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Hirsch, Devaney, and Smale's classic Differential Equations, Dynamical Systems, and an Introduction to Chaos has been used by professors as the primary text for undergraduate and graduate level courses covering differential equations. It provides a theoretical approach to dynamical systems and chaos written for a diverse student population among the fields of mathematics, science, and engineering. Prominent experts provide everything students need to know about dynamical systems as students seek to develop sufficient mathematical skills to analyze the types of differential equations that arise in their area of study. The authors provide rigorous exercises and examples clearly and easily by slowly introducing linear systems of differential equations. Calculus is required as specialized advanced topics not usually found in elementary differential equations courses are included, such as exploring the world of discrete dynamical systems and describing chaotic systems. Classic text by three of the world's most prominent mathematicians Continues the tradition of expository excellence Contains updated material and expanded applications for use in applied studies Cover......Page 1 Differential Equations, Dynamical Systems, and an Introduction to Chaos......Page 2 Copyright......Page 3 Table of Contents......Page 4 Preface to Third Edition......Page 10 Preface......Page 12 1.1 The Simplest Example......Page 16 1.2 The Logistic Population Model......Page 19 1.3 Constant Harvesting and Bifurcations......Page 22 1.4 Periodic Harvesting and Periodic Solutions......Page 25 1.5 Computing the Poincaré Map......Page 26 1.6 Exploration: A Two-Parameter Family......Page 30 Exercises......Page 31 2 Planar Linear Systems......Page 36 2.1 Second-Order Differential Equations......Page 38 2.2 Planar Systems......Page 39 2.3 Preliminaries from Algebra......Page 41 2.4 Planar Linear Systems......Page 44 2.5 Eigenvalues and Eigenvectors......Page 45 2.6 Solving Linear Systems......Page 48 2.7 The Linearity Principle......Page 51 Exercises......Page 52 3.1 Real Distinct Eigenvalues......Page 54 3.2 Complex Eigenvalues......Page 59 3.3 Repeated Eigenvalues......Page 62 3.4 Changing Coordinates......Page 64 Exercises......Page 72 4.1 The Trace–Determinant Plane......Page 76 4.2 Dynamical Classification......Page 79 Case 2......Page 82 Case 3......Page 85 Exercises......Page 86 5.1 Preliminaries from Linear Algebra......Page 88 5.2 Eigenvalues and Eigenvectors......Page 97 5.3 Complex Eigenvalues......Page 100 5.4 Bases and Subspaces......Page 103 5.5 Repeated Eigenvalues......Page 108 5.6 Genericity......Page 115 Exercises......Page 118 6.1 Distinct Eigenvalues......Page 122 6.2 Harmonic Oscillators......Page 129 6.3 Repeated Eigenvalues......Page 135 6.4 The Exponential of a Matrix......Page 138 6.5 Nonautonomous Linear Systems......Page 145 Exercises......Page 150 7 Nonlinear Systems......Page 154 7.1 Dynamical Systems......Page 155 7.2 The Existence and Uniqueness Theorem......Page 157 7.3 Continuous Dependence of Solutions......Page 162 7.4 The Variational Equation......Page 164 7.5 Exploration: Numerical Methods......Page 168 7.6 Exploration: Numerical Methods and Chaos......Page 171 Exercises......Page 172 8.1 Some Illustrative Examples......Page 174 8.2 Nonlinear Sinks and Sources......Page 180 8.3 Saddles......Page 183 8.4 Stability......Page 189 8.5 Bifurcations......Page 190 8.6 Exploration: Complex Vector Fields......Page 197 Exercises......Page 199 9.1 Nullclines......Page 202 9.2 Stability of Equilibria......Page 207 9.3 Gradient Systems......Page 217 9.4 Hamiltonian Systems......Page 221 9.5 Exploration: The Pendulum with Constant Forcing......Page 224 Exercises......Page 225 10.1 Limit Sets......Page 228 10.2 Local Sections and Flow Boxes......Page 231 10.3 The Poincaré Map......Page 233 10.4 Monotone Sequences in Planar Dynamical Systems......Page 235 10.5 The Poincaré–Bendixson Theorem......Page 237 10.6 Applications of Poincaré–Bendixson......Page 240 10.7 Exploration: Chemical Reactions that Oscillate......Page 243 Exercises......Page 244 11.1 Infectious Diseases......Page 248 11.2 Predator–Prey Systems......Page 252 11.3 Competitive Species......Page 259 11.4 Exploration: Competition and Harvesting......Page 265 11.5 Exploration: Adding Zombies to the SIR Model......Page 266 Exercises......Page 267 12.1 An RLC Circuit......Page 272 12.2 The Liénard Equation......Page 276 12.3 The van der Pol Equation......Page 278 12.4 A Hopf Bifurcation......Page 285 12.5 Exploration: Neurodynamics......Page 287 Exercises......Page 288 13.1 Newton's Second