Nonlocal bifurcations / Yu. Ilyashenko, Weigu Li.

Format:

Book

Author/Creator:

Ilʹi︠a︡shenko, I︠U︡. S., author.

Li, Weigu, author.

Series:

Mathematical surveys and monographs ; no. 66.

Mathematical surveys and monographs, 0076-5376 ; volume 66

Language:

English

Subjects (All):

Bifurcation theory.

Physical Description:

1 online resource (303 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [1999]

Language Note:

English

Summary:

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 exposed 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.

Contents:

""Contents""; ""Preface""; ""Chapter 1. Introduction""; ""Â1. Structural stability and Morseâ€?Smale systems""; ""Â2. Equivalence and local bifurcations in generic one-parameter families""; ""Â3. Homoclinic trajectories of nonhyperbolic singular points""; ""Â4. Homoclinic trajectories of nonhyperbolic cycles""; ""Â5. Homoclinic loops of hyperbolic fixed points and other contours""; ""Â6. Summary of results""; ""Chapter 2. Preliminaries""; ""Â1. Prevalence""; ""Â2. Attractors, their dimensions and projections""; ""Â3. Smale horseshoe for high school students""

""Â4. Some results in hyperbolic theory""""Â5. Normal forms for local families""; ""Chapter 3. Bifurcations in the Plane""; ""Â1. Bifurcations of homoclinic loops of planar saddles""; ""Â2. Homoclinic orbit of a saddlenode""; ""Â3. Semistable cycles breaking saddle connections""; ""Chapter 4. Homoclinic Orbits of Nonhyperbolic Singular Points""; ""Â1. Homoclinic orbit of a saddlenode: the case of a nodal hyperbolic part""; ""Â2. Lemma on the hyperbolicity of the product of linear maps""; ""Â3. Homoclinic orbit of a saddlenode: the case of a saddle hyperbolic part""

""Â4. Several homoclinic orbits of a saddlenode""""Â5. Birkhoffâ€?Smale theorem""; ""Chapter 5. Homoclinic Tori and Klein Bottles of Nonhyperbolic Periodic Orbits: Noncritical Case""; ""Â1. The topological and smooth structure of the union of homoclinic orbits""; ""Â2. Persistence of noncritical homoclinic tori and Klein bottles""; ""Â3. The rotation number as a function of the parameter in the family of diffeomorphisms of the circle""; ""Â4. Bifurcations on a noncritical homoclinic torus of a generic saddlenode family""; ""Â5. The blue sky catastrophe on the Klein bottle""

""Â6. Generalized Smale horseshoe existence theorem""""Â7. Several noncritical homoclinic tori or Klein bottles of a nonhyperbolic cycle""; ""Â8. Generation of a strange attractor via the bifurcation of a twisted homoclinic surface""; ""Chapter 6. Homoclinic Torus of a Nonhyperbolic Periodic Orbit: Semicritical Case""; ""Â1. Theorem on the generation of a strange attractor""; ""Â2. Lemmas on limit maps, density and volume contraction""; ""Â3. Rotation sets and periodic points of circle endomorphisms""; ""Â4. Homoclinic orbits of circle endomorphisms""

""Chapter 7. Bifurcations of Homoclinic Trajectories of Hyperbolic Saddles""""Â1. The homoclinic trajectory of a hyperbolic saddle with three real eigenvalues in R[sup(3)]""; ""Â2. The homoclinic trajectory of a hyperbolic saddle with two complex eigenvalues in R[sup(3)]""; ""Â3. Homoclinic orbits of hyperbolic saddles in high-dimensional spaces""; ""Chapter 8. Elements of Hyperbolic Theory""; ""Â1. Hyperbolic sets and their properties""; ""Â2. Introduction to symbolic dynamics""; ""Â3. Hyperbolic fixed point theorem""

""Â4. Sufficient conditions for the existence of a Smale horseshoe""

Notes:

Description based upon print version of record.

Includes bibliographical references.

Description based on print version record.

ISBN:

1-4704-1293-4