Numerical methods for bifurcations of dynamical equilibria /

Numerical methods for bifurcations of dynamical equilibria /

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Bibliographic Details
Author / Creator:Govaerts, Willy J. F.
Imprint:Philadelphia, Pa. : Society for Industrial and Applied Mathematics, ©2000.
Description:xxii, 362 pages : illustrations ; 26 cm
Language:English
Subject:
Differentiable dynamical systems.
Differential equations -- Numerical solutions.
Bifurcation theory.
Bifurcation theory.
Differentiable dynamical systems.
Differential equations -- Numerical solutions.
Format: E-Resource Book
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/12577321
Hidden Bibliographic Details
ISBN:0898714427
9780898714425
9780898719543
0898719542
Notes:Includes bibliographical references and index.
Access is restricted to subscribing institutions.
Also available in print version.
Summary:Dynamical systems arise in all fields of applied mathematics. The author focuses on the description of numerical methods for the detection, computation, and continuation of equilibria and bifurcation points of equilibria of dynamical systems. This subfield has the particular attraction of having links with the geometric theory of differential equations, numerical analysis, and linear algebra.
Other form:Print version: Govaerts, Willy J.F. Numerical methods for bifurcations of dynamical equilibria. Philadelphia, Pa. : Society for Industrial and Applied Mathematics, ©2000 0898714427
Publisher's no.:OT66 SIAM
Table of Contents:
  • Preface
  • Notation
  • Introduction
  • 1. Examples and Motivation
  • 1.1. Nonlinear Equations and Dynamical Systems
  • 1.2. Examples from Population Dynamics
  • 1.2.1. Stable and Unstable Equilibria
  • 1.2.2. A Set of Bifurcation Points
  • 1.2.3. A Cusp Catastrophe
  • 1.2.4. A Hopf Bifurcation
  • 1.3. An Example from Combustion Theory
  • 1.3.1. Finite Element Discretization
  • 1.3.2. Finite Difference Discretization
  • 1.3.3. Numerical Continuation: Motivation by an Example
  • 1.4. An Example of Symmetry Breaking
  • 1.5. Linear and Nonlinear Stability
  • 1.6. Exercises
  • 2. Manifolds and Numerical Continuation
  • 2.1. Manifolds
  • 2.1.1. Definitions
  • 2.1.2. The Tangent Space
  • 2.1.3. Examples
  • 2.2. Branches and Limit Points
  • 2.3. Numerical Continuation
  • 2.3.1. Natural Parameterization
  • 2.3.2. Pseudoarclength Continuation
  • 2.3.3. Steplength Control
  • 2.3.4. Convergence of Newton Iterates
  • 2.3.5. Some Practical Considerations
  • 2.4. Notes and Further Reading
  • 2.5. Exercises
  • 3. Bordered Matrices
  • 3.1. Introduction: Motivation by Cramer's Rule
  • 3.2. The Construction of Nonsingular Bordered Matrices
  • 3.3. The Singular Value Inequality
  • 3.4. The Schur Inverse as Defining System for Rank Deficiency
  • 3.5. Invariant Subspaces of Parameter-Dependent Matrices
  • 3.6. Numerical Methods for Bordered Linear Systems
  • 3.6.1. Backward Stability
  • 3.6.2. Algorithm BEM for One-Bordered Systems
  • 3.6.3. Algorithm BEMW for Wider-Bordered Systems
  • 3.7. Notes and Further Reading
  • 3.8. Exercises
  • 4. Generic Equilibrium Bifurcations in One-Parameter Problems
  • 4.1. Limit Points
  • 4.1.1. The Moore-Spence System for Quadratic Turning Points
  • 4.1.2. Quadratic Turning Points by Direct Bordering Methods
  • 4.1.3. Detection of Quadratic Turning Points
  • 4.1.4. Continuation of Limit Points
  • 4.2. Example: A One-Dimensional Continuous Brusselator
  • 4.2.1. The Model and Its Discretization
  • 4.2.2. Turning Points in the Brusselator Model
  • 4.3. Classical Methods for the Computation of Hopf Points
  • 4.3.1. Hopf Points
  • 4.3.2. Regular Systems with 3N + 2 Equations
  • 4.3.3. Regular Systems with 2N + 2 Equations