Law......Page 292 13.2 Conservative Systems......Page 295 13.3 Central Force Fields......Page 297 13.4 The Newtonian Central Force System......Page 300 13.5 Kepler's First Law......Page 305 13.6 The Two-Body Problem......Page 308 13.7 Blowing up the Singularity......Page 309 13.8 Exploration: Other Central Force Problems......Page 313 13.9 Exploration: Classical Limits of Quantum Mechanical Systems......Page 314 Exercises......Page 316 14 The Lorenz System......Page 320 14.1 Introduction......Page 321 14.2 Elementary Properties of the Lorenz System......Page 323 14.3 The Lorenz Attractor......Page 327 14.4 A Model for the Lorenz Attractor......Page 331 14.5 The Chaotic Attractor......Page 336 14.6 Exploration: The Rössler Attractor......Page 341 Exercises......Page 342 15.1 Introduction......Page 344 15.2 Bifurcations......Page 349 15.3 The Discrete Logistic Model......Page 352 15.4 Chaos......Page 355 15.5 Symbolic Dynamics......Page 359 15.6 The Shift Map......Page 364 15.7 The Cantor Middle-Thirds Set......Page 366 15.8 Exploration: Cubic Chaos......Page 369 15.9 Exploration: The Orbit Diagram......Page 370 Exercises......Page 371 16.1 The Shilnikov System......Page 376 16.2 The Horseshoe Map......Page 383 16.3 The Double Scroll Attractor......Page 390 16.4 Homoclinic Bifurcations......Page 392 16.5 Exploration: The Chua Circuit......Page 396 Exercises......Page 398 17.1 The Existence and Uniqueness Theorem......Page 400 17.2 Proof of Existence and Uniqueness......Page 402 17.3 Continuous Dependence on Initial Conditions......Page 409 17.4 Extending Solutions......Page 412 17.5 Nonautonomous Systems......Page 416 17.6 Differentiability of the Flow......Page 419 Exercises......Page 422 Bibliography......Page 426 C......Page 430 I......Page 431 R......Page 432 Z......Page 433 Differential Equations, Dynamical Systems, and an Introduction to Chaos, Third Edition (2013) 433pp. 978-0-12-382010-5 Cover 1 Differential Equations, Dynamical Systems, and an Introduction to Chaos 2 Copyright 3 Table of Contents 4 Preface to Third Edition 10 Preface 12 1 First-Order Equations 16 1.1 The Simplest Example 16 1.2 The Logistic Population Model 19 1.3 Constant Harvesting and Bifurcations 22 1.4 Periodic Harvesting and Periodic Solutions 25 1.5 Computing the Poincaré Map 26 1.6 Exploration: A Two-Parameter Family 30 Exercises 31 2 Planar Linear Systems 36 2.1 Second-Order Differential Equations 38 2.2 Planar Systems 39 2.3 Preliminaries from Algebra 41 2.4 Planar Linear Systems 44 2.5 Eigenvalues and Eigenvectors 45 2.6 Solving Linear Systems 48 2.7 The Linearity Principle 51 Exercises 52 3 Phase Portraits for Planar Systems 54 3.1 Real Distinct Eigenvalues 54 3.2 Complex Eigenvalues 59 3.3 Repeated Eigenvalues 62 3.4 Changing Coordinates 64 Exercises 72 4 Classification of Planar Systems 76 4.1 The Trace–Determinant Plane 76 4.2 Dynamical Classification 79 Case 1 82 Case 2 82 Case 3 85 4.3 Exploration: A 3D Parameter Space 86 Exercises 86 5 Higher-Dimensional Linear Algebra 88 5.1 Preliminaries from Linear Algebra 88 5.2 Eigenvalues and Eigenvectors 97 5.3 Complex Eigenvalues 100 5.4 Bases and Subspaces 103 5.5 Repeated Eigenvalues 108 5.6 Genericity 115 Exercises 118 6 Higher-Dimensional Linear Systems 122 6.1 Distinct Eigenvalues 122 6.2 Harmonic Oscillators 129 6.3 Repeated Eigenvalues 135 6.4 The Exponential of a Matrix 138 6.5 Nonautonomous Linear Systems 145 Exercises 150 7 Nonlinear Systems 154 7.1 Dynamical Systems 155 7.2 The Existence and Uniqueness Theorem 157 7.3 Continuous Dependence of Solutions 162 7.4 The Variational Equation 164 7.5 Exploration: Numerical Methods 168 7.6 Exploration: Numerical Methods and Chaos 171 Exercises 172 8 Equilibria in Nonlinear Systems 174 8.1 Some Illustrative Examples 174 8.2 Nonlinear Sinks and Sources 180 8.3 Saddles 183 8.4 Stability 189 8.5 Bifurcations 190 8.6 Exploration: Complex Vector Fields 197 Exercises 199 9 Global Nonlinear Techniques 202 9.1 Nullclines 202 9.2 Stability of Equilibria 207 9.3 Gradient Systems 217 9.4 Hamiltonian Systems 221 9.5 Exploration: The Pendulum with Constant Forcing 224 Exercises 225 10 Closed Orbits and Limit Sets 228 10.1 Limit Sets 228 10.2 Local Sections and Flow Boxes 231 10.3 The Poincaré Map 233 10.4 Monotone Sequences in Planar Dynamical Systems 235 10.5 The Poincaré–Bendixson Theorem 237 10.6 Applications of Poincaré–Bendixson 240 10.7 Exploration: Chemical Reactions that Oscillate 243 Exercises 244 11 Applications in Biology 248 11.1 Infectious Diseases 248 11.2 Predator–Prey Systems 252 11.3 Competitive Species 259 11.4 Exploration: Competition and Harvesting 265 11.5 Exploration: Adding Zombies