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Hardcover ISBN:  9780821804971 
Product Code:  SURV/66 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
eBook ISBN:  9781470412937 
Product Code:  SURV/66.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Hardcover ISBN:  9780821804971 
eBook ISBN:  9781470412937 
Product Code:  SURV/66.B 
List Price:  $254.00 $191.50 
MAA Member Price:  $228.60 $172.35 
AMS Member Price:  $203.20 $153.20 

Book DetailsMathematical Surveys and MonographsVolume: 66; 1999; 286 ppMSC: Primary 34; 58
This book studies nonlocal bifurcations that occur on the boundary of the domain of MorseSmale 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.
ReadershipGraduate students and research mathematicians working in ordinary differential equations; physicists, engineers, computer scientists and mathematical biologists.

Table of Contents

Chapters

1. Introduction

2. Preliminaries

3. Bifurcations in the plane

4. Homoclinic orbits of nonhyperbolic singular points

5. Homoclinic tori and Klein bottles of nonhyperbolic periodic orbits: Noncritical case

6. Homoclinic torus of a nonhyperbolic periodic orbit: Semicritical case

7. Bifurcations of homoclinic trajectories of hyperbolic saddles

8. Elements of hyperbolic theory

9. Normal forms for local families: Hyperbolic case

10. Normal forms for unfoldings of saddlenodes


Additional Material

Reviews

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


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This book studies nonlocal bifurcations that occur on the boundary of the domain of MorseSmale 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.
Graduate students and research mathematicians working in ordinary differential equations; physicists, engineers, computer scientists and mathematical biologists.

Chapters

1. Introduction

2. Preliminaries

3. Bifurcations in the plane

4. Homoclinic orbits of nonhyperbolic singular points

5. Homoclinic tori and Klein bottles of nonhyperbolic periodic orbits: Noncritical case

6. Homoclinic torus of a nonhyperbolic periodic orbit: Semicritical case

7. Bifurcations of homoclinic trajectories of hyperbolic saddles

8. Elements of hyperbolic theory

9. Normal forms for local families: Hyperbolic case

10. Normal forms for unfoldings of saddlenodes

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