《实用分歧和稳定分析》求取 ⇩

1Introduction and Prerequisites1

1.1 A Nonmathematical Introduction1

1.2Stationary Points and Stability(ODEs)6

1.2.1 Trajectories and Equilibria6

1.2.2 Deviations7

1.2.3 Stability9

1.2.4 Linear Stability;Duffing Equation11

1.2.5 Degenerate Cases;Parameter Dependence18

1.2.6 Generalizations20

1.3 Limit Cycles22

1.4 Waves27

1.5Maps31

1.5.1 Occurrence of Maps32

1.5.2 Stability of Fixed Points33

1.5.3 Cellular Automata34

1.6Some Fundamental Numerical Methods36

1.6.1 Newton's Method37

1.6.2 Integration of ODEs40

1.6.3 Calculating Eigenvalues41

1.6.4 ODE Boundary-Value Problems42

1.6.5 Further Tools43

2Basic Nonlinear Phenomena45

2.1 A Preparatory Example45

2.2 Elementary Definitions48

2.3 Buckling and Oscillation of a Bean50

2.4 Turning Points and Bifurcation Points:The Geometric View54

2.5 Turning Points and Bifurcation Points:The Algebraic View62

2.6 Hopf Bifurcation68

2.7 Bifurcation of Periodic Orbits75

2.8 Convection Described by Lorenz's Equation78

2.9 Hopf Bifurcation and Stability86

2.10 Generic Branching93

2.11 Bifurcation in the Presence of Symmetry104

3Practical Problems109

3.1 Readily Available Tools and Limited Results109

3.2 Principal Tasks110

3.3 What Else Can Happen113

3.4 Marangoni Convection116

3.5 The Art and Science of Parameter Study120

4Principles of Continuation125

4.1 Ingredients of Predictor-Corrector Methods126

4.2 Homotopy127

4.3Predictors129

4.3.1 ODE Methods;Tangent Predictor129

4.3.2 Polynomial Extrapolation;Secant Predictor131

4.4Parameterizations133

4.4.1 Parameterization by Adding an Equation133

4.4.2 Arclength and Pseudo Arclength135

4.4.3 Local Parameterization135

4.5 Correctors137

4.6 Step Controls141

4.7 Practical Aspects144

5Calculation of the Branching Behavior of Nonlinear Equations147

5.1 Calculating Stability147

5.2 Branching Test Functions151

5.3 Indirect Methods for Calculating Branch Points156

5.4Direct Methods for Calculating Branch Points162

5.4.1 The Branching System163

5.4.2 An Electrical Circuit168

5.4.3 A Family of Test Functions171

5.4.4 Direct Versus Indirect Methods172

5.5Branch Switching178

5.5.1 Constructing a Predictor via the Tangent178

5.5.2 Predictors Based on Interpolation182

5.5.3 Correctors with Selective Properties184

5.5.4 Symmetry Breaking187

5.5.5 Coupled Cell Reaction188

5.5.6 Parameterization by Irregularity192

5.5.7 Other Methods193

5.6Methods for Calculating Specific Branch Points196

5.6.1 A Special Implementation for the Branching System197

5.6.2 Regular Systems for Bifurcation Points199

5.6.3 Methods for Turning Points200

5.6.4 Methods for Hopf Bifurcation Points201

5.6.5 Other Methods202

5.7 Concluding Remarks202

5.8 Two-Parameter Problems203

6Calculating Branching Behavior of Boundary-Value Problems209

6.1 Enlarged Boundary-Value Problems210

6.2 Calculation of Branch Points218

6.3 Stepping Down for an Implementation224

6.4 Branch Switching and Symmetry225

6.5 Trivial Bifurcation233

6.6 Testing Stability237

6.7 Hopf Bifurcation in PDEs241

6.8 Heteroclinic Orbits245

7Stability of Periodic Solutions249

7.1 Periodic Solutions of Autonomous Systems250

7.2 The Monodromy Matrix253

7.3 The Poincaré Map256

7.4Mechanisms of Losing Stability261

7.4.1 Branch Points of Periodic Solutions262

7.4.2 Period Doubling267

7.4.3 Bifurcation into Torus274

7.5Calculating the Monodromy Matrix279

7.5.1 A Posteriori Calculation279

7.5.2 Monodromy Matrix as a By-Product of Shooting281

7.5.3 Numerical Aspects282

7.6 Calculating Branching Behavior283

7.7 Phase Locking290

7.8 Further Examples and Phenomena295

8Qualitative Instruments299

8.1 Significance299

8.2 Construction of Nornal Forms300

8.3 A Program Toward a Classification303

8.4 Singularity Theory for One Scalar Equation305

8.5The Elementary Catastrophes314

8.5.1 The Fold315

8.5.2 The Cusp315

8.5.3 The Swallowtail316

8.6 Zeroth-Order Reaction in a CSTR319

8.7 Center Manifolds322

9Chaos327

9.1 Flows and Attractors328

9.2 Examples of Strange Attractors335

9.3Routes to Chaos338

9.3.1 Route via Torus Bifurcation338

9.3.2 Period-Doubling Route339

9.3.3 Intermittency339

9.4 Phase Space Construction340

9.5 Fractal Dimensions342

9.6Liapunov Exponents346

9.6.1 Liapunov Exponents for Maps346

9.6.2 Liapunov Exponents for ODEs347

9.6.3 Characterization of Attractors350

9.6.4 Computation of Liapunov Exponents351

9.6.5 Liapunov Exponents of Time Series353

9.7 Power Spectra355

A．Appendices359

A.1Some Basic Glossary359

A.2 Some Basic Facts from Linear Algebra360

A.3 Some Elementary Facts from ODEs362

A.4 Inplicit Function Theorem364

A.5 Special Invariant Manifolds365

A.6 Numerical Integration of ODEs366

A.7 Symmetry Groups368

A.8 Numerical Software and Packages369

List of Major Examples371

References373

Index395