Dynamics of Hierarchical Systems: An Evolutionary Approach

ISBN-13: 9783642696947 / Angielski / Miękka / 2011 / 397 str.

John S. Nicolis

Opis

The main aim of these lectures is to tri gger the interest of the restless under- graduate student of physical, mathematical, engineering, or biological sciences in the new and exciting multidisciplinary area of the evolution of "large-scale" dynamical systems. This text grew out of a synthesis of rather heterogeneous mate- rial that I presented on various occasions and in different contexts. For example, from lectures given since 1972 to first- and final-year undergraduate and first- year graduate students at the School of Engineering of the University of Patras and from informal seminars offered to an international group of graduate and post- doctoral students and faculty members at the University of Stuttgart in the aca- demic year 1982-1983. Those who search for rigor or even formality in this book are bound to be rather disappointed. My intention is to start from "scratch" if possible, keeping the rea- soning heuristic and tied as closely as possible to physical intuition; I assume as prerequisites just basic knowledge of (classical) physics (at the level of the Berkeley series or the Feynman lectures), calculus, and some elements of probabil- ity theory. This does not mean that I intended to write an easy book, but rather to eliminate any difficulty for an eager reader who, in spite of incomplete for- malistic training, would like to become acquainted with the physical ideas and con- cepts underlying the evolution and dynamics of complex systems.

