**Mathematical Surveys and Monographs**

Volume: 66;
1999;
286 pp;
Hardcover

MSC: Primary 34; 58;

**Print ISBN: 978-0-8218-0497-1
Product Code: SURV/66**

List Price: $95.00

AMS Member Price: $76.00

MAA Member Price: $85.50

**Electronic ISBN: 978-1-4704-1293-7
Product Code: SURV/66.E**

List Price: $89.00

AMS Member Price: $71.20

MAA Member Price: $80.10

# Nonlocal Bifurcations

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*Yu. Ilyashenko; Weigu Li*

This book studies nonlocal bifurcations that occur on the boundary of the domain of Morse-Smale systems in the space of all dynamical systems. These bifurcations provide a series of fascinating new scenarios for the transition from simple dynamical systems to complicated ones. The main effects are the generation of hyperbolic periodic orbits, nontrivial hyperbolic invariant sets and the elements of hyperbolic theory. All results are rigorously proved and explained in a uniform way. The foundations of normal forms and hyperbolic theories are presented from the very first stages. The proofs are preceded by heuristic descriptions of the ideas. The book contains new results, and many results have not previously appeared in monograph form.

#### Readership

Graduate students and research mathematicians working in ordinary differential equations; physicists, engineers, computer scientists and mathematical biologists.

#### Reviews & Endorsements

This book is clearly the most complete collection in existence of results for nonlocal bifurcations, excluding homoclinic tangencies. It is clearly written and would serve as a very good introduction to this field. The bibliography is surprisingly extensive, which is important since articles in this field are scattered over very many journals.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Nonlocal Bifurcations

- Contents vii8 free
- Preface ix10 free
- Chapter 1. Introduction 116 free
- §1. Structural stability and Morse–Smale systems 116
- §2. Equivalence and local bifurcations in generic one-parameter families 1025
- §3. Homoclinic trajectories of nonhyperbolic singular points 1732
- §4. Homoclinic trajectories of nonhyperbolic cycles 1934
- §5. Homoclinic loops of hyperbolic fixed points and other contours 2237
- §6. Summary of results 2641

- Chapter 2. Preliminaries 3146
- Chapter 3. Bifurcations in the Plane 6984
- Chapter 4. Homoclinic Orbits of Nonhyperbolic Singular Points 8398
- §1. Homoclinic orbit of a saddlenode: the case of a nodal hyperbolic part 8398
- §2. Lemma on the hyperbolicity of the product of linear maps 87102
- §3. Homoclinic orbit of a saddlenode: the case of a saddle hyperbolic part 93108
- §4. Several homoclinic orbits of a saddlenode 102117
- §5. Birkhoff–Smale theorem 104119

- Chapter 5. Homoclinic Tori and Klein Bottles of Nonhyperbolic Periodic Orbits: Noncritical Case 109124
- §1. The topological and smooth structure of the union of homoclinic orbits 109124
- §2. Persistence of noncritical homoclinic tori and Klein bottles 114129
- §3. The rotation number as a function of the parameter in the family of diffeomorphisms of the circle 120135
- §4. Bifurcations on a noncritical homoclinic torus of a generic saddlenode family 127142
- §5. The blue sky catastrophe on the Klein bottle 131146
- §6. Generalized Smale horseshoe existence theorem 138153
- §7. Several noncritical homoclinic tori or Klein bottles of a nonhyperbolic cycle 142157
- §8. Generation of a strange attractor via the bifurcation of a twisted homoclinic surface 153168

- Chapter 6. Homoclinic Torus of a Nonhyperbolic Periodic Orbit: Semicritical Case 159174
- Chapter 7. Bifurcations of Homoclinic Trajectories of Hyperbolic Saddles 181196
- Chapter 8. Elements of Hyperbolic Theory 211226
- Chapter 9. Normal Forms for Local Families: Hyperbolic Case 235250
- Chapter 10. Normal Forms for Unfoldings of Saddlenodes 253268
- Bibliography 281296