Dynamical Systems with Applications using Python

Preface......Page 3
Contents......Page 9
1 Tutorial Intro to Python......Page 15
1.1 The IDLE Integrated Development Environment  for Python......Page 16
1.1.1 Tutorial One: Using Python as a Powerful Calculator......Page 18
1.1.2 Tutorial Two: Simple Programming with Python......Page 20
1.1.3 Tutorial Three: Simple Plotting Usingthe Turtle Module......Page 23
1.2 Anaconda, Spyder and the Libraries,Sympy, Numpy, and Matplotlib......Page 28
1.2.1 Tutorial One: A Tutorial Introduction to Sympy......Page 29
1.2.2 Tutorial Two: A Tutorial Introduction to Numpy and Matplotlib......Page 32
1.2.3 Tutorial Three: Simple Programming, SolvingODEs, and More Detailed Plots......Page 34
1.3 Exercises......Page 41
2 Differential Equations......Page 46
2.1.1 Linear Differential Equations......Page 47
2.1.2 Separable Differential Equations......Page 48
2.1.3 Exact Differential Equations......Page 52
2.1.4 Homogeneous Differential Equations......Page 53
2.2 Applications to Chemical Kinetics......Page 57
2.3 Applications to Electric Circuits......Page 61
2.4 Existence and Uniqueness Theorem......Page 66
2.5 Python Programs......Page 70
2.6 Exercises......Page 72
3 Planar Systems......Page 78
3.1 Canonical Forms......Page 79
3.1.1 Real Distinct Eigenvalues......Page 81
3.1.2 Complex Eigenvalues (λ=αi β)......Page 82
3.1.3 Repeated Real Eigenvalues......Page 83
3.2 Eigenvectors Defining Stable and Unstable Manifolds......Page 85
3.3 Phase Portraits of Linear Systems in the Plane......Page 87
3.4 Linearization and Hartman's Theorem......Page 91
3.5 Constructing Phase Plane Diagrams......Page 92
3.6 Python Programs......Page 100
3.7 Exercises......Page 103
4 Interacting Species......Page 108
4.1 Competing Species......Page 109
4.2 Predator-Prey Models......Page 112
4.3 Other Characteristics Affecting Interacting Species......Page 117
4.4 Python Programs......Page 120
4.5 Exercises......Page 121
5 Limit Cycles......Page 125
5.1 Historical Background......Page 126
5.2 Existence and Uniqueness of Limit Cycles in the Plane......Page 129
5.3 Nonexistence of Limit Cycles in the Plane......Page 135
5.4 Perturbation Methods......Page 139
5.5 Python Programs......Page 148
5.6 Exercises......Page 151
6 Hamiltonian Systems, Lyapunov Functions & Stability......Page 156
6.1 Hamiltonian Systems in the Plane......Page 157
6.2 Lyapunov Functions and Stability......Page 162
6.3 Python Programs......Page 168
6.4 Exercises......Page 169
7 Bifurcation Theory......Page 173
7.1 Bifurcations of Nonlinear Systems in the Plane......Page 174
7.1.1 A Saddle-Node Bifurcation......Page 175
7.1.3 A Pitchfork Bifurcation......Page 177
7.2 Normal Forms......Page 180
7.3 Multistability and Bistability......Page 184
7.4 Python Programs......Page 188
7.5 Exercises......Page 190
8 3D Autonomous Systems & Chaos......Page 195
8.1 Linear Systems and Canonical Forms......Page 196
8.2 Nonlinear Systems and Stability......Page 200
8.3.1 The Rössler Attractor......Page 204
8.3.2 Chaos......Page 206
8.4.1 The Lorenz Equations......Page 209
8.4.2 Chua's Circuit......Page 211
8.4.3 The Belousov-Zhabotinski (BZ) Reaction......Page 214
8.5 Python Programs......Page 217
8.6 Exercises......Page 221
9 Poincaré Maps & Nonautonomous Systems in the Plane......Page 225
9.1 Poincaré Maps......Page 226
9.2 Hamiltonian Systems with Two Degrees of Freedom......Page 231
9.3 Nonautonomous Systems in the Plane......Page 237
9.4 Python Programs......Page 245
9.5 Exercises......Page 249
10 Local & Global Bifurcations......Page 254
10.1 Small-Amplitude Limit CycleBifurcations......Page 255
10.2 Gröbner Bases......Page 261
10.3 Melnikov Integrals and Bifurcating Limit Cyclesfrom a Center......Page 267
10.4 Bifurcations Involving Homoclinic Loops......Page 269
10.5 Python Programs......Page 271
10.6 Exercises......Page 275
11 Second Part of Hilbert 16th Problem......Page 279
11.1 Statement of Problem and Main Results......Page 280
11.2 Poincaré Compactification......Page 283