Spis treści

1. Introduction.- 1.1 What This Book Is About.- 1.2 Statement of the Problem.- 1.3 Some Preliminary Definitions of Complexity and Organization.- 1.3.1 Complexity.- 1.3.2 Organization.- 2. Preliminaries from Nonlinear Dynamics and Statistical Physics.- 2.1 Symmetries and Conservation Principles.- 2.2 Instabilities at the Root of Broken Symmetries, Dissipation, and Irreversibility for Low-Dimensional (Not Statistical) Dynamical Systems.- 2.2.1 The Role of Gravitation.- 2.2.2 Comments on the Role of Coupling Among the Four Basic Interactions in Evolution.- 2.2.3 The Overdamped Nonlinear Oscillator: A Case of Spontaneously Breaking Symmetry.- 2.2.4 The Laser: A Case of Broken Symmetry.- 2.2.5 The Rotating Pendulum: A Case of Bifurcation Leading to Spontaneous Symmetry Breaking.- 2.2.6 Broken Symmetry Through a Hysteresis-Like Process.- 2.2.7 Essentials of Stability Theory.- a) General Criterion.- b) Specific Analyses.- 2.2.8 Behavior of a Two-Dimensional Dynamical System in the Vicinity of Singular Points (Steady States).- 2.2.9 First Encounter with Nontrivial Dissipative Systems: The Concept of the Attractor in Two Dimensions (Limit Cycle).- 2.3 Elements of Statistical Physics and Their Relevance to Evolutionary Phenomena.- 2.3.1 Some Characteristics of Stochastic Systems.- 2.3.2 Informational Entropy, Physical Entropy, Thermodynamic Entropy.- 2.3.3 Entropy of a Perfect Gas at Thermodynamic Equilibrium.- 2.3.4 Entropy of a Photon Gas at Thermodynamic Equilibrium.- 2.3.5 Elements of Newtonian Big Bang Cosmology.- 2.3.6 Expansion of a Mixture of Matter and Radiation. Differential Cooling and Entropy Production.- 2.3.7 The Concept of Complexity.- a) Structural Complexity and Its Relationship to the Stability of a System.- b) Algorithmic Complexity.- 2.4 Concluding Remarks.- 3. The Role of Spherical Electromagnetic Waves as Information Carriers.- 3.1 Radiation from Accelerated Charge in Vacuo. The Concept of “Self”-Force. Thermodynamics of Electromagnetic Radiation.- 3.1.1 Radiation in Vacuum.- 3.1.2 The Concept of Self-Force.- 3.1.3 Thermodynamics of Electromagnetic Radiation.- 3.2 Electromagnetic Wave Propagation in Dispersive Media and Lossy Media.- 3.3 Analysis of a Spherical Wave in Terms of Elemental “Rays”. The Mode Theory of Wave Propagation. Excitable Modes (Degrees of Freedom) in a Closed Cavity.- 3.3.1 Spectral Decomposition of a Spherical Wave.- 3.3.2 The Wave-Guide Mode Theory of Wave Propagation.- 3.3.3 A Cavity Resonator.- 3.4 The Entropy of Electromagnetic Radiation. Information Received by an Electromagnetic Wave Impinging on a Finite Aperture. Ambiguity of Perception.- 4. Elements of Information and Coding Theory, with Applications.- 4.1 Information Transfer and the Concept of Channel Capacity for Discrete and Continuous Memoryless Signals.- 4.2 Some Ideas from Coding Theory Instrumental in Minimizing Reception Error.- 4.3 Some Efficient Coding Algorithms for Source-Channel Matching and Single-Error Detection and Correction.- 4.3.1 Coding for Source-Channel Matching.- 4.3.2 Coding for Error Detection and Correction.- 4.4 Information Sources with Memory. Markov Chains.- 4.5 Specific Examples of Some Useful Channels and Calculations of Their Capacities.- 4.5.1 Capacity of a Homogeneously Turbulent Channel.- 4.5.2 The Lossless Channel.- 4.5.3 The Deterministic Channel.- 4.5.4 The Uniform Channel.- 4.5.5 The Binary Symmetrical Channel.- 4.5.6 The Binary “Erasure” Channel.- 4.5.7 Capacity of an Optical Channel.- 4.5.8 Role of Quantum Noise in an Optical Channel.- 4.5.9 An Introduction to the “Genetic Channel” and the Genetic Code.- 4.5.10 The Phase-Locked Loop in the Absence and Presence of Noise.- 4.6 Modeling of Stochastic Time Series.- 4.7 Communication Between Two Hierarchical Systems Modeled by Controlled Markov Chains.- 4.7.1 Introduction: Elaboration of the Nature of Hierarchical Systems.- 4.7.2 Dynamics at the Base Levels Q, Q’ and the Underlying Game.- 4.7.3 A Semi-Markov Chain Model for the Hierarchical Levels W and W’.- 4.7.4 The Control Problem.- a) Biological Rhythms Underlying the Games.- b) Description of the Communication and Control Processes.- c) Selection of Control Mechanisms.- 4.7.5 Computer Simulation.- 4.7.6 Biological Relevance of the Model.- 4.8 Emergence of New Hierarchical Levels in a Self-Organizing System.- 4.8.1 Formulation of the Problem.- 4.8.2 Creation of a New Hierarchical Level.- 4.8.3 A Comment on Typical Cases of “Psychosomatic Disturbances”.- 5. Elements of Game Theory, with Applications.- 5.1 Constant-Sum Games.- 5.1.1 Both Players Have a Dominant Strategy.- 5.1.2 Only One Player Has a Dominant Strategy.- 5.1.3 Neither Player Has a Dominant Strategy.- 5.1.4 Mixed Strategies.- 5.2 Non-Constant-Sum Games.- 5.2.1 Non-Constant-Sum “Negotiable” Games.- 5.2.2 Non-Constant-Sum, Nonnegotiable “Paradoxical” Games.- 5.3 Competing Species.- 5.4 Survival and Extinction.- 5.5 Some Elementary Knowledge from Genetics: Selection and Fitness.- 5.6 Games Between Animals Adopting Specific Modes of Behavior (Roles). Concepts of Evolutionarily Stable Strategy.- 5.7 The Game of Competitive-Cooperative Production and Exchange. The Concept of “Parasite” at a Symbolic Level.- 5.8 Epidemiology of Rumors.- 6. Stochasticiky Due to Deterministic Dynamics in Three- or Higher-Dimensional Space: Chaos and Strange Attractors.- 6.1 A Reappraisal of Classical Statistical Mechanics. The Kolmogorov-Arnold-Moser Theorem.- 6.2 Dynamics in Three-Dimensional State Space (Three Degrees of Freedom). Steady States, Limit Cycles, Attracting Tori.- 6.3 Strange Attractors.- 6.3.1 One-Dimensional Maps on the Interval. The “Logistic” Model.- 6.3.2 Fractal Dimensionality. The Cantor Set.- 6.3.3 The Concept of the Lyapounov Exponents for the Period-Doubling and Chaotic Regimes.- 6.3.4 A Typical Three-Dimensional Strange Attractor. The Lorenz Model.- 6.3.5 The Rate of Information Production by the Lorenz Attractor.- 6.4 Parameters Characterizing the Average Behavior of Strange Attractors: Dimensions, Entropies, and Lyapounov Exponents.- 6.4.1 The Concept of Information Dimension.- 6.4.2 The Concept of Characteristic Lyapounov Exponents and Their Relation to Information Dimension.- 6.4.3 The Concept of Metric (Kolmogorov-Sinai) Entropy and Its Relation to Information Dimension.- 6.5 A Possible Role for Chaos in Reliable Information Processing.- 6.5.1 Theoretical Considerations and General Discussion.- 6.5.2 Application: The Electrical Activity of the Brain — Should It Be Chaotic?.- 6.5.3 Experimental Data from EEG Research.- 6.5.4 The Model.- 6.5.5 The Dual Role of Intermittency in Information Processing.- 6.5.6 The Origin of Conflict in Communicating Hierarchical Systems.- 6.6 Comments on the Effects of Internal Fluctuations and External Noise on the Stability Properties of Dynamical Systems.- 7. Epilogue: Relevance of Chaos to Biology and Related Fields.- 7.1 Computational Complexity.- 7.2 Towards a Dynamic Theory of Language.- 7.2.1 The Nature of the Problem.- 7.2.2 Structural and Functional Hierarchical Levels.- 7.2.3 An Evolutionary Linguistic Model: Digits and Patterns.- a) Total Entrainment.- b) Part Entrainment, Part “Jittery” Phase Locking.- c) Chaos.- 7.2.4 Unresolved Problems: Communication Between Two Hierarchical Systems.- 7.3 Concluding Remarks.- A. A View of the Role of External Noise at a Neuronal Hierarchical Level.- A.1 Introduction to the Problem.- A.2 Organization Through Weak Stationary-Amplitude Noise.- A.3 Relevance of the Model to Neuronal and Cognitive Organization.- B. On the Difficulty of Treating the Transaction Between Two Hierarchical Levels with Continuous Nonlinear Dynamics.- B.1 The Level Q of Partner I.- B.2 Homeostasis and Cross-Correlations.- B.3 The Level W of Partner I.- B.4 The Controller.- C. Noisy Entrainment of a Slightly Nonlinear Relaxation Oscillator by an External Harmonic Excitation.- C.1 General Description of the Model.- C.2 A Method for the Study of Entrainment.- C.2.1 Strict Entrainment.- C.2.2 Loose or “Jittery” Entrainment.- C.2.3 Pure “Free-Running” Oscillation.- C.2.4 Free-Running Oscillation.- C.3 Mathematical Treatment and Computer Simulation.- C.4 Behavior of the Oscillator Under an Applied Harmonic Excitation (Entrainment).- References.