient to ensure the existence and uniqueness of the flow ϕ(t,x (t is time, x ∈ S, the phase space) and the qualitative properties which we mention below.

Robert Roussarie

Chapter 2. Limit Periodic Sets

Abstract

As explained at the end of the previous chapter, the most difficult problem in the study of bifurcations in a family of vector fields on a surface of genus 0 is the control of the periodic orbits. In fact, in generic smooth families the periodic orbits will be isolated for each value of the parameter. For analytic families we have two possibilities for each orbit: it may be isolated or belong to a whole annulus of periodic orbits. In this last case and for the parameter values for which the system has infinitely many periodic orbits, the vector field has a local analytic first integral and the nearby vector fields in the family may be studied by the perturbation theory introduced in Chapter 4. They have in general isolated periodic orbits. The interest in the study of isolated periodic orbits is also justified by tradition and by applications.

Robert Roussarie

Chapter 3. The 0-Parameter Case

Abstract

As an introduction to the theory of bifurcations, in this chapter we want to consider individual vector fields, i.e., families of vector fields with a 0-dimensional parameter space. We will present two fundamentals tools: the desingularization and the asymptotic expansion of the return map along a limit periodic set. In the particular case of an individual vector field these techniques give the desired final result: the desingularization theorem says that any algebraically isolated singular point may be reduced to a finite number of elementary singularities by a finite sequence of blow-ups. If X is an analytic vector field on S ², then the return map of any elementary graphic has an isolated fixed point. As a consequence, in this special case there is no accumulation of limit cycles in the phase space. In other words, the cyclicity of each limit periodic set is less than one and any analytic vector field on the sphere has only a finite number of limit cycles.

Robert Roussarie

Chapter 4. Bifurcations of Regular Limit Periodic Sets

Abstract

In this chapter, (X _λ) will be a smooth or analytic (in Section 3) family of vector fields on a phase space S, with parameter λ ∈ P, as in Chapter 1. Periodic orbits and elliptic singular points which are limits of sequences of limit cycles are called regular limit periodic sets. The reason for this terminology is that for such a limit periodic set r one can define local return maps on transversal segments, which are as smooth as the family itself. The limit cycles near r will be given by a smooth equation and the theory of bifurcations of limit cycles from r will reduce to the theory of unfoldings of differentiable functions. In fact, we will just need the Preparation Theorem and not the whole Catastrophe Theory to treat finite codimension unfoldings.

Robert Roussarie

Chapter 5. Bifurcations of Elementary Graphics

Abstract

After the regular limit periodic sets, the simplest limit periodic sets are the elementary graphics.

Robert Roussarie

Chapter 6. Desingularization Theory and Bifurcation of Non-elementary Limit Periodic Sets

Abstract

In the study of the Bogdanov-Takens unfolding, we introduced in 4.3.5.2 the following formulas of rescaling in the phase-space and in the parameter space:

$$ x = {r^2}\bar x,y = {r^3}\bar y,\mu = - {r^4},\nu = {r^2}\bar \nu . $$

Robert Roussarie

Backmatter

Titel: Bifurcation of Planar Vector Fields and Hilbert’s Sixteenth Problem
verfasst von: Robert Roussarie
Verlag: Birkhäuser Basel
Electronic ISBN: 978-3-0348-8798-4
Print ISBN: 978-3-0348-9778-5
DOI: https://doi.org/10.1007/978-3-0348-8798-4

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

Chapter 1. Families of Two-dimensional Vector Fields

Chapter 2. Limit Periodic Sets

Chapter 3. The 0-Parameter Case

Chapter 4. Bifurcations of Regular Limit Periodic Sets

Chapter 5. Bifurcations of Elementary Graphics

Chapter 6. Desingularization Theory and Bifurcation of Non-elementary Limit Periodic Sets

Backmatter