Bifurcation, Catastrophe, and Turbulence

Abstract

Bifurcation occurs in a parametrised dynamical system when a change in a parameter causes an equilibrium to split into two. Catastrophe occurs when the stability of an equilibrium breaks down, causing the system to jump into another state. The elementary theory concerns dynamical systems with steady state equilibria (point attractors), and the non-elementary theory concerns systems with dynamic equilibria (periodic attractors and strange attractors). In the elementary case Thom [72] has used singularities to classify both bifurcation and catastrophe, and this has led to a great variety of applications [22]. We illustrate the contrasting styles of application in biology and physics by describing two recent examples. The first is a model by Seif [59] concerning hyperthyroidism, and the second is a model by Schaeffer [58] concerning Taylor cells in fluid flow.