Untangling Complex Systems, page 112
transient chaos 370
universality of chaos 326–7
transistor device 457; FET 457–8; FinFET 459, 459;
unstable nodes/spirals/star 80
graphene 459; miniaturization 458; MOSFET
unsupervised learning 351
458, 458; problems 458–9
UV see ultraviolet (UV)
translation process 182
Traveling Salesman Problem (TSP) 416–17, 464, 466
velocity (V) systems 170
Treatise of Man (book) 11
Verhulst, P.-F. 322, 322n1
Tree of Scientific Knowledge 16, 16–17
Vernier caliper 518, 519
trigger waves 271–3, 272
viscous fingering 390, 393
truncation error (T) 533
visual cortex 442–3
tryptophan molecules 173, 173
visual sensory system, human eye 440; amacrine cells
TSP (Traveling Salesman Problem) 416–17, 464, 466
442; bipolar cell in 442; Blue/Green/Red
tubulin, protein 265
photoreceptor proteins 441; ganglion cells in
Turing, A. 13, 250, 250n3, 486
442; intensity of light 441; photoreceptor cells
Turing bifurcation 111
440, 442; retina 440, 440; rods and cones 440;
Turing machine 354, 416
structure 440, 440; visual cortex 442–3
Turing patterns 242, 246–51; in chemical laboratory Volterra, V. 117n1
251–5; crime hotspots 263; diffusion
von Koch, H. 381
coefficients values 305; formation 255; in
von Neumann architecture, computer 28, 28, 354, 457, 472
nature 255–69; in one-dimensional space
von Neumann, J. 353, 457, 469, 486
301; parameters 304, 305; qualitative description 249; RD model 256, 258, 262; from Wan, E. A. 350
Schnackenberg’s model 309
waveform 513, 514
wave segments 274n13
Ulam, S. 469
The Wealth of Nations (book) 147
ultimate laptop 491
Wheeler, J. A. 6n6
ultradian periodic processes 186
William of Ockham 8
ultrasensitive sigmoidal feedback 179–80
Wolfram, S. 469
ultraviolet (UV) 75; irradiation 341, 374; -visible radiation Works and Days (poem) 5
334, 340
uncertainties propagation 523–5, 524
Z e itschrift für Physik (paper) 32n8
uncertainty principle 352, 419
Zhabotinsky, A. 198
uncompetitive ligands 170, 170
zygote 256–7, 257
Document Outline
Cover
Half Title
Title Page
Copyright Page
Dedication
Table of Contents
Preface
Acknowledgments
About the Author
Chapter 1: Introduction 1.1 The Never-Ending Journey to Discovering the Secrets of Nature 1.1.1 The “Practical Period”
1.1.2 The “Philosophical Period”
1.1.3 The “Experimental Period”
1.1.4 The “Computational Period”
1.2 What Is Science, Today?
1.3 Purpose and Contents of This Book
1.4 Key Questions
1.5 Key Words
1.6 Hints for Further Reading
Chapter 2: Reversibility or Irreversibility? That Is the Question! 2.1 Introduction
2.2 The Thermodynamic Approach 2.2.1 The Classical Definition of Entropy
2.2.2 The Statistical Definition of Entropy
2.2.3 The Logical Definition of Entropy
2.3 An Exhausting Fight against Entropy 2.3.1 The Maxwell’s Demon
2.3.2 A First Mechanical Attempt
2.3.3 Another Mechanical Attempt
2.3.4 The Involvement of Artificial Intelligence: A “Thought Experiment”
2.3.5 The Embodiment of Maxwell’s Demon: A “Real Experiment”
2.3.6 The Surprising Behavior of Small Systems
2.3.7 There Is Still an Open Question
2.4 Key Questions
2.5 Key Words
2.6 Hints for Further Reading
2.7 Exercises
2.8 Solutions to the Exercises
Chapter 3: Out-of-Equilibrium Thermodynamics 3.1 Introduction
3.2 Definition of the Entropy Change for an Out-of-Equilibrium System 3.2.1 Heat Conduction
3.2.2 Chemical Reactions
3.2.3 Diffusion
3.2.4 Migration
3.2.5 Generalization
3.3 Non-equilibrium Thermodynamics in Linear Regime 3.3.1 Fourier’s Law: The Law of Heat Conduction
3.3.2 Ohm’s Law: The Law of Electrical Conduction
3.3.3 Poiseuille’s Law: The Law of Laminar Flow of Fluids
