FROM EQUILIBRIUM TO CHAOS

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1 FROM EQUILIBRIUM TO CHAOS Practica! Bifurcation and Stability Analysis RÜDIGER SEYDEL Institut für Angewandte Mathematik und Statistik University of Würzburg Würzburg, Federal Republic of Germany ELSEVIER New York Amsterdam London

2 Contents Preface Notation ix xüi 1 Introduction and Prerequisites A Nonmathematical Introduction Fundamentals of Stationary Points and Stability (ODEs) Trajectories and Equilibria Deviations Stability Linear Stability; Duffing Equation Degenerate Cases, Parameter Dependence Generalizations Fundamentals of Limit Cycles and Waves Van der Pol Equation Waves Some Fundamental Numerical Methods 29 2 Basic Nonlinear Phenomena A Preparatory Example Elementary Definitions Bückling and Oscillation of a Beam Turning Points and Bifurcation Points: The Geometrie View Tuming Points and Bifurcation Points: The Algebraic View 52 ; iii

3 IV Contents 2.6 Hopf Bifurcation Convection Described by Lorenz's Equation Hopf Bifurcation and Stability Generic Branching Bifurcation in the Presence of Symmetry 91 3 Practical Problems Readily Available Tools and Limited Results Principal Tasks What Else Can Happen Marangoni Convection The Art and Science of Parameter Study Principles of Continuation Predictors ODE-Methods; Tangent Predictor Polynomial Extrapolation; Secant Predictor Parameterizations Parameterization by Adding an Equation Arclength and Pseudo Arclength Local Parameterization Correctors Step Controls Continuation Subject to Constraints More Practical Aspects Calculation of the Branching Behavior of Nonlinear Equations Calculating Stability Branching Test Functions General Methods for Calculating Branch Points Indirect Methods Direct Methods An Electrical Circuit 750

4 5.3.4 A Family of Test Functions Temperature Distribution in a Reacting Material Direct Versus Indirect Methods Branch Switching Constructing a Predictor via the Tangent Predictors Based on Interpolation Correctors with Selective Properties Symmetry Breaking Coupled Cell Reaction Parameterization by Irregularity Other Methods Methods for Calculating Specific Branch Points A Special Implementation for the Branching System Regulär Systems for Bifurcation Points Methods for Turning Points Methods for Hopf Bifurcation Points Other Methods Concluding Remarks Two-Parameter Problems Calculating Branching Behavior of ODE Boundary-Value Problems Enlarged Boundary-Value Problems Calculation of Branch Points Branching System Catalytic Reaction Initial Approximation and Branching Test Function Stepping Down for an Implementation Branch Switching and Symmetry Trivial Bifurcation Testing Stability Hopf Bifurcation in PDEs 229

5 VI Contents 7 Stability of Periodic Solutions Periodic Solutions of Autonomous Systems The Monodromy Matrix The Poincare Map Mechanisms of Losing Stability Branch Points of Periodic Solutions Period Doubling Bifurcation into Torus Calculating the Monodromy Matrix A Posteriori Calculation Monodromy Matrix as By-product of Shooting Numerical Aspects Calculating Branching Behavior Further Examples and Phenomena Qualitative Instruments Significance Singularity Theory for One Scalar Equation The Elementary Catastrophes The Fold (Requires One Parameter, y) The Cusp (Requires Two Parameters, 5, ) The Swallowtail (Requires Three Parameters, y, 8, ) Zeroth-Order Reaction in a CSTR Center Manifolds Chaos Flows and Attractors Examples of Strange Attractors Routes to Chaos Route via Torus Bifurcation Period Doubling Route Intermittency 317

6 9.4 Characterization of Strange Attractors Fractal Dimension Liapunov Exponents Power Spectra 326 Appendixes 329 Appendix 1. Some Elementary Facts from ODEs 329 Appendix 2. Implicit Function Theorem 331 Appendix 3. Some Basic Facts from Linear Algebra 332 Appendix 4. Runge-Kutta-Fehlberg Methods 333 Appendix 5. Transformation into Standard Form 334 Appendix 6. Numerical Software and Packages 336 Appendix 7 Basic Groups 337 References 339 Index 359

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