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Centre manifolds of forced dynamical systems

Published online by Cambridge University Press:  17 February 2009

S. M. Cox  and
A. J. Roberts
S. M. Cox
Affiliation:
Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853, U.S.A.
A. J. Roberts
Affiliation:
Department of Applied Mathematics, University of Adelaide, S.A. 5000, Australia.
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Abstract

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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.

Type
Research Article
Information
The ANZIAM Journal , Volume 32 , Issue 4 , April 1991 , pp. 401 - 436
Copyright
Copyright © Australian Mathematical Society 1991

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