Dynamics of Hierarchical Systems
An Evolutionary Approach
Published on 1. June 1986
Book
Hardback
XV, 397 pages
978-3-540-13323-0 (ISBN)
Description
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.
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Content
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.