J Austral Math Soc Ser B 32 pp401--436, 1991.

Centre manifolds of forced dynamical systems

S. M. Cox and A. J. Roberts

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.

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Authors

S. M. Cox

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

A. J. Roberts

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