3.3.4 Fick’s Law: The Law of Diffusion
3.3.5 Generalization: Symmetry Principle and Onsager Reciprocal Relations
3.3.6 An Experimental Proof of the Reciprocal Relations
3.3.7 Cross-Diffusion
3.3.8 Thermal Diffusion
3.4 Evolution of Out-of-Equilibrium Systems in Linear Regime 3.4.1 The Case of Heat Conduction
3.4.2 The Case of Diffusion
3.5 The Theorem of Minimum Entropy Production in Linear Regime 3.5.1 A Single Force and Flow
3.5.2 The Case of More Than One Force and One Flow
3.6 Evolution of Out-of-Equilibrium Systems in Nonlinear Regime 3.6.1 Chemical Reactions
3.6.2 The Glansdorff-Prigogine Stability Criterion
3.7 The Chemical Transformations and the Linear Regime 3.7.1 Onsager’s Reciprocal Relations for Chemical Reactions
3.7.2 A Particular Case
3.8 The Evolution of Chemical Reactions in Open Systems 3.8.1 The Mono-Dimensional Case
3.8.2 The Bi-Dimensional Case
3.8.3 The Multi-Dimensional Case
3.9 Key Questions
3.10 Key Words
3.11 Hints for Further Reading
3.12 Exercises
3.13 Solutions to the Exercises
Chapter 4: An Amazing Scientific Voyage: From Equilibrium up to Self-Organization through Bifurcations 4.1 Introduction
4.2 Bifurcations 4.2.1 Saddle-Node Bifurcation
4.2.2 Trans-Critical Bifurcation 4.2.2.1 From a Lamp to a Laser: An Example of Trans-Critical Bifurcation
4.2.3 Pitchfork Bifurcation 4.2.3.1 Chiral Symmetry Breaking
4.2.4 Hopf Bifurcations
4.3 Key Questions
4.4 Key Words
4.5 Hint for Further Reading
4.6 Exercises
4.7 Solutions to the Exercises
Chapter 5: The Emergence of Temporal Order in the Ecosystems 5.1 Introduction
5.2 Predator-Prey Relationship: The Lotka-Volterra Model
5.3 Entropy Production in the Lotka-Volterra Model
5.4 More about Predator-Prey Relationships
5.5 Other Relationships within an Ecosystem
5.6 Mathematical Modeling of Symbiotic Relationships 5.6.1 Antagonism
5.6.2 Mutualism
5.7 Key Questions
5.8 Key Words
5.9 Hints for Further Reading
5.10 Exercises
5.11 Solutions to the Exercises
Chapter 6: The Emergence of Temporal Order in the Economy 6.1 Introduction
6.2 The Economic Problem
6.3 Linear and Circular Economy
6.4 The Law of Supply and Demand
6.5 The Business Cycles 6.5.1 Goodwin’s Predator-Prey Model
6.5.2 The Multiplier and Accelerator Model
6.5.3 Other Models
6.5.4 The Real Business Cycles
6.6 Key Questions
6.7 Key Words
6.8 Hints for Further Reading
6.9 Exercises
6.10 Solutions to the Exercises
Chapter 7: The Emergence of Temporal Order within a Living Being 7.1 Introduction
7.2 Metabolic Events 7.2.1 Michaelis-Menten Kinetics
7.2.2 Hill Kinetics
7.2.3 The Nonlinearity of Allosteric Enzymes
7.2.4 Glycolysis
7.3 Cellular Signaling Processes 7.3.1 The Simplest Signal Transduction System
7.3.2 Signal Transduction Systems with Positive Feedback
7.4 Epigenetic Events
7.5 Biological Rhythms
7.6 Amplification and Adaptation in Regulatory and Sensory Systems 7.6.1 Magnitude Amplification
7.6.2 Sensitivity Amplification
7.6.3 Adaptation
7.7 Key Questions
7.8 Key Words
7.9 Hints for Further Reading
7.10 Exercises
7.11 Solutions to the Exercises
Chapter 8: The Emergence of Temporal Order in a Chemical Laboratory 8.1 Introduction
8.2 The Discovery of Oscillating Chemical Reactions
8.3 The Systematic Design of Chemical Oscillators 8.3.1 Excitability
8.3.2 Oscillations
8.3.3 In Practice
8.4 Primary “Oscillators” 8.4.1 Oregonator Model: The “Primary Oscillator” of Coproduct Autocontrol
8.4.2 The Modified Lotka-Volterra or Predator-Prey “Primary Oscillator”
8.4.3 The “Flow Control Primary Oscillator”
8.4.4 The Composite System: A Chemical Equilibrium Coupled to a “Primary Oscillator”
8.4.5 “Delayed Negative Feedback Oscillator”