  • 4.3.4. Regular Systems with N + 2 Equations
  • 4.3.5. Zero-Sum Pairs of Real Eigenvalues
  • 4.3.6. Hopf Points by Complex Arithmetic
  • 4.4. Tensor Products and Bialternate Products
  • 4.4.1. Tensor Products
  • 4.4.2. Condensed Tensor Products
  • 4.4.3. The Natural Involution in C[superscript n] [times] C[superscript n]
  • 4.4.4. The Bialternate Product of Matrices
  • 4.4.5. The Jordan Structure of the Bialternate Product Matrix
  • 4.5. Hopf Points with Bialternate Product Methods
  • 4.5.1. Reconstruction of the Eigenstructure
  • 4.5.2. Double Borders and Detection of Double Hopf Points
  • 4.6. Computation of Hopf Points: Examples
  • 4.6.1. Zero-Sum Pairs of Eigenvalues in the Catalytic Oscillator Model
  • 4.6.2. The Clamped Hodgkin-Huxley Equations
  • 4.6.3. Discretization and Generalized Eigenvalue Problems
  • 4.7. Notes and Further Reading
  • 4.8. Exercises
  • 5. Bifurcations Determined by the Jordan Form of the Jacobian
  • 5.1. Bogdanov-Takens Points and Their Generalizations
  • 5.1.1. Introduction
  • 5.1.2. Numerical Computation of BT Points
  • 5.1.3. Local Analysis of BT Matrices
  • 5.1.4. Transversality and Genericity
  • 5.1.5. Test Functions for BT Points
  • 5.1.6. Example: A Curve of BT Points in the Catalytic Oscillator Model
  • 5.2. ZH Points and Their Generalizations
  • 5.2.1. Transversality and Genericity for Simple Hopf
  • 5.2.2. Transversality and Genericity for ZH
  • 5.2.3. Detection of ZH Points
  • 5.3. DH Points and Resonant DH Points
  • 5.3.1. Introduction
  • 5.3.2. Defining Functions for Multiple Hopf Points
  • 5.3.3. Branch Switching at a DH Point
  • 5.3.4. Resonant DH Points
  • 5.3.5. The Stratified Set of Hopf Points Near a Point with One-to-One Resonance
  • 5.4. Example: The Lateral Pyloric Neuron
  • 5.5. Notes and Further Reading
  • 5.6. Exercises
  • 6. Singularity Theory
  • 6.1. Contact Equivalence of Nonlinear Mappings
  • 6.2. The Numerical Lyapunov-Schmidt Reduction
  • 6.3. Classification of Singularities by Codimension
  • 6.3.1. Introduction and Basic Properties
  • 6.3.2. Singularities from R into R
  • 6.3.3. Singularities from R[superscript 2] into R
  • 6.3.4. Singularities from R[superscript 2] into R[superscript 2]
  • 6.3.5. A Table of k-Singularities
  • 6.3.6. Example: Intersection of a Surface with Its Tangent Plane
  • 6.3.7. Example: A Point on a Rolling Wheel
  • 6.4. Unfolding Theory
  • 6.5. Example: The Continuous Flow Stirred Tank Reactor
  • 6.5.1. Description of the Model
  • 6.5.2. Numerical Computation of a Cusp Point
  • 6.5.3. The Universal Unfolding of a Cusp Point
  • 6.5.4. Example: Unfolding a Cusp in the CSTR
  • 6.5.5. Pairs of Nondegeneracy Conditions: An Example
  • 6.6. Numerical Methods for k-Singularities
  • 6.6.1. The Codimension-1 Singularity from R into R
  • 6.6.2. Singularities from R into R with Codimension Higher than 1
  • 6.6.3. Singularities from R[superscript 2] into R
  • 6.6.4. Singularities from R[superscript 2] into R[superscript 2]
  • 6.7. Notes and Further Reading
  • 6.8. Exercises
  • 7. Singularity Theory with a Distinguished Bifurcation Parameter
  • 7.1. Singularities with a Distinguished Bifurcation Parameter
  • 7.2. Classification of ([lambda] - [kappa])-Singularities from R into R
  • 7.3. Classification of ([lambda] - [kappa])-Singularities from R[superscript 2] into R[superscript 2]
  • 7.4. Numerical Methods for ([lambda] - [kappa])-Singularities
  • 7.4.1. Numerical Methods for ([lambda] - [kappa])-Singularities with Corank 1
  • 7.4.2. Numerical Methods for ([lambda] - [kappa])-Singularities with Corank 2
  • 7.5. Interpretation of Simple Singularities with Corank 1
  • 7.6. Examples in Low-Dimensional Spaces
