The principle of local activity explains the emergence of complex patterns in a homogeneous medium. At first defined in the theory of nonlinear electronic circuits in a mathematically rigorous way, it can be generalized and proven at least for the class of nonlinear reaction–diffusion systems in physics, chemistry, biology, and brain research. Recently, it was realized by memristors for nanoelectronic device applications. In general, the emergence of complex patterns and structures is explained by symmetry breaking in homogeneous media, which is caused by local activity. This book argues that the principle of local activity is really fundamental in science, and can even be identified in quantum cosmology as symmetry breaking of local gauge symmetries generating the complexity of matter and forces in our universe. Applications are considered in economic, financial, and social systems with the emergence of equilibrium states, symmetry breaking at critical points of phase transitions and risky acting at the edge of chaos.

Preface vii

1. The Local Activity P ri nciple and the Emergenc of Complexity 1

1.1 Mathematical Definition of Local Activity 1

1.2 The Local Activity Theorem 9

1.3 Local Activity i the Origin of Complexity 15

2. Local Activity and Edge of Chaos in Computer Visualization 23

2.1 Local Activity and Edge of Chaos of the Brus elator Equations 23

2.2 Local Activity and Edge of Chaos of the Gierer- Meinhardt Equations 35

2.3 Local Activity and Edge of Chaos of the FitzHugh- Nagumo Equations 4

2.4 Local Activity and Edge of Chaos of the Hodgkin- Huxley Equations 75

2.5 Local Activity and Edge of Chaos of the Oregonator Equations 4

3. The Local Activity Principle and the Expansion of the Universe 105

3.1 Mathematical Definition of Symmetry 105

3.2 Symmetries in the Quantum World 107

3.3 Global and Local Symmetrie 10

3.4 Local Gauge Symmetries and Symmetry Breaking 115

4. The Local Activity Principle and t he Dynamic of Matter 147

4.1 The Local Activity Principle of Pattern Formation 147

4.2 The Local Activity Principle and Prigogin 's Dissipative Structures 149

4.3 The Local Activity Principle and Hakn ' yn rgetics 165

5. T he Local Activity Principle and the Evolution of Life 171

5.1 The Local Activity Principle of Turing's Morphogenesis 172

5.2 The Local Activity Principle in Systems Biology 174

5.3 The LocaJ Activity Principle in Brain Research 192

6. T he Local Activit , P rinciple and the Co-evolution of Technology 205

6.1 The LocaJ Activity P rinciple of Cellular Automata 205

6.2 The Local Activity Principle of eural Networks 269

6.3 The Local Activity P rinciple of Memristors 287

6.4 T he Local Activity Principle of Global Information Tetworks 312

7. T he Local Activity P rinciple and Innovation in the Economy and Society 327

7.1 The Local Activity Principle in Sociodynamics 328

7.2 The Local Activity P rinciple and Emerging Riks 338

7.3 T he Local Activity Principle in Financial Dynamic 355

7.4 The Local Activity Principle in Innovation Dynamics 376

7.5 T he Local Activity Principle of Sustainable Entrepreneurship 386

8 The lVIessage of the Local Activity Principle 393

8.1 T he Local Activity Principle in Culture and Philosophy 394

8.2 What can we Learn from the Local Activity Principle in the Age of Globalization? 406

References 409

Author Index 423

Subject Index 425

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