Low-Dimensional Modelling of Dynamical Systems

Abstract

Consider briefly the equations of fluid dynamics-they describe the enormous wealth of detail in all the interacting physical elements of a fluid flow-whereas in applications we want to deal with a description of just that which is interesting. In a wide variety of situations, simple approximate models are needed to perform practical simulations and make forecasts. I review the derivation, from a mathematical description of the detailed dynamics, of accurate, complete and useful low-dimensional models of the interesting dynamics in a system. The development of centre manifold theory and associated techniques puts this modelling process on a firm basis. As in Guckenheinmer & Holmes (1983,S2.5): "... these new methods will really be conventional perturbation style analyses interpreted geometrically..." But the geometric viewpoint of dynamical systems theory greatly enriches our approach by providing a rationale for also deriving correct initial conditions, forcing and boundary conditions for the models-all essential elements of a model.