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 : Society for Industrial and Applied Mathematics, c2000.
Description:xxii, 362 p. : ill. ; 26 cm.
Language:English
Subject:
Differentiable dynamical systems.
Differential equations -- Numerical solutions.
Bifurcation theory.
Bifurcation theory.
Differentiable dynamical systems.
Differential equations -- Numerical solutions.
Format: Print Book
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/4214274
Hidden Bibliographic Details
ISBN:0898714427 (pbk.)
Notes:Includes bibliographical references (p. 343-358) and index.
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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