Nonlinear Dynamics and Chaos with Student Solutions Manual : With Applications to Physics, Biology, Chemistry, and Engineering, Second Edition

"This textbook is aimed at newcomers to nonlinear dynamics and chaos, especially students taking a first course in the subject. The presentation stresses analytical methods, concrete examples, and geometric intuition. The theory is developed systematically, starting with first-order differential equations and their bifurcations, followed by phase plane analysis, limit cycles and their bifurcations, and culminating with the Lorenz equations, chaos, iterated maps, period doubling, renormalization, fractals, and strange attractors. A unique feature of the book is its emphasis on applications. These include mechanical vibrations, lasers, biological rhythms, superconducting circuits, insect outbreaks, chemical oscillators, genetic control systems, chaotic waterwheels, and even a technique for using chaos to send secret messages. In each case, the scientific background is explained at an elementary level and closely integrated with mathematical theory. In the twenty years since the first edition of this book appeared, the ideas and techniques of nonlinear dynamics and chaos have found application to such exciting new fields as systems biology, evolutionary game theory, and sociophysics. This second edition includes new exercises on these cutting-edge developments, on topics as varied as the curiosities of visual perception and the tumultuous love dynamics in Gone With the Wind"--Provided by publisher. Read more... Abstract: An accessible and clearly-written introduction to chaos and nonlinear systems, with plenty of examples, illustrations and applications to science and engineering. Read more... Cover 1 Half Title 2 Title 4 Copyright 5 CONTENTS 6 Preface to the Second Edition 10 Preface to the First Edition 12 1 Overview 16 1.0 Chaos, Fractals, and Dynamics 16 1.1 Capsule History of Dynamics 17 1.2 The Importance of Being Nonlinear 19 1.3 A Dynamical View of the World 24 Part I One-Dimensional Flows 28 2 Flows on the Line 30 2.0 Introduction 30 2.1 A Geometric Way of Thinking 31 2.2 Fixed Points and Stability 33 2.3 Population Growth 36 2.4 Linear Stability Analysis 39 2.5 Existence and Uniqueness 41 2.6 Impossibility of Oscillations 43 2.7 Potentials 45 2.8 Solving Equations on the Computer 47 Exercises for Chapter 2 51 3 Bifurcations 60 3.0 Introduction 60 3.1 Saddle-Node Bifurcation 61 3.2 Transcritical Bifurcation 66 3.3 Laser Threshold 69 3.4 Pitchfork Bifurcation 71 3.5 Overdamped Bead on a Rotating Hoop 77 3.6 Imperfect Bifurcations and Catastrophes 85 3.7 Insect Outbreak 89 Exercises for Chapter 3 95 4 Flows on the Circle 110 4.0 Introduction 110 4.1 Examples and Definitions 110 4.2 Uniform Oscillator 112 4.3 Nonuniform Oscillator 113 4.4 Overdamped Pendulum 118 4.5 Fireflies 120 4.6 Superconducting Josephson Junctions 124 Exercises for Chapter 4 130 Part II Two-Dimensional Flows 138 5 Linear Systems 140 5.0 Introduction 140 5.1 Definitions and Examples 140 5.2 Classification of Linear Systems 146 5.3 Love Affairs 154 Exercises for Chapter 5 157 6 Phase Plane 161 6.0 Introduction 161 6.1 Phase Portraits 161 6.2 Existence, Uniqueness, and Topological Consequences 164 6.3 Fixed Points and Linearization 166 