Dynamical Systems with Applications Using Mathematica®

This textbook, now in its second edition, provides a broad introduction to the theory and practice of both continuous and discrete dynamical systems with the aid of the Mathematica software suite. Taking a hands-on approach, the reader is guided from basic concepts to modern research topics. Emphasized throughout are numerous applications to biology, chemical kinetics, economics, electronics, epidemiology, nonlinear optics, mechanics, population dynamics, and neural networks. The book begins with an efficient tutorial introduction to Mathematica, enabling new users to become familiar with the program, while providing a good reference source for experts. Working Mathematica notebooks will be available at: http://library.wolfram.com/infocenter/Books/9563/ The author has focused on breadth of coverage rather than fine detail, with theorems and proofs being kept to a minimum, though references are included for the inquisitive reader. The book is intended for senior undergraduate and graduate students as well as working scientists in applied mathematics, the natural sciences, and engineering. Many of the chapters are especially useful as reference material for senior undergraduate independent project work. New to the second edition: Since the first printing of this book in 2007, Mathematica has evolved from version 6.0 to version 11.2 in 2017. Accordingly, the second edition has been thoroughly updated and new material has been added. There are many more applications, examples and exercises, all with solutions, and new sections on series solutions of ordinary differential equations and Newton fractals, have been added. There are also new chapters on delay differential equations, image processing, binary oscillator computing, and simulation with Wolfram SystemModeler. Praise for the first edition: "[This book's] content and presentation style convey the excitement that has drawn many students and researchers to dynamical systems in the first place." --Dynamical Systems Magazine "This book presents an original, cheap and powerful solution to the problem of analysis of large data sets." --Studia Universitatis Babes'-Bolyai Mathematica "The one-liner programs come to life when typed in, and the growing programming skill lends itself to inventing [one's] own extensions to the supplied problems." --Datafile, The Journal of the HPCC.-- Provided by publisher Preface 6 1 A Tutorial Introduction to Mathematica 18 1.1 A Quick Tour of Mathematica 19 1.2 Tutorial One: The Basics (One Hour) 21 1.3 Tutorial Two: Plots and Differential Equations (One Hour) 23 1.4 The Manipulate Command and Simple Mathematica Programs 25 1.5 Hints for Programming 28 1.6 Mathematica Exercises 29 Bibliography 32 2 Differential Equations 33 2.1 Simple Differential Equations and Applications 34 2.2 Applications to Chemical Kinetics 43 2.3 Applications to Electric Circuits 47 2.4 Existence and Uniqueness Theorem 53 2.5 Mathematica Commands in Text Format 56 2.6 Exercises 57 Bibliography 60 3 Planar Systems 61 3.1 Canonical Forms 62 3.2 Eigenvectors Defining Stable and Unstable Manifolds 67 3.3 Phase Portraits of Linear Systems in the Plane 70 3.4 Linearization and Hartman's Theorem 74 3.5 Constructing Phase Plane Diagrams 75 3.6 Mathematica Commands 84 3.7 Exercises 85 Bibliography 88 4 Interacting Species 89 4.1 Competing Species 89 4.2 Predator-Prey Models 92 4.3 Other Characteristics Affecting Interacting Species 98 4.4 Mathematica Commands 100 4.5 Exercises 100 Bibliography 103 5 Limit Cycles 104 5.1 Historical Background 105 5.2 Existence and Uniqueness of Limit Cycles in the Plane 108 5.3 Nonexistence of Limit Cycles in the Plane 115 5.4 Perturbation Methods 117 5.5 Mathematica Commands 127 5.6 Exercises 128 Bibliography 131 6 Hamiltonian Systems, Lyapunov Functions, and Stability 133 6.1 Hamiltonian Systems in the Plane 133 6.2 Lyapunov Functions and Stability 138 6.3 Mathematica Commands 145 6.4 Exercises 145 Bibliography 147 7 Bifurcation Theory 148 7.1 Bifurcations of Nonlinear Systems in the Plane 149 7.2 Normal Forms 155 7.3 Multistability and Bistability 159 7.4 Mathematica Commands 162 7.5 Exercises 163 Bibliography 166 8 Three-Dimensional Autonomous Systems and Chaos 167 8.1 Linear Systems and Canonical Forms 168 8.2 Nonlinear Systems and Stability 172 8.3 The Rössler System and Chaos 176 8.4 The Lorenz Equations, Chua's Circuit, and the Belousov-Zhabotinski Reaction 181 8.5 Mathematica Commands 188 8.6 Exercises 191 Bibliography 194 9 Poincaré Maps and Nonautonomous Systems in the Plane 196 9.1 Poincaré Maps 197 9.2 Hamiltonian Systems with Two Degrees of Freedom 203 9.3 Nonautonomous Systems in the Plane 206 9.4 Mathematica Commands 215 9.5 Exercises 216 Bibliography 219 10 Local and