SIMPLE MODELS, CATASTROPHES AND CYCLES
FOREWORD One of the main tasks of IIASA's Regional Issues Project is to develop a theoretical modeling apparatus suitable for characterizing the cycles, oscillations, and discontinuities observed in the dynamics of urban housing, transportation, and industrial development. Furthermore, in order to maintain analytic and computational tractability, a great premium is placed upon the "simplicity" of the models . This report addresses many of these issues from a theoretical modeling standpoint ,
... wing by precept, as well as by example, the mathematical methods associated with questions of model simplification, catastrophes, and cycles. In the report a specific regional development model is discussed as a fundamentally dynamic problem. It is shown that oscillatory rather than steady-state behavior of metropolitan populations and income levels is to be expected, and that such behavior has actually been observed in the United States for the period 1940-77. AKE E. ANDERSSON Leader Regional Issues Project 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 wh y 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.