European Regional Science Association
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European Regional Science Association
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The abstract for paper number 528:
Dimitrios S. Dendrinos, Dept. of Architecture and Urban Design, University of Kansas, , USA, Michael Sonis, Bar-Ilan University, , Israel
Socio-Spatial Dynamics and Discrete Non-linear Probabilistic Chains
This paper reviews the recent developments that occur in the field of Discrete Socio-Spatial Dynamics (SS Dynamics) during the decade after the publication of the book by D.S. Dendrinos, M. Sonis, 1990, “Chaos and Socio-Spatial dynamics”, Springer Verlag. SS Dynamics means the redistribution dynamics of m different statistical populations (stocks) relatively distributed between n different locations (or choosing n different choice alternatives). Examples of such stocks could be m different populations or labor types; distinct capital stocks (classified, for example, according to vintage); financial capital(currencies); different types of economic outputs (products) or any economic, social, political and other of socio-spatial variables, or combinations of them. The main purpose of this paper is to reevaluate the logical basis of m-populations/n locations SS Dynamics from the view point of new theory of non-linear discrete probabilistic chains. The linear probabilistic chains are well-known Markov chains. The non-linear probabilistic chains in the generalizations of linear Markov chains in the case that set of transitional frequencies do not stable in any statistical meaning, but possible consider the dynamics of finite discrete probability distributions. The general probabilistic chain can be generated by the iteration of transformations of this simplex of all probabilistic vectors of fixed dimension into itself. The paper presents the analytical description of all such transformations. The asymptotical behavior of non-linear probabilistic chains includes the bifurcation behavior much richer then the ergodic properties of Markov chains – quasi-periodic motion and different ways to chaos. In the case of existence of attractors or repellers the shifted Shannon entropy changes regularly (increasing or decreasing) along of the orbit of discrete dynamics. The different forms of probabilistic chain are useful for the statistical evaluation of relative dynamics of many socio-economic stocks, such as migration, population, capital, labor, etc. On this basis the elements of the forecasting of probabilistic chains are developed in detail.
The important tool of the dynamic bifurcation analysis provide the critical bifurcation manifolds with the important behavior property: if fixed point of SS Dynamics crosses the bifurcation manifold than the new qualitatively different kind of dynamics will appeared. Three types of critical bifurcation manifolds are analytically described for any discrete SS Dynamics: divergence and flip bifurcation hyper-planes and saddle type flutter bifurcation manifold generated by the movement of simplexes.
Moreover, in the vicinity of fixed points the linear Jacobi approximation of non-linear probabilistic chain generates linear quasi-Markov chain. Furthermore, there exists the discrete time analogue of the variation principle: for each individual relative discrete SS Dynamics these is an information functional of universal Shannon entropy form defined on the set of all autonomous n-dimensional SS dynamics such that its maximization in this set will give the preset individual SS Dynamics.
KEYWORDS: Relative Discrete Socio-Spatial m populations/n locations (SSD) Dynamics; Forecasting procedures for SS Dynamics; Non-linear probabilistic Chains; Linear Bifurcation Analysis and Critical Bifurcation Manifolds; Linear approximations of SS Dynamics; Discrete Variation Calculus for SS Dynamics.
Unfortunately full paper has not been submitted.