to the SIR Model 266 Exercises 267 12 Applications in Circuit Theory 272 12.1 An RLC Circuit 272 12.2 The Liénard Equation 276 12.3 The van der Pol Equation 278 12.4 A Hopf Bifurcation 285 12.5 Exploration: Neurodynamics 287 Exercises 288 13 Applications in Mechanics 292 13.1 Newton's Second Law 292 13.2 Conservative Systems 295 13.3 Central Force Fields 297 13.4 The Newtonian Central Force System 300 13.5 Kepler's First Law 305 13.6 The Two-Body Problem 308 13.7 Blowing up the Singularity 309 13.8 Exploration: Other Central Force Problems 313 13.9 Exploration: Classical Limits of Quantum Mechanical Systems 314 13.10 Exploration: Motion of a Glider 316 Exercises 316 14 The Lorenz System 320 14.1 Introduction 321 14.2 Elementary Properties of the Lorenz System 323 14.3 The Lorenz Attractor 327 14.4 A Model for the Lorenz Attractor 331 14.5 The Chaotic Attractor 336 14.6 Exploration: The Rössler Attractor 341 Exercises 342 15 Discrete Dynamical Systems 344 15.1 Introduction 344 15.2 Bifurcations 349 15.3 The Discrete Logistic Model 352 15.4 Chaos 355 15.5 Symbolic Dynamics 359 15.6 The Shift Map 364 15.7 The Cantor Middle-Thirds Set 366 15.8 Exploration: Cubic Chaos 369 15.9 Exploration: The Orbit Diagram 370 Exercises 371 16 Homoclinic Phenomena 376 16.1 The Shilnikov System 376 16.2 The Horseshoe Map 383 16.3 The Double Scroll Attractor 390 16.4 Homoclinic Bifurcations 392 16.5 Exploration: The Chua Circuit 396 Exercises 398 17 Existence and Uniqueness Revisited 400 17.1 The Existence and Uniqueness Theorem 400 17.2 Proof of Existence and Uniqueness 402 17.3 Continuous Dependence on Initial Conditions 409 17.4 Extending Solutions 412 17.5 Nonautonomous Systems 416 17.6 Differentiability of the Flow 419 Exercises 422 Bibliography 426 Index 430 A 430 B 430 C 430 D 431 E 431 F 431 G 431 H 431 I 431 K 432 L 432 M 432 N 432 O 432 P 432 R 432 S 433 T 433 U 433 V 433 W 433 Z 433

Hirsch, Devaney, and Smale’s classic Differential Equations, Dynamical Systems, and an Introduction to Chaos has been used by professors as the primary text for undergraduate and graduate level courses covering differential equations. It provides a theoretical approach to dynamical systems and chaos written for a diverse student population among the fields of mathematics, science, and engineering. Prominent experts provide everything students need to know about dynamical systems as students seek to develop sufficient mathematical skills to analyze the types of differential equations that arise in their area of study. The authors provide rigorous exercises and examples clearly and easily by slowly introducing linear systems of differential equations. Calculus is required as specialized advanced topics not usually found in elementary differential equations courses are included, such as exploring the world of discrete dynamical systems and describing chaotic systems.

• Classic text by three of the world’s most prominent mathematicians
• Continues the tradition of expository excellence
• Contains updated material and expanded applications for use in applied studies

Pour la première fois, François, Mick et Annie vont passer les grandes vacances chez leur tante à Kernach. Ils y font la connaissance de leur cousine Claude, un vrai garçon manqué. Habituée à la solitude, elle est d'abord très distante vis-à-vis d'eux. Mais les quatre enfants ne tardent pas à devenir inséparables. Accompagnés du fidèle chien Dagobert, ils partent à la découverte du trésor qu'indique une vieille carte trouvée sur l'île de Kernach. Ils doivent cependant décoder les indices au plus vite, car ils ne sont pas les seuls à rechercher le trésor... [Payot.ch] "Differential Equations, Dynamical Systems, and an Introduction to Chaos, now in its third edition, covers the dynamical aspects of ordinary differential equations. It explores the relations between dynamical systems and certain fields outside pure mathematics, and continues to be the standard textbook for advanced undergraduate and graduate courses in this area.""Written for students with a background in calculus and elementary linear algebra, the text is rigorous yet accessible and contains examples and explorations to reinforce learning."--Back cover Julian, Dick, Anne, George and Timmy the dog find excitement and adventure wherever they go in Enid Blyton's most popular series. In their first adventure, the Famous Five find a shipwreck off Kirrin Island. But where is the treasure? The Famous Five are on the trail, looking for clues, but they're not alone. Someone else has got the same idea! Time is running out for the Famous Five -- who will follow the clues and get to the treasure first?