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Bifurcation theory of functional differential equations / Shangjiang Guo, Jianhong Wu.



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Guo, Shangjiang, author.
Publication date:
New York : Springer Verlag [2013]
  • Book
  • ix, 289 pages : illustrations ; 24 cm.
Includes bibliographical references (pages 275-286) and index.
  • 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.
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).
Also issued online.
Wu, Jianhong, 1964- author.
Applied Mathematical Sciences, 0066-5452 ; Volume 184
Applied mathematical sciences (Springer-Verlag New York Inc.) ; v. 184.

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