Contents ix Exercises 6.5 246 6.6. Rate of Change of Volume 247 Exercises 6.6 249 6.7. Poincar´ e Map 251 Exercises 6.7 261 6.8. Applications 262 Exercises 6.8 271 6.9. Theory and Proofs 272 Chapter 7. Chaotic Attractors 285 7.1. Attractors 285 Exercises 7.1 289 7.2. Chaotic Attractors 291 Exercise 7.2 296 7.3. Lorenz System 297 Exercises 7.3 312 7.4. R¨ ossler Attractor 313 Exercises 7.4 316 7.5. Forced Oscillator 317 Exercises 7.5 319 7.6. Lyapunov Exponents 320 Exercises 7.6 328 7.7. Test for Chaotic Attractors 329 Exercises 7.7 331 7.8. Applications 331 7.9. Theory and Proofs 336 Part 2. Iteration of Functions Chapter 8. Iteration of Functions as Dynamics 343 8.1. One-Dimensional Maps 343 8.2. Functions with Several Variables 349 Chapter 9. Periodic Points of One-Dimensional Maps 353 9.1. Periodic Points 353 Exercises 9.1 362 9.2. Iteration Using the Graph 362 Exercises 9.2 366 9.3. Stability of Periodic Points 367 Exercises 9.3 382 9.4. Critical Points and Basins 386

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