| Preface | p. XI |
| Introduction | p. i |
| What is a dynamical system? | p. 1 |
| State vectors | p. 1 |
| The next instant: discrete time | p. 2 |
| The next instant: continuous time | p. 4 |
| Summary | p. 6 |
| Problems | p. 6 |
| Examples | p. 8 |
| Mass and spring | p. 8 |
| RLC circuits | p. 10 |
| Pendulum | p. 11 |
| Your bank account | p. 16 |
| Economic growth | p. 17 |
| Pushing buttons on your calculator | p. 19 |
| Microbes | p. 22 |
| Predator and prey | p. 24 |
| Newton's Method | p. 26 |
| Euler's method | p. 28 |
| "Random" number generation | p. 31 |
| Problems | p. 32 |
| What we want; what we can get | p. 35 |
| Linear Systems | p. 37 |
| One dimension | p. 37 |
| Discrete time | p. 37 |
| Continuous time | p. 45 |
| Summary | p. 48 |
| Problems | p. 49 |
| Two (and more) dimensions | p. 50 |
| Discrete time | p. 51 |
| Continuous time | p. 57 |
| The nondiagonalizable case* | p. 81 |
| Problems | p. 88 |
| Examplification: Markov chains | p. 91 |
| Introduction | p. 91 |
| Markov chains as linear systems | p. 93 |
| The long term | p. 96 |
| Problems | p. 98 |
| Nonlinear Systems 1: Fixed Points | p. 101 |
| Fixed points | p. 102 |
| What is a fixed point? | p. 102 |
| Finding fixed points | p. 103 |
| Stability | p. 104 |
| Problems | p. 108 |
| Linearization | p. 110 |
| One dimension | p. 110 |
| Two and more dimensions | p. 117 |
| Problems | p. 126 |
| Lyapunov functions | p. 128 |
| Linearization can fail | p. 128 |
| Energy | p. 130 |
| Lyapunov's method | p. 133 |
| Gradient systems | p. 137 |
| Problems | p. 144 |
| Examplification: Iterative methods for solving equations | p. 146 |
| Problems | p. 150 |
| Nonlinear Systems 2: Periodicity and Chaos | p. 153 |
| Continuous time | p. 154 |
| One dimension: no periodicity | p. 154 |
| Two dimensions: the Poincare-Bendixson theorem | p. 155 |
| The Hopfbifurcation | p. 161 |
| Higher dimensions: the Lorenz system and chaos | p. 163 |
| Problems | p. 167 |
| Discrete time | p. 168 |
| Periodicity | p. 169 |
| stability of periodic points | p. 173 |
| Bifurcation | p. 175 |
| Sarkovskii's theorem | p. 188 |
| Chaos and symbolic dynamics | p. 202 |
| Problems | p. 216 |
| Examplification: Riffle shuffles and the shift map | p. 218 |
| Riffle shuffles | p. 218 |
| The shift map | p. 219 |
| Shifting and shuffling | p. 223 |
| Shuffling again and again | p. 226 |
| Problems | p. 229 |
| Fractals | p. 231 |
| Cantor's set | p. 231 |
| Symbolic representation of Cantor's set | p. 232 |
| Cantor's set in conventional notation | p. 233 |
| The link between the two representations | p. 236 |
| Topological properties of the Cantor set | p. 236 |
| In what sense a fractal? | p. 240 |
| Problems | p. 241 |
| Biting out the middle in the plane | p. 242 |
| Sierpiriski's triangle | p. 242 |
| Koch's snowflake | p. 243 |
| Problems | p. 245 |
| Contraction mapping theorems | p. 246 |
| Contraction maps | p. 246 |
| Contraction mapping theorem on the real line | p. 247 |
| Contraction mapping in higher dimensions | p. 249 |
| Contractive affine maps: the spectral norm* | p. 250 |
| Other metric spaces | p. 254 |
| Compact sets and Hausdorff distance | p. 255 |
| Problems | p. 258 |
| Iterated function systems | p. 260 |
| From point maps to set maps | p. 260 |
| The union of set maps | p. 262 |
| Examples revisited | p. 265 |
| IFSs defined | p. 270 |
| Working backward | p. 271 |
| Problems | p. 276 |
| Algorithms for drawing fractals | p. 276 |
| A deterministic algorithm | p. 277 |
| Dancing on fractals279 | |
| A randomized algorithm | p. 283 |
| Problems | p. 286 |
| Fractal dimension | p. 287 |
| Covering with balls | p. 287 |
| Definition of dimension | p. 290 |
| Simplifying the definition | p. 292 |
| Just-touching similitudes and dimension | p. 300 |
| Problems | p. 306 |
| Examplification: Fractals in nature | p. 307 |
| Dimension of physical fractals | p. 308 |
| Estimating surface area | p. 312 |
| Image analysis | p. 314 |
| Problems | p. 314 |
| Complex Dynamical Systems | p. 317 |
| Julia sets | p. 317 |
| Definition and examples | p. 317 |
| Escape-time algorithm | p. 324 |
| Other Julia sets | p. 326 |
| Problems | p. 327 |
| The Mandelbrot set | p. 327 |
| Definition and various views | p. 327 |
| Escape-time algorithm | p. 332 |
| Problems | p. 332 |
| Examplification: Newton's method revisited | p. 333 |
| Problems | p. 336 |
| Examplification: Complex bases | p. 336 |
| Place value revisited | p. 336 |
| ifss revisited | p. 338 |
| Problems | p. 340 |
| Background Material | p. 341 |
| Linear algebra | p. 341 |
| Much ado about 0 | p. 341 |
| Linear independence | p. 342 |
| Eigenvalues/vectors | p. 342 |
| agonalization | p. 343 |
| Jordan canonical form | p. 343 |
| Basic linear transformations of the plane | p. 344 |
| Complex numbers | p. 347 |
| Calculus | p. 349 |
| Intermediate and mean value theorems | p. 349 |
| Partial derivatives | p. 350 |
| Differential equations | p. 351 |
| Equations | p. 351 |
| What is a differential equation? | p. 351 |
| Standard notation | p. 352 |
| Computing | p. 355 |
| Differential equations | p. 355 |
| Analytic solutions | p. 356 |
| Numerical solutions | p. 357 |
| Triangle Dance | p. 365 |
| About the accompanying software | p. 366 |
| Bibliography | p. 369 |
| Index | p. 371 |
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