11.3 Global Results for Liénard Systems......Page 289
11.4 Local Results for Liénard Systems......Page 298
11.5 Python Programs......Page 299
11.6 Exercises......Page 300
12 Delay Differential Equations......Page 305
12.1 Introduction and the Method of Steps......Page 306
12.2 Applications in Biology......Page 312
12.3 Applications in Nonlinear Optics......Page 318
12.4 Other Applications......Page 321
12.5 Python Programs......Page 323
12.6 Exercises......Page 328
13 Linear Discrete Dynamical Systems......Page 334
13.1 Recurrence Relations......Page 335
13.2 The Leslie Model......Page 340
13.3 Harvesting and Culling Policies......Page 344
13.4 Python Programs......Page 349
13.5 Exercises......Page 350
14 Nonlinear Discrete Dynamical Systems......Page 354
14.1 The Tent Map and Graphical Iterations......Page 355
14.2 Fixed Points and Periodic Orbits......Page 360
14.3 The Logistic Map, Bifurcation Diagram,and Feigenbaum Number......Page 367
14.4 Gaussian and Hénon Maps......Page 375
14.5 Applications......Page 380
14.6 Python Programs......Page 383
14.7 Exercises......Page 387
15 Complex Iterative Maps......Page 392
15.1 Julia Sets and the Mandelbrot Set......Page 393
15.2 Boundaries of Periodic Orbits......Page 398
15.3 The Newton Fractal......Page 402
15.4 Python Programs......Page 403
15.5 Exercises......Page 406
16 Electromagnetic Waves & Optical Resonators......Page 409
16.1 Maxwell's Equations and Electromagnetic Waves......Page 410
16.2 Historical Background......Page 412
16.3 The Nonlinear SFR Resonator......Page 418
16.4 Chaotic Attractors and Bistability......Page 419
16.5 Linear Stability Analysis......Page 423
16.6 Instabilities and Bistability......Page 426
16.7 Python Programs......Page 430
16.8 Exercises......Page 434
17 Fractals & Multifractals......Page 439
17.1 Construction of Simple Examples......Page 440
17.2 Calculating Fractal Dimensions......Page 447
17.3 A Multifractal Formalism......Page 454
17.4 Multifractals in the Real World and Some Simple Examples......Page 458
17.5 Python Programs......Page 465
17.6 Exercises......Page 470
18 Image Processing with Python......Page 477
18.1 Image Processing and Matrices......Page 478
18.2 The Fast Fourier Transform......Page 483
18.3 The Fast Fourier Transform on Images......Page 490
18.4 Exercises......Page 493
19 Chaos Control & Synchronization......Page 496
19.1 Historical Background......Page 497
19.2 Controlling Chaos in the Logistic Map......Page 502
19.3 Controlling Chaos in the Hénon Map......Page 503
19.4 Chaos Synchronization......Page 510
19.5 Python Programs......Page 514
19.6 Exercises......Page 518
20 Neural Networks......Page 523
20.1 Introduction......Page 524
20.2 The Delta Learning Rule and Backpropagation......Page 530
20.3 The Hopfield Network and Lyapunov Stability......Page 535
20.4 Neurodynamics......Page 545
20.5 Python Programs......Page 549
20.6 Exercises......Page 554
21 Binary Oscillator Computing......Page 560
21.1 Brain Inspired Computing......Page 561
21.2 Oscillatory Threshold Logic......Page 566
21.3 Applications and Future Work......Page 574
21.4 An Assay for Neuronal Degradation......Page 580
21.5 Python Programs......Page 582
21.6 Exercises......Page 587
22 Coursework & Examination-Type Questions......Page 593
22.1 Examples of Coursework Questions......Page 594
22.2 Examination 1......Page 609
22.3 Examination 2......Page 612
22.4 Examination 3......Page 615
23.1 Chapter 1......Page 621
23.2 Chapter 2......Page 624
23.3 Chapter 3......Page 625
23.4 Chapter 4......Page 627
23.5 Chapter 5......Page 628
23.7 Chapter 7......Page 630
23.8 Chapter 8......Page 632
23.10 Chapter 10......Page 633
23.11 Chapter 11......Page 634
23.13 Chapter 13......Page 636
23.14 Chapter 14......Page 638
23.15 Chapter 15......Page 639
23.17 Chapter 17......Page 640
23.19 Chapter 19......Page 641
23.21 Chapter 21......Page 642
23.22 Chapter 22......Page 643
IDLE Python Programs......Page 647
Anaconda Python Programs......Page 648
Index......Page 652