Chaotic Behaviour in Spatial Systems and Forecasting

Abstract

Since the last twenty years the world of “chaotic” phenomena has gained importance in all fields of sciences. Nowadays it even seems that regular trajectories are only exceptional events in the dynamics of complex systems. The refinements of theoretical and numerical tools and the improvement of empirical reserach methods provide the possibility of scrutinizing up to now hidden details of a system. The understanding of not only the “coarse grained” structure of a system but of the interactions of its different subsystems requires an expansion of the phase space of variables. The in general nonlinear interactions among the different variables, for instance in socio-economic systems, result in mathematical models which exhibit a variety of instabilities (Weidlich and Haag, 1983). Dissipation and fluctuation are always present and are responsible for the complex spatio-temporal patterns one can observe in models belonging to different research fields. Phase transitions are regular events if the control parameters of a system pass so called critical values.