The Journal of the Australian Mathematical Society. Series B. Applied Mathematics

Research Article

Centre manifolds of forced dynamical systems

S. M. Coxa1 and A. J. Robertsa2

a1 Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853, U.S.A.

a2 Department of Applied Mathematics, University of Adelaide, S.A. 5000, Australia.

Abstract

Centre manifolds arise in a rational approach to the problem of forming low-dimensional models of dynamical systems with many degrees of freedom. When a dynamical system with a centre manifold is subject to a small forcing, F, there are two effects: to displace the centre manifold; and to alter the evolution thereon. We propose a formal scheme for calculating the centre manifold of such a forced dynamical system. Our formalism permits the calculation of these effects, with errors of order |F|2. We find that the displacement of the manifold allows a reparameterisation of its description, and we describe two “natural” ways in which this can be carried out. We give three examples: an introductory example; a five-mode model of the atmosphere to display the quasi-geostrophic approximation; and the forced Kuramoto-Sivashinsky equation.

(Received April 11 1990)

(Revised July 16 1990)