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Bifurcation theory of functional differential equations

Author: Shangjiang Guo; Jianhong Wu
Publisher: [Place of publication not identified] : [publisher not identified], 2013.
Series: Applied mathematical sciences (Springer-Verlag New York Inc.)
Edition/Format:   eBook : Document : EnglishView all editions and formats
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  Read more...
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Genre/Form: Electronic books
Additional Physical Format: Print version:
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Shangjiang Guo; Jianhong Wu
ISBN: 9781461469926 1461469929 1461469910 9781461469919 9781299857247 1299857248
OCLC Number: 862201288
Description: 1 online resource (ix, 289 pages) : illustrations.
Contents: Preface; Contents; 1 Introduction to Dynamic Bifurcation Theory; 1.1 Introduction; 1.2 Topological Equivalence; 1.3 Structural Stability; 1.4 Codimension-One Bifurcations of Equilibria; 1.4.1 Fold Bifurcation; 1.4.2 Poincaré-Andronov-Hopf Bifurcation; 1.5 Transcritical and Pitchfork Bifurcations of Equilibria; 1.6 Bifurcations of Closed Orbits; 1.7 Homoclinic Bifurcation; 1.8 Heteroclinic Bifurcation; 1.9 Two-Parameter Bifurcations of Equilibria; 1.9.1 Bogdanov-Takens Bifurcation; 1.9.2 Cusp Bifurcation; 1.9.3 Fold-Hopf Bifurcation; 1.9.4 Bautin Bifurcation; 1.9.5 Hopf-Hopf Bifurcation. 1.10 Some Other Bifurcations2 Introduction to Functional Differential Equations; 2.1 Infinite Dynamical Systems Generated by Time Lags; 2.2 The Framework for DDEs; 2.2.1 Definitions; 2.2.2 An Operator Equation; 2.2.3 Spectrum of the Generator; 2.2.4 An Adjoint Operator; 2.2.5 A Bilinear Form; 2.2.6 Neural Networks with Delay: A Case Studyon Characteristic Equations; 2.2.6.1 General Additive Neural Networks with Delay; 2.2.6.2 Special Case: Two Neurons; 2.3 General Framework of NFDEs; 3 Center Manifold Reduction; 3.1 Some Examples of Ordinary Differential Equations. 3.2 Invariant Manifolds of RFDEs3.3 Center Manifold Theorem; 3.4 Calculation of Center Manifolds; 3.4.1 The Hopf Case; 3.4.2 The Fold-Hopf Case; 3.4.3 The Double Hopf Case; 3.5 Center Manifolds with Parameters; 3.6 Preservation of Symmetry; 4 Normal Form Theory; 4.1 Introduction; 4.2 Unperturbed Vector Fields; 4.2.1 The Poincaré-Birkhoff Normal Form Theorem; 4.2.2 Computation of Normal Forms; 4.2.2.1 The Matrix Method; 4.2.2.2 The Adjoint Operator Method; 4.2.3 Internal Symmetry; 4.3 Perturbed Vector Fields; 4.3.1 Normal Form for Hopf Bifurcation; 4.3.2 Norm Form Theorem. 4.3.3 Preservation of External Symmetry4.4 RFDEs with Symmetry; 4.4.1 Basic Assumptions; 4.4.2 Computation of Symmetric Normal Forms; 4.4.3 Nonresonance Conditions; 5 Lyapunov-Schmidt Reduction; 5.1 The Lyapunov-Schmidt Method; 5.2 Derivatives of the Bifurcation Equation; 5.3 Equivariant Equations; 5.4 The Steady-State Equivariant Branching Lemma; 5.5 Generalized Hopf Bifurcation of RFDE; 5.6 Equivariant Hopf Bifurcation of NFDEs; 5.7 Application to a Delayed van der Pol Oscillator; 5.8 Applications to a Ring Network; 5.9 Coupled Systems of NFDEs and Lossless Transmission Lines. 5.10 Wave Trains in the FPU Lattice6 Degree Theory; 6.1 Introduction; 6.2 The Brouwer Degree; 6.3 The Leray-Schauder Degree; 6.4 Global Bifurcation Theorem; 6.5 S1-Equivariant Degree; 6.5.1 Differentiability Case; 6.5.2 Nondifferentiability Case; 6.6 Global Hopf Bifurcation Theory of DDEs; 6.7 Application to a Delayed Nicholson Blowflies Equation; 6.7.1 The Nicholson Blowflies Equation; 6.7.2 The Global Hopf Bifurcation Theorem of Wei-Li; 6.7.3 Nicholson's Blowflies Equation Revisited: Onset and Termination of Nonlinear Oscillations; 6.8 Rotating Waves and Circulant Matrices.
Series Title: Applied mathematical sciences (Springer-Verlag New York Inc.)
Responsibility: Shangjiang Guo, Jianhong Wu.

Abstract:

This book summarizes effective and general approaches and frameworks in the investigation of bifurcation phenomena for functional differential equations (FDEs). It provides all the tools from  Read more...

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"The book contains a comprehensive list of references to the subject and can be particularly helpful for readers who are interested in the mathematical details that arise in the study of bifurcations Read more...

 
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