Untangling Complex Systems Read online

  7.3.2

  Signal Transduction Systems with Positive Feedback ................... 177

  7.4

  Epigenetic Events ........................................................................................ 181

  7.5

  Biological Rhythms ..................................................................................... 185

  7.6

  Amplification and Adaptation in Regulatory and Sensory Systems ........... 187

  7.6.1

  Magnitude Amplification ............................................................... 187

  7.6.2

  Sensitivity Amplification ............................................................... 188

  7.6.3 Adaptation ...................................................................................... 190

  7.7

  Key Questions ............................................................................................. 191

  x

  Contents

  7.8

  Key Words ................................................................................................... 192

  7.9

  Hints for Further Reading ........................................................................... 192

  7.10 Exercises ...................................................................................................... 192

  7.11

  Solutions to the Exercises ............................................................................ 193

  Chapter 8 The Emergence of Temporal Order in a Chemical Laboratory ............................... 197

  8.1 Introduction ................................................................................................. 197

  8.2

  The Discovery of Oscillating Chemical Reactions ..................................... 197

  8.3

  The Systematic Design of Chemical Oscillators ......................................... 199

  8.3.1 Excitability .....................................................................................202

  8.3.2 Oscillations ....................................................................................203

  8.3.3

  In Practice ......................................................................................203

  8.4

  Primary “Oscillators” ..................................................................................205

  8.4.1

  Oregonator Model: The “Primary Oscillator” of Coproduct

  Autocontrol ....................................................................................205

  8.4.2

  The Modified Lotka-Volterra or Predator-Prey “Primary

  Oscillator” ......................................................................................208

  8.4.3

  The “Flow Control Primary Oscillator” ........................................ 212

  8.4.4

  The Composite System: A Chemical Equilibrium Coupled

  to a “Primary Oscillator” ............................................................... 214

  8.4.5

  “Delayed Negative Feedback Oscillator” ...................................... 216

  8.5

  Overview and Hints for Further Reading ....................................................220

  8.6

  Key Questions ............................................................................................. 222

  8.7

  Key Words ................................................................................................... 222

  8.8 Exercises ...................................................................................................... 222

  8.9

  Solutions to the Exercises ............................................................................ 227

  Chapter 9 The Emergence of Order in Space ........................................................................... 241

  9.1 Introduction ................................................................................................. 241

  9.2

  The Reaction-Diffusion Model ................................................................... 241

  9.3

  Turing Patterns ............................................................................................246

  9.4

  Turing Patterns in a Chemical Laboratory .................................................. 251

  9.5

  Turing Patterns in Nature ............................................................................ 255

  9.5.1

  Biology: The Development of Embryos ......................................... 256

  9.5.2

  Biology: Regeneration of Tissues...................................................260

  9.5.3

  Biology: Phyllotaxis ....................................................................... 261

  9.5.4

  Biology: Animal Markings ............................................................ 261

  9.5.5

  Ecology, Sociology, and Economy ................................................. 263

  9.5.6 Geomorphology ............................................................................. 263

  9.5.7

  The Next Development of Turing’s Theory: The

  Mechanochemical Patterning ........................................................264

  9.6

  Chemical Waves ..........................................................................................269

  9.6.1

  Propagator-Controller Model ......................................................... 270

  9.6.1.1 Phase Waves ................................................................... 271

  9.6.1.2 Trigger Waves ................................................................. 271

  9.6.2

  Shapes of Chemical Waves ............................................................ 273

  9.6.2.1 Mono- and Bi-Dimensional Waves ................................ 273

  9.6.2.2 Three-Dimensional Waves ............................................. 274

  9.6.2.3 Effect of Curvature ......................................................... 274

  Contents

  xi

  9.7

  “Chemical” Waves in Biology ..................................................................... 275

  9.7.1

  Waves in a Neuron ....................................................................... 275

  9.7.2

  The Fisher-Kolmogorov Equation ................................................ 279

  9.7.3

  Waves in Our Brain ...................................................................... 281

  9.7.4

  Waves in Our Heart ...................................................................... 282

  9.7.5

  Calcium Waves ............................................................................. 283

  9.7.6

  cAMP Waves: The Case of Dictyostelium Discoideum...............284

  9.7.7

  Spreading of Species, Epidemics and … Fads .............................285

  9.8

  Liesegang Patterns.......................................................................................285

  9.9

  Liesegang Phenomena in Nature .................................................................288

  9.9.1

  In Geology ....................................................................................288

  9.9.2

  In Biology ..................................................................................... 289

  9.10


  A Final Note: The Reaction-Diffusion Structures in Art and

  Technology .................................................................................................. 289

  9.10.1 Reaction-Diffusion Processes as Art ........................................... 289

  9.10.2 Reaction-Diffusion Processes in Technology ..............................290

  9.11

  Key Questions .............................................................................................290

  9.12

  Key Words ................................................................................................... 291

  9.13

  Hints for Further Reading ........................................................................... 291

  9.14 Exercises ...................................................................................................... 291

  9.15

  Solutions to the Exercises ............................................................................296

  Chapter 10 The Emergence of Chaos in Time ............................................................................ 317

  10.1 Introduction ................................................................................................. 317

  10.2

  Nonlinearity and Chaos: The Case of the Double Pendulum ..................... 317

  10.3

  Nonlinearity and Chaos: The Case of the Population Growth and the

  Logistic Map................................................................................................ 321