8.5 Overview and Hints for Further Reading
8.6 Key Questions
8.7 Key Words
8.8 Exercises
8.9 Solutions to the Exercises
Chapter 9: The Emergence of Order in Space 9.1 Introduction
9.2 The Reaction-Diffusion Model
9.3 Turing Patterns
9.4 Turing Patterns in a Chemical Laboratory
9.5 Turing Patterns in Nature 9.5.1 Biology: The Development of Embryos
9.5.2 Biology: Regeneration of Tissues
9.5.3 Biology: Phyllotaxis
9.5.4 Biology: Animal Markings
9.5.5 Ecology, Sociology, and Economy
9.5.6 Geomorphology
9.5.7 The Next Development of Turing’s Theory: The Mechanochemical Patterning
9.6 Chemical Waves 9.6.1 Propagator-Controller Model 9.6.1.1 Phase Waves
9.6.1.2 Trigger Waves
9.6.2 Shapes of Chemical Waves 9.6.2.1 Mono- and Bi-Dimensional Waves
9.6.2.2 Three-Dimensional Waves
9.6.2.3 Effect of Curvature
9.7 “Chemical” Waves in Biology 9.7.1 Waves in a Neuron
9.7.2 The Fisher-Kolmogorov Equation
9.7.3 Waves in Our Brain
9.7.4 Waves in Our Heart
9.7.5 Calcium Waves
9.7.6 cAMP Waves: The Case of Dictyostelium Discoideum
9.7.7 Spreading of Species, Epidemics and … Fads
9.8 Liesegang Patterns
9.9 Liesegang Phenomena in Nature 9.9.1 In Geology
9.9.2 In Biology
9.10 A Final Note: The Reaction-Diffusion Structures in Art and Technology 9.10.1 Reaction-Diffusion Processes as Art
9.10.2 Reaction-Diffusion Processes in Technology
9.11 Key Questions
9.12 Key Words
9.13 Hints for Further Reading
9.14 Exercises
9.15 Solutions to the Exercises
Chapter 10: The Emergence of Chaos in Time 10.1 Introduction
10.2 Nonlinearity and Chaos: The Case of the Double Pendulum
10.3 Nonlinearity and Chaos: The Case of the Population Growth and the Logistic Map
10.4 The Universality of Chaos
10.5 Convection
10.6 The Entropy Production in the Nonlinear Regime: The Case of Convection
10.7 The “Butterfly Effect” 10.7.1 The Complexity of Convection in the Terrestrial Atmosphere
10.7.2 The Lorenz’s Model
10.7.3 The Sensitivity to the Initial Conditions
10.7.4 The Hydrodynamic Photochemical Oscillator
10.8 Aperiodic Time Series 10.8.1 How Do We Recognize Chaotic Time Series? 10.8.1.1 Time Delay τ
10.8.1.2 Embedding Dimension m
10.8.1.3 Lyapunov Exponents
10.8.1.4 Kolmogorov-Sinai Entropy
10.8.1.5 Correlation Dimension
10.8.1.6 Permutation Entropy
10.8.1.7 Surrogate Data
10.8.1.8 Short-Term Predictability and Long-Term Unpredictability
10.8.2 Prediction of the Chaotic Time Series 10.8.2.1 Artificial Neural Networks
10.9 Mastering Chaos 10.9.1 Applications 10.9.1.1 Communication by Chaotic Dynamics
10.9.1.2 Computing by Chaotic Dynamics
10.10 Key Questions
10.11 Key Words
10.12 Hints for Further Reading
10.13 Exercises
10.14 Solutions to the Exercises
Chapter 11: Chaos in Space: The Fractals 11.1 Introduction
11.2 What Is a Fractal?
11.3 Fractal Dimension
