Complex and Adaptive Dynamical Systems: A Primer

by Claudius Gros

Overview

We are living in an ever more complex world, an epoch where human actions can accordingly acquire far-reaching potentialities. Complex and adaptive dynamical systems are ubiquitous in the world surrounding us and require us to adapt to new realities and the way of dealing with them.

This primer has been developed with the aim of conveying a wide range of "Commons-sense" Knowledge in the field of quantitative complex system science at an introductory level, providing an entry point to this both fascinating and vitally important subject. The approach is modular and phenomenology driven. Examples of emerging phenomena of generic importance treated in this book are:

The small world phenomenon in social and scale-free networks.

Phase transitions and self-organized criticality in adaptive systems.

Life at the edge of chaos and coevolutionary avalanches resulting from the unfolding of all living.

The concept of living dynamical systems and emotional diffusive control within cognitive system theory.

Technical course prerequisites are a basic knowledge of ordinary and partial differential equations and of statistics. Each chapter comes with exercises and suggestions for further reading - solutions to the exercises are also provided.

This second edition adds a new chapter on quantifiying/measuring complexity in given systems, together with an introduction to information theory, has an expanded exercises and solutions section, and contains both revised and additional subsections.

Product Details

ISBN-13:	9783642047060
Publisher:	Springer-Verlag New York, LLC
Publication date:	09/24/2010
Sold by:	Barnes & Noble
Format:	eBook
File size:	6 MB

About the Author

Prof. Claudius Gros holds a chair in theoretical physics at the Johann Wolfgang von Goethe University in Frankfurt am Main, Germany. After completing a PhD in theoretical condensed-matter physics in 1988 at the ETH in Zurich, Switzerland, Prof. Gros went as Chester-Davies fellow to Indiana University, US, before returning to Germany. His present research interests include Cognitive System Theory and Solid State Physics, with emphasis on high-temperature superconductors.

