Simple Models, Catastrophes and Cycles

Casti, J.L. (1984). Simple Models, Catastrophes and Cycles. IIASA Research Report (Reprint). IIASA, Laxenburg, Austria: RR-85-002. Reprinted from Kybernetes, 13(4):213-229 [1984] 10.1108/eb005693.

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Abstract

It is often observed in practice that the essential behavior of mathematical models involving many variables can be captured by a much smaller model involving a few variables. Further, the simple model very often displays oscillatory behavior of some sort, especially when critical problem parameters are varied in certain ranges. This paper attempts to supply arguments from the theory of dynamical systems for why oscillatory behavior is so frequently observed and to show how such behavior emerges as a natural consequence of focusing attention upon so-called "essential" variables in the process of model simplification. The relationship of model simplification and oscillatory behavior is shown to be inextricably intertwined with the problems of bifurcation and catastrophe in that the oscillations emerge when critical system parameters, i.e. those retained in the simple model, pass through critical regions. The importance of the simplification, oscillation and bifurcation pattern is demonstrated here by consideration of several examples from the environmental economic and urban areas.

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