  10.4

  The Universality of Chaos ........................................................................... 326

  10.5 Convection ................................................................................................... 327

  10.6

  The Entropy Production in the Nonlinear Regime: The Case of

  Convection ................................................................................................... 332

  10.7

  The “Butterfly Effect” ................................................................................. 334

  10.7.1 The Complexity of Convection in the Terrestrial Atmosphere ..... 334

  10.7.2 The Lorenz’s Model ..................................................................... 335

  10.7.3 The Sensitivity to the Initial Conditions ...................................... 338

  10.7.4 The Hydrodynamic Photochemical Oscillator .............................340

  10.8

  Aperiodic Time Series ................................................................................. 342

  10.8.1 How Do We Recognize Chaotic Time Series? ............................. 343

  10.8.1.1 Time Delay τ ............................................................... 343

  10.8.1.2 Embedding Dimension m ............................................344

  10.8.1.3 Lyapunov Exponents ................................................... 345

  10.8.1.4 Kolmogorov-Sinai Entropy .........................................346

  10.8.1.5 Correlation Dimension ................................................ 347

  10.8.1.6 Permutation Entropy ................................................... 347

  10.8.1.7 Surrogate Data ............................................................348

  10.8.1.8 Short-Term Predictability and Long-Term

  Unpredictability ..........................................................348

  10.8.2 Prediction of the Chaotic Time Series ......................................... 349

  10.8.2.1 Artificial Neural Networks ......................................... 350

  xii

  Contents

  10.9

  Mastering Chaos........................................................................................ 352

  10.9.1 Applications ............................................................................... 354

  10.9.1.1 Communication by Chaotic Dynamics ..................... 354

  10.9.1.2 Computing by Chaotic Dynamics............................. 354

  10.10

  Key Questions ........................................................................................... 355

  10.11

  Key Words ................................................................................................. 355

  10.12

  Hints for Further Reading ......................................................................... 356

  10.13 Exercises .................................................................................................... 356

  10.14

  Solutions to the Exercises ..........................................................................360

  Chapter 11 Chaos in Space: The Fractals ................................................................................... 379

  11.1 Introduction ............................................................................................... 379

  11.2

  What Is a Fractal? ...................................................................................... 381

  11.3

  Fractal Dimension ..................................................................................... 383

  11.4

  Fractals That Are Not Perfectly Self-Similar............................................ 385

  11.5

  The Fractal-like Structures in Nature ....................................................... 386

  11.6

  The Dimensions of Fractals That Are Not Perfectly Self-Similar ............ 388

  11.7

  A Method for Generating Fractal-like Structures in the Lab .................... 389

  11.8

  Dendritic Fractals ...................................................................................... 392

  11.9 Multifractals .............................................................................................. 394

  11.9.1

  Analysis of the Complex Images ............................................... 394

  11.9.2

  Analysis of the Complex Time Series ........................................ 395

  11.10

  Diffusion in Fractals ................................................................................. 395

  11.11

  Chemical Reactions on Fractals and Fractal-like Kinetics in Cells ......... 396

  11.12

  Power Laws or Stretched Exponential Functions? .................................... 399

  11.13

  Why Does Chaos Generate Fractals? ........................................................ 401

  11.14

  Chaos, Fractals, and Entropy ....................................................................402

  11.15

  Key Questions ...........................................................................................403

  11.16

  Key Words .................................................................................................403

  11.17

  Hints for Further Reading .........................................................................404

  11.18 Exercises ....................................................................................................404

  11.19

  Solutions to the Exercises ..........................................................................405

  Chapter 12 Complex Systems ..................................................................................................... 415

  12.1

  The Natural Complexity Challenges ......................................................... 415

  12.2

  The Computational Complexity of the Natural Comple
x Systems ........... 415

  12.3

  If It Were NP = P, Would Be the Complexity Challenges Surely Won? .. 419

  12.4

  The Features of Complex Systems ............................................................ 420

  12.4.1 Networks .................................................................................... 420

  12.4.2

  Out-of-Equilibrium Systems ...................................................... 426

  12.4.2.1 The Thermodynamics of Thermal Radiation ........... 426

  12.4.2.2 The Fate of the Solar Thermal Radiation and the

  Climate Change ........................................................ 430

  12.4.2.3 Solar Radiation and Life on Earth ............................ 431

  12.4.2.4 Solar Radiation as an Energy Source for Life on

  Earth ......................................................................... 433

  12.4.2.5 Solar Radiation as Information Source for Life

  on Earth .................................................................... 437

  12.4.3

  Emergent Properties ...................................................................444

  Contents

  xiii

  12.5

  Key Questions ...........................................................................................446

  12.6

  Key Words .................................................................................................446

  12.7

  Hints for Further Reading ......................................................................... 447

  12.8 Exercises .................................................................................................... 447

  12.9

  Solutions to the Exercises .......................................................................... 450

  Chapter 13 How to Untangle Complex Systems? ....................................................................... 457

  13.1 Introduction ............................................................................................... 457

  13.2

  Improving Electronic Computers .............................................................. 457

  13.3

  Natural Computing .................................................................................... 461

  13.3.1 Computing Inspired by Natural Information Systems ................. 462

  13.3.1.1

  Artificial Life, Systems Chemistry, Systems

  Biology, and Synthetic Biology ................................ 462

  13.3.1.2

  Membrane Computing ..............................................463