Table of Contents

1 Graph Theory and Small-World Networks 1

1.1 Graph Theory and Real-World Networks 1

1.1.1 The Small-World Effect 1

1.1.2 Basic Graph-Theoretical Concepts 3

1.1.3 Properties of Random Graphs 8

1.2 Generalized Random Graphs 14

1.2.1 Graphs with Arbitrary Degree Distributions 14

1.2.2 Probability Generating Function Formalism 19

1.2.3 Distribution of Component Sizes 21

1.3 Robustness of Random Networks 24

1.4 Small-World Models 28

1.5 Scale-Free Graphs 30

Exercises 36

Further Reading 37

2 Chaos, Bifurcations and Diffusion 39

2.1 Basic Concepts of Dynamical Systems Theory 39

2.2 The Logistic Map and Deterministic Chaos 45

2.3 Dissipation and Adaption 50

2.3.1 Dissipative Systems and Strange Attractors 50

2.3.2 Adaptive Systems 55

2.4 Diffusion and Transport 59

2.4.1 Random Walks, Diffusion and Lévy Flights 60

2.4.2 The Langevin Equation and Diffusion 63

2.5 Noise-Controlled Dynamics 66

2.5.1 Stochastic Escape 66

2.5.2 Stochastic Resonance 69

2.6 Dynamical Systems with Time Delays 73

Exercises 76

Further Reading 77

3 Complexity and Information Theory 79

3.1 Probability Distribution Functions 79

3.1.1 The Law of Large Numbers 83

3.1.2 Time Series Characterization 84

3.2 Entropy and Information 87

3.2.1 Information Content of a Real-World Time Series 93

3.2.2 Mutual Information 95

3.3 Complexity Measures 100

3.3.1 Complexity and Predictability 101

3.3.2 Algorithmic and Generative Complexity 104

Exercises 106

Further Reading 107

4 Random Boolean Networks 109

4.1 Introduction 109

4.2 Random Variables and Networks 111

4.2.1 Boolean Variables and Graph Topologies 111

4.2.2 Coupling Functions 113

4.2.3 Dynamics 115

4.3 The Dynamics of Boolean Networks 116

4.3.1 The Flow of Information Through the Network 117

4.3.2 The Mean-Field Phase Diagram 119

4.3.3 The Bifurcation Phase Diagram 121

4.3.4 Scale-Free Boolean Networks 125

4.4 Cycles and Attractors 127

4.4.1 Quenched Boolean Dynamics 127

4.4.2 The K = 1 Kauffman Network 130

4.4.3 The K = 2 Kauffman Network 132

4.4.4 The K = N Kauffman Network 132

4.5 Applications 135

4.5.1 Living at the Edge of Chaos 135

4.5.2 The Yeast Cell Cycle 136

4.5.3 Application to Neural Networks 139

Exercises 140

Further Reading 141

5 Cellular Automata and Self-Organized Criticality 145

5.1 The Landau Theory of Phase Transitions 145

5.2 Criticality in Dynamical Systems 150

5.2.1 1/f Noise 154

5.3 Cellular Automata 155

5.3.1 Conway's Game of Life 156

5.3.2 The Forest Fire Model 157

5.4 The Sandpile Model and Self-Organized Criticality 159

5.5 Random Branching Theory 161

5.5.1 Branching Theory of Self-Organized Criticality 161

5.5.2 Galton-Watson Processes 166

5.6 Application to Long-Term Evolution 168

Exercises 175

Further Reading 176

6 Darwinian Evolution, Hypercycles and Game Theory 179

6.1 Introduction 179

6.2 Mutations and Fitness in a Static Environment 181

6.3 Deterministic Evolution 185

6.3.1 Evolution Equations 185

6.3.2 Beanbag Genetics - Evolutions Without Epistasis 189

6.3.3 Epistatic Interactions and the Error Catastrophe 191

6.4 Finite Populations and Stochastic Escape 195

6.4.1 Strong Selective Pressure and Adaptive Climbing 196

6.4.2 Adaptive Climbing Versus Stochastic Escape 199

6.5 Prebiotic Evolution 201

6.5.1 Quasispecies Theory 201

6.5.2 Hypercycles and Autocatalytic Networks 202

6.6 Coevolution and Game Theory 206

Exercises 211

Further Reading 212

7 Synchronization Phenomena 215

7.1 Frequency Locking 215

7.2 Synchronization of Coupled Oscillators 216

7.3 Synchronization with Time Delays 223

7.4 Synchronization via Aggregate Averaging 225

7.5 Synchronization via Causal Signaling 229

7.6 Synchronization and Object Recognition in Neural Networks 233

7.7 Synchronization Phenomena in Epidemics 236

Exercises 240

Further Reading 241

8 Elements of Cognitive Systems Theory 243

8.1 Introduction 243

8.2 Foundations of Cognitive Systems Theory 245

8.2.1 Basic Requirements for the Dynamics 245

8.2.2 Cognitive Information Processing Versus Diffusive Control 249

8.2.3 Basic Layout Principles 251

8.2.4 Learning and Memory Representations 253

8.3 Motivation, Benchmarks and Diffusive Emotional Control 257

8.3.1 Cognitive Tasks 257

8.3.2 Internal Benchmarks 258

8.4 Competitive Dynamics and Winning Coalitions 261

8.4.1 General Considerations 262

8.4.2 Associative Thought Processes 266

8.4.3 Autonomous Online Learning 270

8.5 Environmental Model Building 273

8.5.1 The Elman Simple Recurrent Network 273

8.5.2 Universal Prediction Tasks 277

Exercises 279

Further Reading 280

9 Solutions 283

Solutions to the Exercises of Chapter 1 283

Solutions to the Exercises of Chapter 2 289

Solutions to the Exercises of Chapter 3 292

Solutions to the Exercises of Chapter 4 298

Solutions to the Exercises of Chapter 5 301

Solutions to the Exercises of Chapter 6 305

Solutions to the Exercises of Chapter 7 309

Solutions to the Exercises of Chapter 8 312

Index 315

Editorial Reviews

The author combines in the book theoretical elements with their applications, realizing an interesting work with potential practical applications in the domain of complex and adaptive dynamical systems. The book can be read without advanced knowledge in the domain and can be useful to researchers dealing with complex and adaptive dynamical systems.” (Gheorghe Tigan, zbMATH 1317.93001, 2015)

From the Publisher