6.4 Rabbits versus Sheep 171 6.5 Conservative Systems 175 6.6 Reversible Systems 179 6.7 Pendulum 183 6.8 Index Theory 189 Exercises for Chapter 6 196 7 Limit Cycles 213 7.0 Introduction 213 7.1 Examples 214 7.2 Ruling Out Closed Orbits 216 7.3 Poincaré−Bendixson Theorem 220 7.4 Liénard Systems 227 7.5 Relaxation Oscillations 228 7.6 Weakly Nonlinear Oscillators 232 Exercises for Chapter 7 245 8 Bifurcations Revisited 259 8.0 Introduction 259 8.1 Saddle-Node, Transcritical, and Pitchfork Bifurcations 259 8.2 Hopf Bifurcations 266 8.3 Oscillating Chemical Reactions 272 8.4 Global Bifurcations of Cycles 279 8.5 Hysteresis in the Driven Pendulum and Josephson Junction 283 8.6 Coupled Oscillators and Quasiperiodicity 295 8.7 Poincaré Maps 300 Exercises for Chapter 8 306 Part III Chaos 326 9 Lorenz Equations 328 9.0 Introduction 328 9.1 A Chaotic Waterwheel 329 9.2 Simple Properties of the Lorenz Equations 338 9.3 Chaos on a Strange Attractor 344 9.4 Lorenz Map 352 9.5 Exploring Parameter Space 356 9.6 Using Chaos to Send Secret Messages 361 Exercises for Chapter 9 367 10 One-Dimensional Maps 374 10.0 Introduction 374 10.1 Fixed Points and Cobwebs 375 10.2 Logistic Map: Numerics 379 10.3 Logistic Map: Analysis 383 10.4 Periodic Windows 387 10.5 Liapunov Exponent 392 10.6 Universality and Experiments 395 10.7 Renormalization 405 Exercises for Chapter 10 413 11 Fractals 424 11.0 Introduction 424 11.1 Countable and Uncountable Sets 425 11.2 Cantor Set 427 11.3 Dimension of Self-Similar Fractals 430 11.4 Box Dimension 435 11.5 Pointwise and Correlation Dimensions 437 Exercises for Chapter 11 442 12 Strange Attractors 448 12.0 Introduction 448 12.1 The Simplest Examples 448 12.2 Hénon Map 454 12.3 Rössler System 459 12.4 Chemical Chaos and Attractor Reconstruction 462 12.5 Forced Double-Well Oscillator 466 Exercises for Chapter 12 473 Answers to Selected Exercises 479 References 489 Author Index 502 Subject Index 506 With Applications to Physics, Biology, Chemistry, and Engineering Content: Cover Half Title Title Copyright CONTENTS Preface to the Second Edition Preface to the First Edition 1 Overview 1.0 Chaos, Fractals, and Dynamics 1.1 Capsule History of Dynamics 1.2 The Importance of Being Nonlinear 1.3 A Dynamical View of the World Part I One-Dimensional Flows 2 Flows on the Line 2.0 Introduction 2.1 A Geometric Way of Thinking 2.2 Fixed Points and Stability 2.3 Population Growth 2.4 Linear Stability Analysis 2.5 Existence and Uniqueness 2.6 Impossibility of Oscillations 2.7 Potentials 2.8 Solving Equations on the Computer Exercises for Chapter 2 3 Bifurcations3.0 Introduction 3.1 Saddle-Node Bifurcation 3.2 Transcritical Bifurcation 3.3 Laser Threshold 3.4 Pitchfork Bifurcation 3.5 Overdamped Bead on a Rotating Hoop 3.6 Imperfect Bifurcations and Catastrophes 3.7 Insect Outbreak Exercises for Chapter 3 4 Flows on the Circle 4.0 Introduction 4.1 Examples and Definitions 4.2 Uniform Oscillator 4.3 Nonuniform Oscillator 4.4 Overdamped Pendulum 4.5 Fireflies 4.6 Superconducting Josephson Junctions Exercises for Chapter 4 Part II Two-Dimensional Flows 5 Linear Systems 5.0 Introduction 5.1 Definitions and Examples 5.2 Classification of Linear Systems5.3 Love Affairs Exercises for Chapter 5 6 Phase Plane 6.0 Introduction 6.1 Phase Portraits 6.2 Existence, Uniqueness, and