Differential Equations, Dynamical Systems, and Linear Algebra (eBook)

Robert L. Devaney, Morris W. Hirsch, Stephen Smale (Autoren)

eBook Download: PDF

1974 | 1. Auflage
358 Seiten
Elsevier Science (Verlag)
978-0-08-087376-3 (ISBN)

Infos

This book is about dynamical aspects of ordinary differential equations and the relations between dynamical systems and certain fields outside pure mathematics. A prominent role is played by the structure theory of linear operators on finite-dimensional vector spaces, the authors have included a self-contained treatment of that subject.

Inhalt

Front Cover 1
Differential Equations, Dynamical Systems, and Linear Algebra 4
Copyright Page 5
Contents 6
Preface 10
CHAPTER 1. FIRST EXAMPLES 14
1. The Simplest Examples 14
2. Linear Systems with Constant Coefficients 22
Notes 26
CHAPTER 2. NEWTON'S EQUATION AND KEPLER'S LAW 27
1. Harmonic Oscillators 28
2. Some Calculus Background 29
3. Conservative Force Fields 30
4. Central Force Fields 32
5. States 35
6. Elliptical Planetary Orbits 36
Notes 40
CHAPTER 3. LINEAR SYSTEMS WITH CONSTANT COEFICIENTS AND REAL EIGENVALUES 42
1. Basic Linear Algebra 42
2. Real Eigenvalues 55
3. Differential Equations with Real, Distinct Eigenvalues 60
4. Complex Eigenvalues 68
CHAPTER 4. LINEAR SYSTEMS WITH CONSTANT COEFFICIENTS AND COMPLEX EIGENVALUES 75
1. Complex Vector Spaces 75
2. Real Operators with Complex Eigenvalues 79
3. Application of Complex Linear Algebra to Differential Equations 82
CHAPTER 5. LINEAR SYSTEMS AND EXPONENTIALS OF OPERATORS 87
1. Review of Topology in Rn 88
2. New Norms for Old 90
3. Exponentials of Operators 95
4. Homogeneous Linear Systems 102
5. A Nonhomogeneous Equation 112
6. Higher Order Systems 115
Notes 121
CHAPTER 6. LINEAR SYSTEMS AND CANONICAL FORMS OF OPERATORS 122
1. The Primary Decomposition 123
2. The S + N Decomposition 129
3. Nilpotent Canonical Forms 135
4. Jordan and Real Canonical Forms 139
5. Canonical Forms and Differential Equations 146
6. Higher Order Linear Equations 151
7. Operators on Function Spaces 155
CHAPTER 7. CONTRACTIONS AND GENERIC PROPERTIES OF OPERATORS 157
1. Sinks and Sources 157
2. Hyperbolic Flows 163
3. Generic Properties of Operators 166
4. The Significance of Genericity 171
CHAPTER 8. FUNDAMENTAL THEORY 172
1. Dynamical Systems and Vector Fields 172
2. The Fundamental Theorem 174
3. Existence and Uniqueness 176
4. Continuity of Solutions in Initial Conditions 182
5. On Extending Solutions 184
6. Global Solutions 186
7. The Flow of a Differential Equation 187
Notes 191
CHAPTER 9. STABILITY OF EQUILIBRIA 193
1. Nonlinear Sinks 193
2. Stability 198
3. Liapunov Functions 205
4. Gradient Systems 212
5. Gradients and Inner Products 217
Notes 222
CHAPTER 10. DIFFERENTIAL EQUATIONS FOR ELECTRICAL CIRCUITS 223
1. An RLC Circuit 224
2. Analysis of the Circuit Equations 228
3. Van der Pol's Equation 230
4. Hopf Bifurcation 240
5. More General Circuit Equations 241
Notes 251
CHAPTER 11. THE POINCARÉ-BENDIXSON THEOREM 252
1. Limit Sets 252
2. Local Sections and Flow Boxes 255
3. Monotone Sequences in Planar Dynamical Systems 257
4. The Poincaré-Bendixson Theorem 261
5. Applications of the Poincaré-Bendixson Theorem 263
Notes 267
CHAPTER 12. ECOLOGY 268
1. One Species 268
2. Predator and Prey 271
3. Competing Species 278
Notes 287
CHAPTER 13. PERIODIC ATTRACTORS 289
1. Asymptotic Stability of Closed Orbits 289
2. Discrete Dynamical Systems 291
3. Stability and Closed Orbits 294
CHAPTER 14. CLASSICAL MECHANICS 300
1. The n-Body Problem 300
2. Hamiltonian Mechanics 303
Notes 308
CHAPTER 15. NONAUTONOMOUS EQUATIONS AND DIFFERENTIABILITY OF FLOWS 309
1. Existence, Uniqueness, and Continuity for Nonautonomous Differential Equations 309
2. Differentiability of the Flow of Autonomous Equations 311
CHAPTER 16. PERTURBATION THEORY AND STRUCTURAL STABILITY 317
1. Persistence of Equilibria 317
2. Persistence of Closed Orbits 322
3. Structural Stability 325
AFTERWORD 332
APPENDIX I: ELEMENTARY FACTS 335
1. Set Theoretic Conventions 335
2. Complex Numbers 336
3. Determinants 337
4. Two Propositions on Linear Algebra 338
APPENDIX II: POLYNOMIALS 341
1. The Fundamental Theorem of Algebra 341
APPENDIX III: ON CANONICAL FORMS 344
1. A Decomposition Theorem 344
2. Uniqueness of S and N 346
3. Canonical Forms for Nilpotent Operators 347
APPENDIX IV: THE INVERSE FUNCTION THEOREM 350
REFERENCES 354
ANSWERS TO SELECTED PROBLEMS 356
Subject Index 368
Pure and Applied Mathematics 372

Produkt-Details

• Erscheint lt. Verlag: 28.6.1974
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Technik
• ISBN-10: 0-08-087376-6 / 0080873766
• ISBN-13: 978-0-08-087376-3 / 9780080873763