  • 7.6.1. Winged Cusps in the CSTR
  • 7.6.2. An Eutrophication Model
  • 7.7. Example: The One-Dimensional Brusselator
  • 7.7.1. Computational Study of a Curve of Equilibria
  • 7.7.2. Computational Study of a Curve of Turning Points
  • 7.7.3. Computational Study of a Curve of Hysteresis Points
  • 7.7.4. Computational Study of a Curve of Transcritical Bifurcation Points
  • 7.7.5. A Winged Cusp on a Curve of Pitchfork Bifurcations
  • 7.7.6. A Degenerate Pitchfork on a Curve of Pitchfork Bifurcations
  • 7.7.7. Computation of Branches of Cusp Points and Quartic Turning Points
  • 7.8. Numerical Branching
  • 7.8.1. Simple Bifurcation Point and Isola Center
  • 7.8.2. Cusp Points in [kappa]-Singularity Theory
  • 7.8.3. Transcritical and Pitchfork Bifurcations in ([lambda] - [kappa])-Singularity Theory
  • 7.8.4. Branching Point on a Curve of Equilibria
  • 7.9. Exercises
  • 8. Symmetry-Breaking Bifurcations
  • 8.1. The Z[subscript 2]-Case: Corank 1 and Symmetry Breaking
  • 8.1.1. Basic Results on Z[subscript 2]-Equivariance
  • 8.1.2. Symmetry Breaking on a Branch of Equilibria: Generic Scenario
  • 8.1.3. The Lyapunov-Schmidt Reduction with Symmetry-Adapted Bordering
  • 8.1.4. The Classification of Z[subscript 2]-Equivariant Germs
  • 8.1.5. Numerical Detection, Computation, and Continuation
  • 8.1.6. Branching and Numerical Study of a Nonsymmetric Branch
  • 8.2. The Z[subscript 2]-Case: Corank 2 and Mode Interaction
  • 8.2.1. Numerical Example: A Corank-2 Point on a Curve of Turning Points
  • 8.2.2. Continuation of Turning Points by Double Bordering
  • 8.2.3. The Z[subscript 2]-Equivariant Reduction by a Symmetry-Adapted Double Bordering
  • 8.2.4. Computation of a Corank-2 Point
  • 8.2.5. Analysis and Computation of the Singularity Properties of a Corank-2 Point
  • 8.2.6. The Z[subscript 2]-Equivariant Classification of Corank-2 Points with Distinguished Bifurcation Parameter
  • 8.3. Rank Drop on a Curve of Singular Points
  • 8.3.1. Corank-1 Singularities in Two State Variable
  • 8.3.2. The Case of a Symmetry-Adapted Bordering
  • 8.3.3. Numerical Example: A Corank-2 Point on a Curve of Cusps
  • 8.4. Other Symmetry Groups
  • 8.4.1. Symmetry-Adapted Bases
  • 8.4.2. The Equivariant Branching Lemma
  • 8.4.3. Example: A System with D[subscript 4]-Symmetry
  • 8.4.4. Numerical Implementation
  • 8.5. Notes and Further Reading
  • 8.6. Exercises
  • 9. Bifurcations with Degeneracies in the Nonlinear Terms
  • 9.1. Principles of Center Manifold Theory
  • 9.1.1. The Homological Equation for Dynamics in the Center Manifold
  • 9.1.2. Normal Form Results
  • 9.1.3. General Remarks on the Computation
  • 9.2. Computation of CPs
  • 9.2.1. The Manifold
  • 9.2.2. A Minimally Extended Defining System
  • 9.2.3. A Large Defining System
  • 9.3. Computation of GH Points
  • 9.3.1. The Manifold
  • 9.3.2. A Minimally Extended Defining System
  • 9.3.3. A Large Defining System
  • 9.4. Examples
  • 9.4.1. A Turning Point of Periodic Orbits in the Hodgkin-Huxley Model
  • 9.4.2. Bifurcations with High Codimension in the LP-Neuron Model
  • 9.4.3. Dynamics of Corruption in Democratic Societies
  • 9.5. Notes and Further Reading
  • 9.6. Exercises
  • 10. An Introduction to Large Dynamical Systems
  • 10.1. A Class of One-Dimensional PDEs
  • 10.1.1. Space Discretization
  • 10.1.2. Integration by Crank-Nicolson
  • 10.1.3. B-stability and the Implicit Midpoint Rule
  • 10.1.4. Numerical Continuation
  • 10.1.5. Solution of Linear Systems
  • 10.1.6. Example: The Nonadiabatic Tubular Reactor
  • 10.2. Bifurcations: Reduction to a Low-Dimensional State Space
  • 10.3. Notes and Further Reading
  • 10.4. Exercises
  • Bibliography
  • Index

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