The Many Facets of Complexity Science: In Memory of Professor Valentin Afraimovich (Nonlinear Physical Science)

This book explores recent developments in theoretical research and data analysis of real-world complex systems, organized in three parts, namely Entropy, information, and complexity functions Multistability, oscillations, and rhythmic synchronization Diffusions, rotation, and convection in fluids The collection of works devoted to the memory of Professor Valentin Afraimovich provides a deep insight into the recent developments in complexity science by introducing new concepts, methods, and applications in nonlinear dynamical systems covering physical problems and mathematical modelling relevant to economics, genetics, engineering vibrations, as well as classic problems in physics, fluid and climate dynamics, and urban dynamics. The book facilitates a better understanding of the mechanisms and phenomena in nonlinear dynamics and develops the corresponding mathematical theory to apply nonlinear design to practical engineering. It can be read by mathematicians, physicists, complex systems scientists, IT specialists, civil engineers, data scientists, and urban planners. Preface 7 Contents 8 The Directional Entropy for Spatially Extended Dynamical Systems 10 1 Introduction 10 2 LDS and Directional Entropy 12 2.1 LDS 13 2.2 Definitions and Properties of Directional Entropy 14 3 Directional Entropy in Lifts Dynamical Systems 15 4 Conclusion 18 References 20 Detecting Regularity with Complexity Functions 22 1 Introduction 22 2 Complexity Functions 23 2.1 Complexity Function 23 2.2 Local Complexity Function 25 3 A Master-Slave Case 26 3.1 A Physical Interpretation of (ε,t)-Separability 27 3.2 Numerical Measurement of the Complexity Function 28 4 Application to the Standard Map 30 4.1 Brief Introduction to the Standard Map 30 4.2 Study Framework 30 4.3 Results on the Trapping Times τ 32 4.4 Summary of the Efficiency of Detecting Regularity with Complexity for the la Standard Map 33 4.5 Influence of the Parameters 35 5 Application to the HMF 36 5.1 Introduction 36 5.2 Introduction of the HMF Model, and the Master-Slave Implementation 37 5.3 Chaos as a Function of the Number of Particles N 39 5.4 Simulations of the HMF 39 6 Conclusion 42 References 43 Local Complexity Functions of the Ehrenfest's Wind-Tree Model 46 References 51 Selective Chaos of Travelling Waves in Feedforward Chains of Bistable Maps 53 1 Introduction 53 2 Unidirectional Systems, Chaos of Travelling Waves and Symbolic Dynamics 54 3 Selective Chaos of Travelling Waves for Finite Rank Approximations 58 3.1 Rank 1 Approximation 59 3.2 Rank 2 Approximation 59 3.3 Rank 3 Approximation 64 4 Concluding Remarks 68 References 69 On Periodic Motions in a van der Pol Oscillator 71 1 Introduction 71 2 A Semi-analytical Method 73 3 Finite Fourier Series 76 4 Semi-analytical Solutions 78 5 Numerical Illustrations 83 6 Conclusions 88 References 88 Hidden Periodic Motions for Brushless Motor with Unsteady Torque Excitation 90 1 Introduction 90 2 Motion Discretization and Mapping Reconstruction for Periodic Motions 92 3 Analytical Bifurcation 95 4 Periodic Motions 98 5 Conclusions 104 References 107 Chunking Rhythmic Synchronization: Bellerophon States and Quantized Clusters of Globally Coupled Phase Oscillators 109 1 Introduction 109 2 Results 111 3 Discussion 119 References 120 Chatter Dynamics and Stability of the Impulsive van der Pol Equation 121 1 Introduction 121 2 Chatter Dynamics 123 3 Stability Analysis 127 3.1 Preliminaries 127 3.2 Basic Conceptions 130 3.3 Stability Criteria 133 4 Applications 137 5 Conclusions 140 References 141 Complex Motions in an Inclined Impact Pair with a Periodic Excitation 143 1 Introduction 143 2 Physical Model 145 3 Flow Switchability 146 3.1 Domains and Boundaries 146 3.2 Switching Results 150 4 Mapping Structures and Periodic Motions 155 4.1 Periodic Motions with Stick 155 4.2 Periodic Motions Without Stick 159 5 Conclusions 164 References 165 Nonlinear Dynamics of Deep Open-Ocean Convection: An Analytical Approach 166 1 Introduction 166 2 Whitehead's `Tank Model' with Saline Upper Layer 168 2.1 Model Description 168 2.2 Model Equations and Non-Dimensional Variables 169 3 Transitional Dynamics: Exact Analytical Solutions 172 3.1 A Generic Model of Transitional Dynamics 173 3.2 Exact Analytical Solutions for Different Dynamic Regimes 177 4 Implications for the Ocean 183 5 Conclusions 185 References 186 On the Necessary Conditions for Preserving the Nonnegative Cone: Mixed Diffusion 189 1 Introduction 189 2 The Preservation of the Nonnegativity of the Solution of the System with Mixed Diffusion 193 References 195 Multi-scale Analysis of Urban Spatial Structures Acquired from OpenStreetMap 197 1 Introduction 197 2 Spatial Graphs of Urban Environments 199 3 Discrete Time Anisotropic Scale-Dependent Random Walks 200 4 Stationary Distributions and Balance Equations of Anisotropic Random Walks 201 5 Exploring Graphs with Random Walks 203 6 The City of Lubbock Is Running Away 204 6.1 Isolation Index 206 6.2 Integration Index 207 7 Discussion and Conclusion 208 References 209