Bifurcations of Planar Vector Fields: Nilpotent Singularities and Abelian Integrals

233 Pages·1991·3.12 MB·English

by  Freddy Dumortier

Lecture Notes in Mathematics 1480
Editors:
A. Dold, Heidelberg
B. Eckmann, Z~rich
.F Takens, Groningen
Subseries:
Instituto de Mathemfitica Pura e Aplicada
Rio de Janeiro, Brazil (vol. )84
Adviser:
C. Camacho
E Dumortier R. Roussarie
J. Sotomayor H. Zotadek
Bifurcatio s of Planar
Vector Fields
Nilpotent Singularities and Abelian Integrals
Springer-Verlag
Berlin Heidelberg NewYork
London Paris Tokyo
Hong Kong Barcelona
tsepaduB
Authors
Freddy Dumortier
Limburgs Universitaire Centrum
Universitaire Campus
3610 Diepenbeck, Belgium
Robert Roussarie
Ddpartement de Math6matiques
Universit6 de Bourgogne
UFR de Sciences et Techniques
Laboratoire de Topologie
(U. A. no. 755 du CNRS), B. R 831
21004 Dijon, France
Jorge Sotomayor
Instituto de Matemfitica Pura e Aplicada
Estrada Dona Castorina 011
CEP 22460 Jardim BotSnico
Rio de Janeiro, Brazil
Henryk Zotadek
Institute of Mathematics
Warsaw University
00-901 Warsaw, Poland
Mathematics Subject Classification (1991): 58F14, 34C05, 34D30
ISBN 3-540-54521-2 Springer-Vertag Berlin Heidelberg New York
ISBN 0-387-54521-2 Springer-Verlag New York Berlin Heidelberg
This work is subject to copyright. All rights are reserved, whether the whole or part of
the material si concerned, specifically the rights of translation, reprinting, re-use of
illustrations, recitation, broadcasting, reproduction on microfilmso r in anyo ther way,
and storage in data banks. Duplication of this publication or parts thereof is permitted
only under the provisions of the German Copyright Law of September 9, 1965, ni its
current version, and permission for use must always be obtained from Springer-Verlag.
Violations are liable for prosecution under the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1991
Printed ni Germany
Typesetting: Camera ready by author
Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr.
46/3140-543210 - Printed on acid-freep aper
PREFACE
The study of bifurcations of families of dynamical systems defined by vector fields (i.e.
ordinary differential equations) depending on real parameters is at present an active area
of theoretical and applied research.
Problems in mathematical biology, fluid dynamics, electrical engineering, among other
applied disciplines, lead to multiparametric vector fields whose bifurcation analysis of
equilibria (singular points) and oscillations (cycles) is required.
The case of planar vector fields, due to the presence of regular as well as singular limit
cycles is the first one, in increasing dimension of phase space, whose study cannot be fully
reduced to the analysis of singularities and zeroes of algebraic equations, particularly when
the number of parameters involved is larger than or equal to two.
The results established in this volume illustrate the diversity of the algebraic, geo-
metric and analytic methods used in the description of the variety of structural patterns
that appear in the bifurcation diagrams of generic three-parameter families of planar vector
fields, around singular points whose linear parts are nilpotent.
The analysis involved in their proofs and in the discussion of the remaining conjectures
points out to the actual limits of established tools for the study of complex bifurcation
problems.
The introductions to the two works which constitute this volume locate precisely, in
the context of the current literature, the specific character of each of their contributions.
The authors
Generic 3.-Parameter Families of Planar Vector Fields,
Unfoldings of Saddle~ Focus and Elliptic Singularities
With Nilpotent Linear Parts
by
F. Dumortier, R. Roussarie, J. Sotomayor
Contents of the volume
Generic 3-parameter families of planar vector fields, unfoldings of saddle, focus and elliptic
singularities with nilpotent linear parts.
by F. Dumortier, R. Roussarie, J. Sotomayor ...................... VI
Table of Contents .................................. VII
Part I Presentation of the Results and Normalization ............... 1
Part tI Rescalings an Analytic Treatment .................... 57
Abelian integrals in unfoldings of codimension 3 singular planar vector fields.
by H. Zoladek ....................................... 165
Table of Contents .................................. 166
Part I The weakened 16-th Hilbert Problem ................... 167
Part II The Saddle and Elliptic Cases ...................... 173
Part III The Focus Case .............................. t93
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
Table of contents
PART I: PRESENTATION OF THE RESULTS AND NORMALIZATION
Chapter I: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
:I.I Position of the problem and statement of results ................ 1
:2.1 Codimension 1-phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
:3.1 Conic structure of the bifurcation set and rescaling .............. 13
:4.1 Organization of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Chapter II: Definitions and notations . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Chapter III: Transformation into normal form . . . . . . . . . . . . . . . . . . . . . 22
Chapter IV: Bifurcations of codimension 1 and 2 . . . . . . . . . . . . . . . . . . . . 28
IV.l: Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
IV.2: Codimension 1 bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . 31
IV.3: Codimension 2 bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . 37
PART Ih RESC, ALINGS AND ANALYTIC TREATMENT
Chapter V: Elementary properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
V.A: Location and nature of critical points . . . . . . . . . . . . . . . . . . . . 57
V.B: Location of the Hopf bifurcations of codimensions 1 and 2 ......... 59