11.4 Fractals That Are Not Perfectly Self-Similar
11.5 The Fractal-like Structures in Nature
11.6 The Dimensions of Fractals That Are Not Perfectly Self-Similar
11.7 A Method for Generating Fractal-like Structures in the Lab
11.8 Dendritic Fractals
11.9 Multifractals 11.9.1 Analysis of the Complex Images
11.9.2 Analysis of the Complex Time Series
11.10 Diffusion in Fractals
11.11 Chemical Reactions on Fractals and Fractal-like Kinetics in Cells
11.12 Power Laws or Stretched Exponential Functions?
11.13 Why Does Chaos Generate Fractals?
11.14 Chaos, Fractals, and Entropy
11.15 Key Questions
11.16 Key Words
11.17 Hints for Further Reading
11.18 Exercises
11.19 Solutions to the Exercises
Chapter 12: Complex Systems 12.1 The Natural Complexity Challenges
12.2 The Computational Complexity of the Natural Complex Systems
12.3 If It Were NP = P, Would Be the Complexity Challenges Surely Won?
12.4 The Features of Complex Systems 12.4.1 Networks
12.4.2 Out-of-Equilibrium Systems 12.4.2.1 The Thermodynamics of Thermal Radiation
12.4.2.2 The Fate of the Solar Thermal Radiation and the Climate Change
12.4.2.3 Solar Radiation and Life on Earth
12.4.2.4 Solar Radiation as an Energy Source for Life on Earth
12.4.2.5 Solar Radiation as Information Source for Life on Earth
12.4.3 Emergent Properties
12.5 Key Questions
12.6 Key Words
12.7 Hints for Further Reading
12.8 Exercises
12.9 Solutions to the Exercises
Chapter 13: How to Untangle Complex Systems? 13.1 Introduction
13.2 Improving Electronic Computers
13.3 Natural Computing 13.3.1 Computing Inspired by Natural Information Systems 13.3.1.1 Artificial Life, Systems Chemistry, Systems Biology, and Synthetic Biology
13.3.1.2 Membrane Computing
13.3.1.3 DNA and RNA Computing
13.3.1.4 Evolutionary Computing
13.3.1.5 Artificial Immune Systems
13.3.1.6 Cellular Automata
13.3.1.7 Artificial Intelligence, Fuzzy Logic, and Robots
13.3.1.8 Protein Computing
13.3.1.9 Amorphous Computing
13.3.1.10 Building Models of Complex Systems: ODEs, Boolean Networks, and Fuzzy Cognitive Maps
13.3.1.11 Agent-Based Modeling
13.3.2 Computing by Exploiting the Physicochemical Laws 13.3.2.1 Thermodynamics
13.3.2.2 Classical Physics
13.3.2.3 Computing with Subatomic Particles, Atoms, and Molecules
13.3.2.4 The “Ultimate Laptop”
13.4 Last Conclusive Thoughts and Perspectives
13.5 Last Motivating Sentences Pronounced by “Important People”
13.6 Key Questions
13.7 Key Words
13.8 Hints for Further Reading
13.9 Exercises
13.10 Solutions of the Exercises
Appendix A: Numerical Solutions of Differential Equations
Appendix B: The Maximum Entropy Method
Appendix C: Fourier Transform of Waveforms
Appendix D: Errors and Uncertainties in Laboratory Experiments
Appendix E: Errors in Numerical Computation
References
Index
Pier Luigi Gentili, Untangling Complex Systems
