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11. Cities as Spatial Chaotic Attractors
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CHAPTER 11 Cities as Spatial Chaotic Attractors Dimitrios S. Dendrinos Developments in the mathematical theory of dynamical stability and bifurcation theory have considerably influenced developments throughout the social sciences. S.ources have been either the field of structural stability emanating from catastrophe theory (see Thorn 1975; Zeeman 1977), mathematical chaos (see Lorenz 1963; May 1976; Guckenheimer and Holmes 1983; Thompson and Stewart 1986; Schuster 1988; Devaney 1989), or the theory of fractals (see Mandelbrot 1977; Peitgen and Richter 1986; Barnsley 1988). The above are only a short list of basic references among many that have appeared over the past 30 years in these fields. Almost without exception, social science fields have been influenced by these developments, including areas of study in archaeology, history, sociology , demography, political science, geography, psychology, and economics. In particular, the fields of economics (micro- and macroeconomics, including economic growth theory, the theory of finance, business cycles theory, technological progress, international trade, and so on; see Anderson, Arrow, and Pines 1988; Barnett, Geweke, and Shall 1989; Barnett and Chen 1990; and Rosser 1991), and spatial analysis (geography, urban and regional analysis, transportation dynamics and traffic flow analysis, and so on), have experienced considerable impact from these mathematical advances. Along with the considerable new (basic as well as marginal) insight gained by such applications, there has also been a revision of past claims in social theory. To some, a call for reconfiguring the social sciences has emerged from insights gained through these newly developed fields. But, together with this call, the realization has settled in that there are, at least at present, certain insurmountable practical as well as theoretical obstacles to such reconfiguration. This paper was written during spring, 1992. 237 238 Chaos Theory in the Social Sciences Contributions At the outset, a few basic points will be made as to the magnitude of contributions to and extent of impact from applications emanating from nonlinear dynamics into social sciences. Then, some remarks will follow, addressing the question of the considerable drawbacks of developing a truly dynamic social science at present. Basic and Marginal Inroads This is not the appropriate forum to survey all the areas where chaos theory has made inroads or to fully assess the impact from or the value of these contributions. Some of the papers in this volume do so in their own special fields. Whereas in some areas, like those in spatial analysis and geography, the contributions might be basic, in other areas, particularly in economics, one may find the contributions made to date to be rather marginal. In economics, the most credible of the attempts to exploit chaos (and catastrophe) theory occurred when existing classical and neoclassical deductionist (causality-based) theories were modified to marginally change preexisting (and widely accepted among disciples of current orthodoxies) assumptions . Such changes were offered as means of showing how chaotic dynamics can arise from currently prevailing theoretical models, and thus enhance these models' utility. There is another side to this coin, however. Marginal extensions of currently accepted paradigms in economic analysis point directly to one of their weaknesses, namely, that under slight but proper modifications in specification , these models can reproduce almost anything that the analyst wishes to produce through theoretical deduction. A scientist might find such accommodations somewhat troublesome. New Dynamical Features The single most important contribution mathematical chaos has made is to demonstrate the possible presence of new dynamical features in social systems that theoreticians had never addressed before. Until very recently, aperiodic movement was an attribute of social science models to avoid and quasi-periodic motion was never even conceived of. Chaos theory showed that through such dynamics one can detect interesting and novel insights in the behavior of social systems. It also supplied a method to classify the various dynamical trajectories obtainable in phase space and to catalog a large number of possible bifurcations. [3.236.139.73] Project MUSE (2024-03-28 19:07 GMT) Cities as Spatial Chaotic Attractors 239 A New Type of Explanation Mathematical chaos further suggests that periodic, quasi-periodic, and nonperiodic trajectories in the phase portrait of interdependent socioeconomic variables can be generated by very simple dynamical equations. Chaos occurs in dimensionally very low systems, where it most often has been studied. High-dimensional chaos is still an elusive subject. Efficiently capturing aperiodic motions can be accomplished with simple dynamical specifications. Efficient dynamic specifications can be viewed as a new method of explanation in the social sciences. Mathematical chaos highlights a topic first addressed about twenty years...