Topological Consequences 6.3 Fixed Points and Linearization 6.4 Rabbits versus Sheep 6.5 Conservative Systems 6.6 Reversible Systems 6.7 Pendulum 6.8 Index Theory Exercises for Chapter 6 7 Limit Cycles 7.0 Introduction 7.1 Examples 7.2 Ruling Out Closed Orbits 7.3 Poincaré−Bendixson Theorem 7.4 Liénard Systems 7.5 Relaxation Oscillations 7.6 Weakly Nonlinear Oscillators Exercises for Chapter 7 8 Bifurcations Revisited8.0 Introduction 8.1 Saddle-Node, Transcritical, and Pitchfork Bifurcations 8.2 Hopf Bifurcations 8.3 Oscillating Chemical Reactions 8.4 Global Bifurcations of Cycles 8.5 Hysteresis in the Driven Pendulum and Josephson Junction 8.6 Coupled Oscillators and Quasiperiodicity 8.7 Poincaré Maps Exercises for Chapter 8 Part III Chaos 9 Lorenz Equations 9.0 Introduction 9.1 A Chaotic Waterwheel 9.2 Simple Properties of the Lorenz Equations 9.3 Chaos on a Strange Attractor 9.4 Lorenz Map 9.5 Exploring Parameter Space 9.6 Using Chaos to Send Secret Messages Exercises for Chapter 910 One-Dimensional Maps 10.0 Introduction 10.1 Fixed Points and Cobwebs 10.2 Logistic Map: Numerics 10.3 Logistic Map: Analysis 10.4 Periodic Windows 10.5 Liapunov Exponent 10.6 Universality and Experiments 10.7 Renormalization Exercises for Chapter 10 11 Fractals 11.0 Introduction 11.1 Countable and Uncountable Sets 11.2 Cantor Set 11.3 Dimension of Self-Similar Fractals 11.4 Box Dimension 11.5 Pointwise and Correlation Dimensions Exercises for Chapter 11 12 Strange Attractors 12.0 Introduction 12.1 The Simplest Examples 12.2 Hénon Map This Textbook Is Aimed At Newcomers To Nonlinear Dynamics And Chaos, Especially Students Taking A First Course In The Subject. The Presentation Stresses Analytical Methods, Concrete Examples, And Geometric Intuition. The Theory Is Developed Systematically, Starting With First-order Differential Equations And Their Bifurcations, Followed By Phase Plane Analysis, Limit Cycles And Their Bifurcations, And Culminating With The Lorenz Equations, Chaos, Iterated Maps, Period Doubling, Renormalization, Fractals, And Strange Attractors. A Unique Feature Of The Book Is Its Emphasis On Applications. These Include Mechanical Vibrations, Lasers, Biological Rhythms, Superconducting Circuits, Insect Outbreaks, Chemical Oscillators, Genetic Control Systems, Chaotic Waterwheels, And Even A Technique For Using Chaos To Send Secret Messages. In Each Case, The Scientific Background Is Explained At An Elementary Level And Closely Integrated With Mathematical Theory. In The Twenty Years Since The First Edition Of This Book Appeared, The Ideas And Techniques Of Nonlinear Dynamics And Chaos Have Found Application To Such Exciting New Fields As Systems Biology, Evolutionary Game Theory, And Sociophysics. This Second Edition Includes New Exercises On These Cutting-edge Developments, On Topics As Varied As The Curiosities Of Visual Perception And The Tumultuous Love Dynamics In Gone With The Wind. Overview -- One-dimensional Flows -- Flows On The Line -- Bifurcations -- Flows On The Circle -- Two-dimensional Flows -- Linear Systems -- Phase Plane -- Limit Cycles -- Bifurcations Revisited -- Chaos -- Lorenz Equations -- One-dimensional Maps -- Fractals -- Strange Attractors. Steven H. Strogatz. Includes Bibliographical References (pages 470-482) And Indexes.