Chaos and Socio-Spatial Dynamics

Presents a discrete in time-space universal map of relative dynamics that is used to unfold an extensive catalogue of dynamic events not previously discussed in mathematical or social science literature. With emphasis on the chaotic dynamics that may ensue, the book describes the evolution on the basis of temporal and locational advantages. It explains nonlinear discrete time dynamic maps primarily through numerical simulations. These very rich qualitative dynamics are linked to evolution processes in socio-spatial systems. Important features include: The analytical properties of the one-stock, two- and three-location map; the numerical results from the one- and two-stock, two- and three-location dynamics; and the demonstration of the map's potential applicability in the social sciences through simulating population dynamics of the U.S. Regions over a two-century period. In addition, this book includes new findings: the Hopf equivalent discrete time dynamics bifurcation; the Feigenbaum slope-sequences; the presence of strange local attractors and containers; switching of extreme states; the presence of different types of turbulence; local and global turbulence. Intended for researchers and advanced graduate students in applied mathematics and an interest in dynamics and chaos. Mathematical social scientists in many other fields will also find this book useful.

Contenu

Prologue.- I. Socio-Spatial Dynamics.- A. Introduction.- 1. A Brief Look at the Literature.- 2. Some Simple Dynamic Models.- 3. The Location-Dependent Elements of a Theory of Socio-Spatial Dynamics.- B. The Four Lenses to View Socio-Spatial Dynamics.- 1. The Absolute-Relative Lens.- 2. Continuous Dynamics.- 3. Discrete Dynamics: The Universal Discrete Relative Dynamics Model.- Conclusions.- II. One Stock, Two Regions.- Summary.- A. The First Iterate and Associated Analytical Properties of the Model: x(t + 1) = 1/1 + AF[x(t)].- 1. Overview.- 2. Fixed-Point Behavior and the Discrete Map.- 3. Competitive Exclusion Equilibria.- 4. Classification of Fundamental Relative Spatial Dynamics.- B. Log-Linear Comparative Advantages Producing Functions: F = x(t)a[l - x(t)]b.- 1. Interpretation and Discussion of the Log-Linear Model.- 2. Intervals of Stability of Equilibria.- 3. Analytical Properties of the Log-Linear Model.- 4. Geometric Description of the Iterative Process.- C. Higher Iterates and Fundamental Bifurcations in Discrete Dynamics.- 1. The Second Iterate and Two-Period Cycles.- 2. Analytical Description of Two-Period Cycles.- 3. Period-Doubling and the Feigenbaum Slope-Sequences.- 4. Domains of Nonexistence of k-Period Cycles (k ? 3) and the Hopf Equivalent Bifurcation.- D. The Exponential Locational Advantages Producing Function.- Conclusions.- III. One Stock, Multiple Locations.- Summary.- A. The General Model.- 1. Analytical Results.- 2. Ranking of Stocks According to Size.- 3. Trajectory Domains for the One-Stock, Three-Location Model.- B. The Log-Linear Comparative Advantages Model.- 1. Analytical Properties.- 2. Numerical Analysis for the One-Stock, Multiple-Location Model.- C. Empirical Evidence.- 1. The Nine U.S. Regions and Their Aggregations.- 2. The Time Step and the Forces at Work.- 3. U.S. Regional Relative Population Instability.- 4. The Statistical Tests of Significance.- D. Border Sequences in the One-Stock, Multiple-Location, Log-Linear Model.- 1. Some General Results.- 2. Areas of State Variable Movement in the One-Stock, Three-Location, Log-Linear Model.- E. One-Stock, Multiple-Location, Discrete-Time, Logistic Growth.- 1. Definition and Central Analytical Properties.- 2. The Jacobi Matrix.- 3. Equilibria.- Conclusions.- IV. Multiple Stocks, Multiple Locations.- Summary.- A. The General Model.- 1. Analytical Results.- 2. The Log-Linear Specification.- B. The Two-Stock, Two-Location Model.- 1. The General Case.- 2. The Log-Linear Specifications.- 3. An Example.- 4. Numerical Results for the Two-Stock, Two-Location, Log-Linear Model.- C. The Two-Stock, I-Location Case.- 1. The General Specifications.- 2. The Log-Linear Specifications.- D. The Two-Stock, Multiple-Location, Discrete-Time, Logistic Growth Model.- 1. Definitions and Analytical Properties.- 2. The Jacobi Block-Matrix.- 3. Dynamic Structure of the Equilibria.- Conclusions.- Epilogue.- 1. Six Central Issues.- 2. Three Areas of Application.- 3. Further Research Suggestions.- Appendix I Second-Order Determinants of the Three-Location, One-Stock Model.- Appendix II The Determinant of the Log-Linear Model.- References.- Author Index.