Written in a style that breaks the barriers between the disciplines, this monograph enables researchers from life science, physics, engineering, or chemistry to access the most recent results in a common language. The resulting review character of this project sets it apart from specialized journals, and allows each volume to respond quickly to new developments.This third volume contains new topics ranging from chaotic computing, via random dice tossing and stochastic limit-cycle oscillators, to a number theoretic example of self-organized criticality, wave localization in complex networks and anomalous diffusion.A first-class board of international scientists advises the editor, such that the carefully selected and invited contributions represent the latest and most relevant findings.
Author(s): Heinz Georg Schuster
Edition: 1
Publisher: Wiley-VCH
Year: 2010
Language: English
Pages: 262
Tags: Математика;Нелинейная динамика;
Reviews of Nonlinear Dynamics and Complexity......Page 4
Contents......Page 8
Preface......Page 14
List of Contributors......Page 16
1.1 Brief History of Computers......Page 18
1.2 The Conceptualization, Foundations, Design and Implementation of Current Computer Architectures......Page 19
1.3 Limits of Binary Computers and Alternative Approaches to Computation: What Lies Beyond Moore’s Law?......Page 20
1.4 Exploiting Nonlinear Dynamics for Computations......Page 21
1.5 General Concept......Page 22
1.6 Continuous-Time Nonlinear System......Page 25
1.7.1 Discrete-Time Nonlinear System......Page 27
1.7.2 Continuous-Time Nonlinear System......Page 30
1.8 Logic from Nonlinear Evolution: Dynamical Logic Outputs......Page 33
1.8.1 Implementation of Half- and Full-Adder Operations......Page 34
1.9 Exploiting Nonlinear Dynamics to Store and Process Information......Page 35
1.9.1 Encoding Information......Page 36
1.9.2 Processing Information......Page 38
1.9.3 Representative Example......Page 41
1.9.4 Implementation of the Search Method with Josephson Junctions......Page 42
1.9.5 Discussions......Page 45
1.10 VLSI Implementation of Chaotic Computing Architectures: Proof of Concept......Page 47
1.11 Conclusions......Page 49
References......Page 51
2.1 Introduction......Page 54
2.2 Model......Page 55
2.2.1 Bounce Map with Dissipation......Page 57
2.3 Phase Space Structure: Poincaré Section......Page 58
2.4 Orientation Flip Diagrams......Page 63
2.5 Bounce Diagrams......Page 70
2.6 Summary and Conclusions......Page 73
2.7 Acknowledgments......Page 74
References......Page 75
3.1 Introduction......Page 76
3.2 Phase Description of Oscillator......Page 78
3.3 Oscillator with White Gaussian Noise......Page 79
3.3.1 Stochastic Phase Equation......Page 80
3.3.2 Derivation......Page 82
3.3.3 Steady Phase Distribution and Frequency......Page 85
3.3.4 Numerical Examples......Page 86
3.4.1 Generalized Stochastic Phase Equation......Page 89
3.4.2 Derivation......Page 92
3.4.3 Steady Phase Distribution and Frequency......Page 94
3.4.4 Numerical Examples......Page 95
3.4.5 Phase Equation in Some Limits......Page 98
3.5.1 Periodically Driven Oscillator with White Gaussian Noise......Page 102
3.5.2 Periodically Driven Oscillator with Ornstein–Uhlenbeck Noise......Page 104
3.5.3 Conjecture......Page 105
3.6 Summary......Page 106
References......Page 107
4 Complex Systems, numbers and Number Theory......Page 108
4.1.1 Benford’s Law and Generalized Benford’s Law......Page 110
4.1.2 Are the First-Digit Frequencies of Prime Numbers Benford Distributed?......Page 112
4.1.3 Prime Number Theorem Versus Size-Dependent Generalized Benford’s Law......Page 115
4.1.4 The Primes Counting Function L(N)......Page 116
4.2 Phase Transition in Numbers: the Stochastic Prime Number Generator......Page 118
4.2.1.1 Network Image and Order Parameter......Page 122
4.2.1.2 Annealed Approximation......Page 124
4.2.1.3 Data Collapse......Page 127
4.2.2 Computational Complexity......Page 128
4.2.2.1 Worst-Case Classification......Page 129
4.2.2.2 Easy-Hard-Easy Pattern......Page 130
4.2.2.3 Average-Case Classification......Page 133
4.3 Self-Organized Criticality in Number Systems: Topology Induces Criticality......Page 134
4.3.2 Division Dynamics and SOC......Page 135
4.3.3 Analytical Developments: Statistical Physics Versus Number Theory......Page 138
4.3.4 A More General Class of Models......Page 141
4.4 Conclusions......Page 142
References......Page 143
5.1 Introduction......Page 148
5.2 Complex Networks......Page 150
5.2.1 Scale-Free and Small-World Networks......Page 151
5.2.2 Clustering......Page 154
5.2.3 Percolation on Networks......Page 155
5.2.4 Simulation of Complex Networks......Page 156
5.3.1 Standard Anderson Model and Quantum Percolation......Page 159
5.3.2 Vibrational Excitations and Oscillations......Page 161
5.3.3 Optical Modes in a Network......Page 163
5.3.4 Anderson Model with Magnetic Field......Page 165
5.4.1 Random Matrix Theory......Page 166
5.4.2 Level Statistics for Disordered Systems......Page 168
5.4.3 Corrected Finite-Size Scaling......Page 170
5.4.4 Finite-Size Scaling with Two Parameters......Page 172
5.5 Localization–Delocalization Transitions in Complex Networks......Page 173
5.5.1 Percolation Networks......Page 174
5.5.2 Small-World Networks without Clustering......Page 175
5.5.3 Scale-Free Networks with Clustering......Page 176
5.5.4 Systems with Constant and Random Magnetic Field......Page 178
5.6 Conclusion......Page 180
References......Page 182
6.1 Introduction......Page 186
6.2 Deterministic Chaos......Page 187
6.2.1 Dynamics of Simple Maps......Page 188
6.2.2 Ljapunov Chaos......Page 190
6.2.3 Entropies......Page 195
6.2.4 Open Systems, Fractals and Escape Rates......Page 202
6.3 Deterministic Diffusion......Page 209
6.3.1 What is Deterministic Diffusion?......Page 210
6.3.2.1 The Diffusion Equation......Page 214
6.3.2.2 Basic Idea of the Escape Rate Formalism......Page 215
6.3.2.3 The Escape Rate Formalism Worked out for a Simple Map......Page 217
6.4 Anomalous Diffusion......Page 222
6.4.1.1 What is Anomalous Diffusion?......Page 223
6.4.1.2 Continuous Time Random Walk Theory......Page 226
6.4.1.3 A Fractional Diffusion Equation......Page 230
6.4.2.1 Cell Migration......Page 233
6.4.2.2 Experimental Results......Page 234
6.4.2.3 Theoretical Modeling......Page 236
6.5 Summary......Page 240
References......Page 241
Color Figures......Page 246
Index......Page 258
