Thom: The Local States of a System


„We propose the following general model to parameterize the local states of a system: the space of observables contains a closed subset K, called the catastrophe set, and as long as the representative point m of the system does not meet K, the local nature of the system does not change. The essential idea introduced here is that the local structure of K, the topological type of its singularities and so forth, is in fact determined by an underlying dynamic defined on a manifold M which is in general impossible to exhibit. The evolution of the system will be defined by a vector field X on M, which will define the macroscopic dynamic. Whenever the point m meets K, there will be a discontinuity in the nature of the system which we will interpret as a change in the previous form, a morphogenesis. Because of the restrictions outlined above on the local structure of K we can, to a certain extent, classify and predict the singularities of the morphogenesis of the system without knowing either the underlying dynamic or the macroscopic evolution defined by X. In fact, in most cases we proceed in the opposite direction: from a macroscopic examination of the morphogenesis of a process and a local and global study of its singularities, we can try to reconstruct the dynamic that generates it. Although the goal is to construct the quantitative global model (M, K, X), this may be difficult or even impossible. However, the local dynamical interpretation of the singularities of the morphogenesis is possible and useful and is an indispensable preliminary to defining the kinematic of the model; and even if a global dynamical evolution is not accessible, our local knowledge will be much improved in the process.“ (p. 7f.) #Thom #system #discontinuity #morphogenesis

Thom, René, Structural Stability and Morphogenesis: An Outline of a General Theory of Models. Boca Raton, FL: CRC Press 2018.