Bifurcation theory of functional differential equations
 Author/Creator
 Guo, Shangjiang, author.
 Language
 English.
 Publication
 New York : Springer Verlag [2013]
 Physical description
 ix, 289 pages : illustrations ; 24 cm.
 Series
 Applied mathematical sciences (SpringerVerlag New York Inc.) ; v. 184.
Access
Available online

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QA380 .G86 2013

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QA380 .G86 2013
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Contributors
 Contributor
 Wu, Jianhong, 1964 author.
Contents/Summary
 Bibliography
 Includes bibliographical references (pages 275286) and index.
 Contents

 Introduction to dynamic bifurcation theory
 Introduction
 Topological equivalence
 Structural stability
 Codimensionone bifurcations of equilibria
 Fold bifurcation
 PoincaréAndronovHopf bifurcation
 Transcritical and Pitchfork bifurcations of equilibria
 Bifurcations of closed orbits
 Homoclinic bifurcation
 Heteroclinic bifurcation
 Twoparameter bifurcations of equilibria
 Bogdanovtakens bifurcation
 Cusp bifurcation
 Foldhopf bifurcation
 Bautin bifurcation
 Hopfhopf bifurcation
 Some other bifurcations
 Introduction to functional differential equations
 Infinite dynamical systems generated by time lags
 The framework for DDEs
 Definitions
 An operator equation
 Spectrum of the generator
 An adjoint operator
 A bilinear form
 Neural networks with delay : a case study on characteristic equations
 General framework of NFDEs
 Center manifold reduction
 Some examples of ordinary differential equations
 Invariant manifolds of RFDEs
 Center manifold theorem
 Calculation of center manifolds
 The Hopf case
 The FoldHopf case
 The Double Hopf case
 Center manifolds with parameters
 Preservation of symmetry
 Normal form theory
 Introduction
 Unperturbed vector fields
 The PoincaréBirkhoff normal form theorem
 Computation of normal forms
 Internal symmetry
 Perturbed vector fields
 Normal form for Hopf bifurcation
 Norm form theorem
 Preservation of external symmetry
 RFDEs with symmetry
 Basic assumptions
 Computation of symmetric normal forms
 Nonresonance conditions
 LyapunovSchmidt reduction
 The LyapunovSchmidt method
 Derivatives of the bifurcation equation
 Equivariant equations
 The steadystate equivariant branching lemma
 Generalized Hopf bifurcation of RFDE
 Equivariant Hopf bifurcation of NFDEs
 Application to a delayed van der pol oscillator
 Applications to a ring network
 Coupled systems of NFDEs and lossless transmission lines
 Wave trains in the FPU lattice
 Degree theory
 Introduction
 The Brouwer degree
 The LeraySchauder degree
 Global bifurcation theorem
 S¹equivariant degree
 Differentiability case
 Nondifferentiability case
 Global Hopf bifurcation theory of DDEs
 Application to a delayed Nicholson Blowflies equation
 The Nicholson Blowflies equation
 The global Hopf Bifurcation theorem of WeiLi
 Nicholson's Blowflies equation revisited : onset and termination of nonlinear oscillations
 Rotating waves and circulant matrices
 Statedependent DDEs
 Local Hopf bifurcation
 Global bifurcation
 Uniform bounds for periods of periodic solutions in a connected component
 Uniform boundedness of periodic solutions
 Global continuation of rapidly oscillating periodic solutions : an example
 Bifurcation in symmetric FDEs
 Introduction
 Fold bifurcation
 Standard fold bifurcation
 Fold bifurcations with Z₂symmetry
 Fold bifurcations with O(2)symmetry
 Hopf bifurcation
 A little history
 Standard Hopf bifurcation
 Equivariant Hopf bifurcation
 Application to Dnequivariant Hopf bifurcation
 Hopf bifurcation in a ring network
 Bogdanovtakens bifurcation
 Center manifold reduction
 Bogdanov normal form
 Normal form of system (7.60) with a fixed equilibrium
 D3equivariant Bogdanovtakens bifurcation
 Double Hopf bifurcation
 References
 Index.
 Summary
 This book provides a crash course on various methods from the bifurcation theory of Functional Differential Equations (FDEs). FDEs arise very naturally in economics, life sciences and engineering and the study of FDEs has been a major source of inspiration for advancement in nonlinear analysis and infinite dimensional dynamical systems. The book summarizes some practical and general approaches and frameworks for the investigation of bifurcation phenomena of FDEs depending on parameters. The book aims to be selfcontained so the readers will find in this book all relevant materials in bifurcation, dynamical systems with symmetry, functional differential equations, normal forms and center manifold reduction. This material was used in graduate courses on functional differential equations at Hunan University (China) and York University (Canada).
Subjects
Bibliographic information
 Publication date
 2013
 Responsibility
 Shangjiang Guo, Jianhong Wu.
 Series
 Applied Mathematical Sciences, 00665452 ; Volume 184
 Note
 Also issued online.
 ISBN
 9781461469919
 1461469910