V.C: Bifurcations along the set SN . . . . . . . . . . . . . . . . . . . . . . . . . 66
V.D: Rotational property with respect to the parameter v ............ 71
V.E: The principal rescaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Chapter VI: The central rescaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
VI.A: Definition and basic properties . . . . . . . . . . . . . . . . . . . . . . . . 85
VI.B: The saddle case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
1. Hopf bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
2. Integrating factor and Abelian integral ................... 88
3. Saddle connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
IIIV
4. Bifurcation point of two saddle connections ................ 99
5. Complete analysis of the saddle case in a large central rescaling chart . . 102
6. Study in some "principal rescaling cone" around the TSC-line ...... 114
VI.C: The elliptic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
VI.D: The focus case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
1. Study along the #2-axis in a large central rescaling chart ......... 120
2. Study along the v-axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . t25
3. Study in a "principal rescaling cone" around the DH-line and the DL-line 133
Chapter VII: Conclusions and discussion of remaining problems ............ 135
VII.A: The genral conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
VII.B: The saddle case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
VII.C: The focus case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
VII.D: The elliptic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
PART I : PRESENTATION OF THE RESULTS AND NORMALIZATION
CHAPTER I : INTRODUCTION
I.I : Position of the problem and statement of results
This paper continues the study of 3-parameter families of vector
fields in the plane, initiated in [DRS]. We present here three unfoldings
of singularities of cedimension .3 These unfoldings, together with those
mentioned or studied in [DRS], provide a complete list of topological
models for all the possible generic local k-parameter families for k s .3
The reader is referred to the introduction of [DRS] for a review of known
facts concerning unfoldings used freely here.
We consider germs of vector fields (at 0 E <I] ) 2 with nilpotent
a
1-jet; that is with a linear part linearly conjugate to y x-8 " Such a germ
has a 2-jet C -conjugate to :
a a (1)
Y 7x + (~x2 + ~xy) 7y
(For the definition of conjugacy, see Chapter II).
In [DRS] we studied the cUS P case corresponding to ~ ~ 0 and ~ = O. Here,
we study the other possibilities of codimension ,3 when ~ = 0 and ~ ~ O.
The germs of vector fields at 0 e ,2 R] whose 1-jets are nilpotent form a
manifold of codimension 2 in the space of all singular germs. Therefore
the 2 sets of conditions : ~ ~ O, ~ = O and ~ = O, ~ ~ 0 define 2 sets of
codimension .3 We want to study 3-parameter local families which
generically unfold a germ belonging to a regular part of these 2 sets,
defined by removing a subset of codimension 4.
For the cusp case (~ ~ O, ~ = O) we defined in [DRS] the regular part to be
the subset of germs whose 4-jet is C~-equivalent to ;
8
Y xx8 + (x2 + xBy) 8_ (2)
- ay
This set is the union of 2 submanifolds E3C+ and EC ,3 obtained by removing
a 2a
the subset of germs whose 4-jet is equivalent to y x~a + x 8y
We know by [T] that the germs in the second set (~ = O, ~ ~ O) have a 4-jet
~ C conjugate to :
Y ~a x + (ElX3 + dx4 + bxy + ax 2 y + ex 3 y )a ~y (3)
with b>O, I = e O, + I.
It was shown in [D] that the topological type of such a germ is determined
by its 3-jet, if ~I ~ 0 and b ~ 24 in case ~I = -i. The regular part of
our set is defined by these conditions and the extra condition
I 5 a e - 3b d ~ 0 (3')
which we will explain in Chapter III, third step. We will also show that
the 4-jet is then C -equivalent to :
3 2 x3y) a
Y x~a + (~i x + bxy + c2x y + f ~y with ~1,2 = + ,I b > 0. (3")
The topological type falls into one of the following categories :
i) The saddle case I : = e ,I any 2 E and b. (a degenerate saddle). Here we
make a distinction according to the sign of c
"2
We denote by 3 Z S+ (~2 = + 1), the subsets of germs with such a 4-jet.
They all have the same topological type.
2) The focus case I : = e -I and O<b<2~. (a degenerate focus). We denote
by 3 E F+ (c2 = ~ i) the corresponding subsets of germs.
3) The elliptic case : I e = -i and b>2v~. Notation E +E3 with 2 = E _+ .I
They all have the same topological type.
c2=i (2=-i
Saddle case Focus cases Elliptic case
Fig. .I : The different topological types
The union of the 6 subsets 3 E S+' 3 EF+' E3 E+ is the regular part of our second
set. Each of them is a submanifold of codimension 3 in the space of singular
germs of vector fields (i.e. which vanish at 0 ~ R] )2 and so, a codimension
5 submanifold of the space of all germs. We want to study generic local
3-parameter families 1 X with X 3 E E 3 E 3 E l( ~ R] )3
' o S+' F+ or E+" "
The genericity condition consists in the transversality of the mapping
(m,l) e RI 2 x ~3 ~ j4XA )m( to the sets E3 S+' E3 F+ and E3 E+" An example of
such a family in each case, called standard family, is given by :
~l = Y ~a x + ( ElX3 + #2 x + >I + y(~ + bx + ~2x2)) ~ ay (4)
where I = (#i' #2' ~)' and of course : b>O, b~2~, ,lE 2 = + .I
The present article is devoted to establish several basic facts, in
support of the following conjecture :
Let l X and Y1 be two local 3-parameter families, with Xo, Yo belonging to
the same set : ~S+' 3 E3 F+ or ~E+" 3 Suppose that both families are transversal
to this set (in the sense defined above). Then they are fiber-C °
equivalent (in the sense of Chapter II).
In particular, this includes that any two standard families X~, ~
with ~o' ~ in the same set ~3 ~3 ~3 will be (fiber-C ) ° equivalent
o S+, F+ or E+