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
CONTENT
Prologue -- I. Socio-Spatial Dynamics -- A. Introduction -- B. The Four Lenses to View Socio-Spatial Dynamics -- 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)] -- B. Log-Linear Comparative Advantages Producing Functions: F = x(t)a[l โ{128}{147} x(t)]b -- C. Higher Iterates and Fundamental Bifurcations in Discrete Dynamics -- D. The Exponential Locational Advantages Producing Function -- Conclusions -- III. One Stock, Multiple Locations -- Summary -- A. The General Model -- B. The Log-Linear Comparative Advantages Model -- C. Empirical Evidence -- D. Border Sequences in the One-Stock, Multiple-Location, Log-Linear Model -- E. One-Stock, Multiple-Location, Discrete-Time, Logistic Growth -- Conclusions -- IV. Multiple Stocks, Multiple Locations -- Summary -- A. The General Model -- B. The Two-Stock, Two-Location Model -- C. The Two-Stock, I-Location Case -- D. The Two-Stock, Multiple-Location, Discrete-Time, Logistic Growth Model -- 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