Global Bifurcations 220 10.1 Small-Amplitude Limit Cycle Bifurcations 221 10.2 Gröbner Bases 226 10.3 Melnikov Integrals and Bifurcating Limit Cycles from a Center 232 10.4 Bifurcations Involving Homoclinic Loops 234 10.5 Mathematica Commands 236 10.6 Exercises 238 Bibliography 240 11 The Second Part of Hilbert's Sixteenth Problem 242 11.1 Statement of Problem and Main Results 243 11.2 Poincaré Compactification 246 11.3 Global Results for Liénard Systems 252 11.4 Local Results for Liénard Systems 261 11.5 Mathematica Commands 262 11.6 Exercises 263 Bibliography 265 12 Delay Differential Equations 268 12.1 Introduction and the Method of Steps 269 12.2 Applications in Biology 275 12.3 Applications in Nonlinear Optics 281 12.4 Other Applications 285 12.5 Exercises 289 Bibliography 292 13 Linear Discrete Dynamical Systems 295 13.1 Recurrence Relations 296 13.2 The Leslie Model 301 13.3 Harvesting and Culling Policies 306 13.4 Mathematica Commands 310 13.5 Exercises 310 Bibliography 314 14 Nonlinear Discrete Dynamical Systems 315 14.1 The Tent Map and Graphical Iterations 316 14.2 Fixed Points and Periodic Orbits 322 14.3 The Logistic Map, Bifurcation Diagram, and Feigenbaum Number 328 14.4 Gaussian and Hénon Maps 335 14.5 Applications 340 14.6 Mathematica Commands 342 14.7 Exercises 344 Bibliography 347 15 Complex Iterative Maps 349 15.1 Julia Sets and the Mandelbrot Set 350 15.2 Boundaries of Periodic Orbits 354 15.3 The Newton Fractal 358 15.4 Mathematica Commands 359 15.5 Exercises 360 Bibliography 361 16 Electromagnetic Waves and Optical Resonators 362 16.1 Maxwell's Equations and Electromagnetic Waves 363 16.2 Historical Background 365 16.3 The Nonlinear SFR Resonator 370 16.4 Chaotic Attractors and Bistability 372 16.5 Linear Stability Analysis 375 16.6 Instabilities and Bistability 379 16.7 Mathematica Commands 384 16.8 Exercises 385 Bibliography 388 17 Fractals and Multifractals 390 17.1 Construction of Simple Examples 391 17.2 Calculating Fractal Dimensions 398 17.3 A Multifractal Formalism 404 17.4 Multifractals in the Real World and Some Simple Examples 409 17.5 Mathematica Commands 417 17.6 Exercises 419 Bibliography 422 18 Image Processing and Analysis with Mathematica 425 18.1 Image Processing and Matrices 426 18.2 The Fast Fourier Transform 429 18.3 The Fast Fourier Transform on Images 433 18.4 Exercises 434 Bibliography 436 19 Chaos Control and Synchronization 437 19.1 Historical Background 438 19.2 Controlling Chaos in the Logistic Map 442 19.3 Controlling Chaos in the Hénon Map 446 19.4 Chaos Synchronization 450 19.5 Mathematica Commands 455 19.6 Exercises 456 Bibliography 458 20 Neural Networks 460 20.1 Introduction 461 20.2 The Delta Learning Rule and Backpropagation 467 20.3 The Hopfield Network and Lyapunov Stability 472 20.4 Neurodynamics 481 20.5 Mathematica Commands 485 20.6 Exercises 487 Bibliography 490 21 Binary Oscillator Computing 492 21.1 Brain Inspired Computing 492 21.2 Oscillatory Threshold Logic 498 21.3 Applications and Future Work 503 21.4 Mathematica Commands 508 21.5 Exercises 510 Bibliography 512 22 An Introduction to Wolfram SystemModeler 515 22.1 Introduction 516 22.2 Electric Circuits 519 22.3 A Mechanical System 521 22.4 Causal (Block Based) Modeling 523 22.5 Exercises 526 Bibliography 528 23 Coursework and Examination-Type Questions 529 23.1 Examples of Coursework Questions 530 23.2 Examination 1 539 23.3 Examination 2 543 23.4 Examination 3 547 24 Solutions to Exercises 552 24.1 Chapter 1 552 24.2 Chapter 2 553 24.3 Chapter 3 554 24.4 Chapter 4 556 24.5 Chapter 5 558 24.6 Chapter 6 560 24.7 Chapter 7 561 24.8 Chapter 8 563 24.9 Chapter 9 564 24.10 Chapter 10 565 24.11 Chapter 11 565 24.12 Chapter 12 567 24.13 Chapter 13 567 24.14 Chapter 14 569 24.15 Chapter 15 571 24.16 Chapter 16 572 24.17 Chapter 17 572 24.18 Chapter 18 573 24.19 Chapter 19 573 24.20 Chapter 20 574 24.21 Chapter 21 574 24.22 Chapter 22 575 24.23 Chapter 23 576 Index 580 This textbook, now in its second edition, provides a broad introduction to the theory and practice of both continuous and discrete dynamical systems with the aid of the Mathematica software suite. Taking a hands-on approach, the reader is guided from basic concepts to modern research topics. Emphasized throughout are numerous applications to biology, chemical kinetics, economics, electronics, epidemiology, nonlinear optics, mechanics, population dynamics, and neural networks. The book begins with an efficient tutorial introduction to Mathematica, enabling new users to become familiar with the program, while providing a good reference source for experts. Working Mathematica notebooks will be available at: http://library.wolfram.com/infocenter/Books/9563/ The author has focused on breadth of coverage rather than fine detail, with theorems and