Includes bibliographical references (pages 275-286) and index.
Contents:
Introduction to dynamic bifurcation theory
Introduction
Topological equivalence
Structural stability
Codimension-one bifurcations of equilibria
Fold bifurcation
Poincaré-Andronov-Hopf bifurcation
Transcritical and Pitchfork bifurcations of equilibria
Bifurcations of closed orbits
Homoclinic bifurcation
Heteroclinic bifurcation
Two-parameter bifurcations of equilibria
Bogdanov-takens bifurcation
Cusp bifurcation
Fold-hopf bifurcation
Bautin bifurcation
Hopf-hopf 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 Fold-Hopf case
The Double Hopf case
Center manifolds with parameters
Preservation of symmetry
Normal form theory
Introduction
Unperturbed vector fields
The Poincaré-Birkhoff 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
Lyapunov-Schmidt reduction
The Lyapunov-Schmidt method
Derivatives of the bifurcation equation
Equivariant equations
The steady-state 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 Leray-Schauder degree
Global bifurcation theorem
S¹-equivariant 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 Wei-Li
Nicholson's Blowflies equation revisited : onset and termination of nonlinear oscillations
Rotating waves and circulant matrices
State-dependent 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 Z₂-symmetry
Fold bifurcations with O(2)-symmetry
Hopf bifurcation
A little history
Standard Hopf bifurcation
Equivariant Hopf bifurcation
Application to Dn-equivariant Hopf bifurcation
Hopf bifurcation in a ring network
Bogdanov-takens bifurcation
Center manifold reduction
Bogdanov normal form
Normal form of system (7.60) with a fixed equilibrium
D3-equivariant Bogdanov-takens 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 self-contained 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).