Noise sustained patterns : fluctuations and nonlinearities

Preface vii Acknowledgements ix Chapter 1 Introduction 1 1.1 Stability theory revisited 1 1.1.1 Noise and metastability 1 1.1.2 Linear stability analysis 6 1.2 Instabilities and nonlinear events in everyday life 11 1.2.1 Arms race 11 1.2.2 Distribution of wealth 13 1.2.3 Front nucleation and propagation 14 1.2.4 Supply chain dynamics 16 1.2.5 A hodgepodge of mechanical instabilities 21 1.3 Postscript 24 Chapter 2 Essentials 25 2.1 Probabilistic and information theoretic measures 25 2.2 Matrix manipulations 27 2.2.1 Circulant matrices 37 2.2.2 Singular value decomposition 39 2.3 Delay-differential equations 40 2.4 The fluctuation-dissipation theorem 43 2.5 The Fokker-Planck equation 45 2.6 Numerical techniques for the simulation of stochastic equations 47 2.7 Experimental aspects of generating noise 50 xi xii Contents 2.8 Complex integration 52 Chapter 3 Noise Induced Temporal Phenomena 59 3.1 Escape from metastable states 59 3.2 Stochastic resonance in bistable systems 61 3.2.1 Double-well potential 63 3.2.2 Two state theory 68 3.2.3 The diode resonator 71 3.2.4 The logistic map 74 3.3 Postscript 76 Chapter 4 Adding Spatial Dimensions 77 4.1 Spatiotemporal stochastic resonance 77 4.1.1 Array enhancement 78 4.1.2 Global vs. local dynamics 83 4.1.3 Kink nucleation in a <pA model 85 4.1.4 Kink nucleation in the experiment 91 4.1.5 Decay rates and kink speeds 92 4.1.6 Coupled maps 95 4.2 Doubly stochastic resonance 99 4.3 Spatial patterns 102 4.3.1 Parametric surface waves 105 4.3.1.1 Noise induced bistability of Faraday waves . . 107 4.3.1.2 Granular flow 109 4.3.1.3 Intermittent pattern switching 115 4.4 Postscript 117 Chapter 5 Stochastic Transport Phenomena 119 5.1 Noise-sustained structures in convectively unstable media . . . 119 5.1.1 The Ginzburg-Landau equation 122 5.1.2 Unidirectionally coupled diode resonators 124 5.1.3 Coupled maps 127 5.2 Noise sustained front transmission 129 5.2.1 Propagation failure in cardiac tissue 137 5.2.2 Information theoretic measures revisited: ROC curves 141 5.3 Theory 147 5.3.1 The continuum limit 148 Contents xiii 5.3.2 Kink Brownian motion 149 5.3.3 The Peierls-Nabarro potential 151 5.4 Noise enhanced wave propagation 152 5.4.1 Monostable noise enhanced propagation 160 5.5 Stochastic ratchets and Brownian motors 162 5.6 Postscript 164 Chapter 6 Sundry Topics 167 6.1 Minority game 167 6.1.1 The thermal minority game 171 6.2 Traffic dynamics 177 6.2.1 Time-continuous model 178 6.2.2 Discrete time dynamics 183 6.3 Dithering 187 6.4 Noise in neural networks 192 Chapter 7 Afterthoughts 197 Appendix A Normal Matrices 203 Appendix B Integrating Colored-Noise, Coupled SDEs 205 B.l Exploiting symmetries of coupled differential equations 207 B.2 Coupled Duffing oscillators 210 Appendix C Numerical Implementation of t h e F P T 213 Appendix D Absolute and Convective Instabilities 217 Bibliography 221 Index 237