proofs being kept to a minimum, though references are included for the inquisitive reader. The book is intended for senior undergraduate and graduate students as well as working scientists in applied mathematics, the natural sciences, and engineering. Many of the chapters are especially useful as reference material for senior undergraduate independent project work. New to the second edition: Since the first printing of this book in 2007, Mathematica has evolved from version 6.0 to version 11.2 in 2017. Accordingly, the second edition has been thoroughly updated and new material has been added. There are many more applications, examples and exercises, all with solutions, and new sections on series solutions of ordinary differential equations and Newton fractals, have been added. There are also new chapters on delay differential equations, image processing, binary oscillator computing, and simulation with Wolfram SystemModeler. Praise for the first edition: z[This book's] content and presentation style convey the excitement that has drawn many students and researchers to dynamical systems in the first place.y --Dynamical Systems Magazine zThis book presents an original, cheap and powerful solution to the problem of analysis of large data sets.y --Studia Universitatis Babes'-Bolyai Mathematica zThe one-liner programs come to life when typed in, and the growing programming skill lends itself to inventing [one's] own extensions to the supplied problems.y --Datafile, The Journal of the HPCC This book provides an introduction to the theory of dynamical systems with the ® aid of the Mathematica computer algebra system. It is written for both senior undergraduates and graduate students. The ?rst part of the book deals with c- tinuous systems using ordinary differential equations (Chapters 1–10), the second part is devoted to the study of discrete dynamical systems (Chapters 11–15), and Chapters 16 and 17 deal with both continuous and discrete systems. It should be pointedoutthatdynamicalsystemstheoryisnotlimitedtothesetopicsbutalso- compassespartialdifferentialequations,integralandintegrodifferentialequations, stochastic systems, and time-delay systems, for instance. References [1]–[4] given at the end of the Preface provide more information for the interested reader. The author has gone for breadth of coverage rather than ?ne detail and theorems with proofs are kept at a minimum. The material is not clouded by functional analytic and group theoretical de?nitions, and so is intelligible to readers with a general mathematical background. Some of the topics covered are scarcely covered el- where. Most of the material in Chapters 9, 10, 14, 16, and 17 is at a postgraduate levelandhasbeenin?uencedbytheauthor’sownresearchinterests. Thereismore theory in these chapters than in the rest of the book since it is not easily accessed anywhere else. It has been found that these chapters are especially useful as ref- ence material for senior undergraduate project work. The theory in other chapters of the book is dealt with more comprehensively in other texts, some of which may be found in the references section of the corresponding chapter. Front Matter ....Pages i-xvi A Tutorial Introduction to Mathematica (Stephen Lynch)....Pages 1-15 Differential Equations (Stephen Lynch)....Pages 17-44 Planar Systems (Stephen Lynch)....Pages 45-72 Interacting Species (Stephen Lynch)....Pages 73-87 Limit Cycles (Stephen Lynch)....Pages 89-117 Hamiltonian Systems, Lyapunov Functions, and Stability (Stephen Lynch)....Pages 119-133 Bifurcation Theory (Stephen Lynch)....Pages 135-153 Three-Dimensional Autonomous Systems and Chaos (Stephen Lynch)....Pages 155-183 Poincaré Maps and Nonautonomous Systems in the Plane (Stephen Lynch)....Pages 185-208 Local and Global Bifurcations (Stephen Lynch)....Pages 209-230 The Second Part of Hilbert’s Sixteenth Problem (Stephen Lynch)....Pages 231-256 Delay Differential Equations (Stephen Lynch)....Pages 257-283 Linear Discrete Dynamical Systems (Stephen Lynch)....Pages 285-304 Nonlinear Discrete Dynamical Systems (Stephen Lynch)....Pages 305-338 Complex Iterative Maps (Stephen Lynch)....Pages 339-351 Electromagnetic Waves and Optical Resonators (Stephen Lynch)....Pages 353-380 Fractals and Multifractals (Stephen Lynch)....Pages 381-415 Image Processing and Analysis with Mathematica (Stephen Lynch)....Pages 417-428 Chaos Control and Synchronization (Stephen Lynch)....Pages 429-451 Neural Networks (Stephen Lynch)....Pages 453-484 Binary Oscillator Computing (Stephen Lynch)....Pages 485-507 An Introduction to Wolfram SystemModeler (Stephen Lynch)....Pages 509-522 Coursework and Examination-Type Questions (Stephen Lynch)....Pages 523-545 Solutions to Exercises (Stephen Lynch)....Pages 547-574 Back Matter ....Pages 575-585