Bifurcation of Maps and Applications (eBook)

G. Iooss (Herausgeber)

eBook Download: PDF

1979 | 1. Auflage
231 Seiten
Elsevier Science (Verlag)
978-0-08-087147-9 (ISBN)

Infos

Bifurcation of maps and applications

Inhalt

Front Cover 1
Bifurcation of Maps and Applications 4
Copyright Page 5
Contents 6
Introduction 8
CHAPTER I. Stability or instability of a fixed point of a map in a Banach space 12
CHAPTER II. Bifurcation of fixed points in R 20
1. Fixed points 20
2. Points of period 2 23
3. The Poincaré map - orbital stability 28
CHAPTER III. Hopf bifurcation in R2 38
1. Standard Hopf-bifurcation 38
2. Non-standard Hopf-bifurcation 55
3. Rotation number of the diffeomorphism restricted to the invariant bifurcated closed curve and weak resonance 58
4. Hopf-bifurcation for fields in R2 82
5. Bifurcation into a 2-dimensional invariant torus for a non-autonomous differential equation 89
6. Bifurcation into a 2-dimensional invariant torus for an autonomous differential equation 96
7. Exercise 110
8. Domain of attractivity and uniqueness of the invariant circle 111
CHAPTER IV. Subharmonic bifurcations of fixed points in R2-strong resonance 116
1. The general study 116
2. Subharmonic bifurcations for a non-autonomous differential equation 134
3. Subharmonic bifurcations for an autonomous differential equation 137
4. Relation with the paper of Arnold and comments 138
CHAPTER V. Invariant manifolds and applications 142
1. The hyperbolic case 143
2. The central case 156
3. Application to bifurcation problems 168
4. Applications to differential equations 180
CHAPTER VI. Bifurcation of an invariant circle into an invars 2-torus for a one parameter family of maps 212
1. Introduction. Definitions 213
2. Main theorem and comments 216
3. Center manifold theorem 219
4. Proof of the main theorem. Step 1: Reduction to the dimension 2 222
5. Proof of the main theorem. Step 2: Persistence of invariant circles for µ != 0 228
6. Proof of the main theorem. Step 3: Bifurcation 233
7. An example 237
BIBLIOGRAPHY 240

Produkt-Details

• Erscheint lt. Verlag: 1.1.1979
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Naturwissenschaften
• Themenwelt: Technik
• ISBN-10: 0-08-087147-X / 008087147X
• ISBN-13: 978-0-08-087147-9 / 9780080871479