5 - Catastrophe Theory of Stage Transitions in Metrical and Discrete Stochastic Systems

Abstract

This chapter explains the catastrophe theory of stage transitions in metrical and discrete stochastic systems. Central theoretical constructs in developmental biology and psychology, such as epigenesis and emergence, seemed to defy any attempt at causal modeling. Moreover, the mathematical theory of nonlinear dynamics has provided innovative, rigorous approaches to the empirical study of epigenetical processes. In a bifurcation analysis, a nonlinear system is subjected to smooth variation of its parameters. One might conceive this variation as the result of maturation, giving rise to slow continuous changes in the system parameters. Most of the time this only yields continuous variation of the behavior of the system, but for particular parameter values, the system may undergo a sudden change in which new types of behavior emerge and old behavior types disappear. Such a discontinuous change in a system's behavior marks the point when its dynamics become unstable, after which a spontaneous shift occurs to a new, stable dynamical regime. Hence, sudden transitions in the dynamics of systems undergoing smooth quasi-static parameter variation constitute a hallmark of self-organization. Catastrophe theory deals with the critical points or equilibria of gradient systems, that is, systems whose dynamics are governed by the gradient of a potential. The extension of the program of elementary catastrophe theory to stochastic systems leads to a number of deep and challenging questions which for the most part are still unexplored.