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Nonlocal Bifurcations
Yu. Ilyashenko, Moscow State University, Russia, and Weigu Li, Beijing University, People's Republic of China

Mathematical Surveys and Monographs
1999; 286 pp; hardcover
Volume: 66
ISBN-10: 0-8218-0497-9
ISBN-13: 978-0-8218-0497-1
List Price: US$89
Member Price: US$71.20
Order Code: SURV/66
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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.


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


"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

  • Introduction
  • Preliminaries
  • Bifurcations in the plane
  • Homoclinic orbits of nonhyperbolic singular points
  • Homoclinic tori and Klein bottles of nonhyperbolic periodic orbits: Noncritical case
  • Homoclinic torus of a nonhyperbolic periodic orbit: Semicritical case
  • Bifurcations of homoclinic trajectories of hyperbolic saddles
  • Elements of hyperbolic theory
  • Normal forms for local families: Hyperbolic case
  • Normal forms for unfoldings of saddlenodes
  • Bibliography
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