The Instability of Evolving Systems

Abstract

The differentiation of cells in developmental biology the emergence of order in chemical and biochemical reactions gravitational collapse in astrophysics and the instability of an atomic lattice are all considered in the present article as examples of an equilibrium path of a system which suddenly becomes unstable. The theory of elastic stability offers one set of explanations. Another comes from the more recent catastrophe theory associated with the name of René Thorn. In the following contribution these two approaches are taken a aspects of a unified non-linear bifurcation theory, able to reduce an original multi-dimensional problem to one which can be described in term of a small number of active internal coordinates and a corresponding number of external control parameters. This allow three-dimensional picture to be drawn for many of the most relevant singularities.