The Lorenz Model

Abstract

The three-dimensional set of ordinary differential equations, known as the Lorenz equations were first introduced by Ed Lorenz [2] in 1963 as a model of convection in a two-dimensional cell. Since then other authors, e.g. Haken [1] have shown that the same equations can be derived from the Maxwell-Bloch equations for single-mode lasers with damping. The equations are of mathematical interest because of the wide variety of behaviours that they display -- including chaotic behaviour and the existence of strange attractors -- for different values of the three parameters r, σ and b. It seems that the relevant range of parameters for laser applications is σ < b + 1 which, it must be confessed, is not the parameter range of greatest interest from a mathematical point of view. However, providing 3σ - 1 > 2b attracting chaotic behaviour will still occur [3,4] though it will occur at parameter values for which the equations also have stable stationary points and these may determine most of the observed behaviour of the system.