Simple Models, Catastrophes and Cycles

Casti, J.L. (1983). Simple Models, Catastrophes and Cycles. IIASA Working Paper. IIASA, Laxenburg, Austria: WP-83-087

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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 only a few variables. Further, the simpler 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, bifurcation pattern is demonstrated here by consideration of several examples from the environmental, economic, and urban areas.

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