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Centre manifolds of forced dynamical systems

  • S. M. Cox (a1) and A. J. Roberts (a2)
Abstract
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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[1] D. Armbruster , J. Guckenheimer and P. Holmes Kuramoto-Sivashinsky dynamics on the center-unstable manifold”, SIAM J. Appl. Math. 49 (1989) 676691.

[2] J. Carr “Applications of centre manifold theory” Applied Math. Sci. 35 (1981), Springer.

[3] P. H. Coullet and E. A. Spiegel Amplitude equations for systems with competing instabilities”, SIAM J. Appl. Math. 43 (1983) 776821.

[5] J. M. Hyman and B. Nicolaenko The Kuramoto-Sivashinsky equation: a bridge between PDE's and dynamical systems”, Physica D 18 (1986) 113126.

[6] E. N. Lorenz On the existence of a slow manifold”, J. Atmos. Sci. 43 (1986) 15471557.

[7] E. N. Lorenz and V. Krishnamurthy On the nonexistence of a slow manifold”, J. Atmos. Sci. 44 (1987) 29402949.

[8] G. N. Mercer and A. J. Roberts The application of centre manifold theory to the dispersion of contaminant in channels with varying flow properties”, SIAM J. Appl. Math. (1990) to appear.

[9] A. Mielke On Saint-Venant's problem and Saint-Venant's principle in nonlinear elasticity” in Trends in Appl. of Math, to Mech. eds. J. F. Besseling & W. Eckhaus (1988) 252260.

[11] B. Nicolaenko , B. Scheurer and R. Temam Some global dynamical properties of the Kuramoto-Sivashinsky equations: nonlinear stability and attractors”, Physica D 16 (1985) 155183.

[14] A. J. Roberts , “The utility of an invariant manifold description of the evolution of a dynamical system”, SIAM J. Math. Anal. 20 (1989) 14471458.

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The ANZIAM Journal
  • ISSN: 1446-1811
  • EISSN: 1446-8735
  • URL: /core/journals/anziam-journal
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