Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-30T15:18:34.359Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

[1] Armbruster, D., Guckenheimer, J. and Holmes, P. “Kuramoto-Sivashinsky dynamics on the center-unstable manifold”, SIAM J. Appl. Math. 49 (1989) 676–691.CrossRefGoogle Scholar
[2] Carr, J. “Applications of centre manifold theory” Applied Math. Sci. 35 (1981), Springer.Google Scholar
[3] Coullet, P. H. and Spiegel, E. A. “Amplitude equations for systems with competing instabilities”, SIAM J. Appl. Math. 43 (1983) 776–821.CrossRefGoogle Scholar
[4] Foias, C., Nicolaenko, B., Sell, G. R. and Temam, R. “Inertial manifolds for the Kuramoto-Sivashinsky equation and an estimate of their lowest dimension”, J. Math. Pures et Appl. 67 (1988) 197–226.Google Scholar
[5] Hyman, J. M. and Nicolaenko, B. “The Kuramoto-Sivashinsky equation: a bridge between PDE's and dynamical systems”, Physica D 18 (1986) 113–126.CrossRefGoogle Scholar
[6] Lorenz, E. N. “On the existence of a slow manifold”, J. Atmos. Sci. 43 (1986) 1547–1557.2.0.CO;2>CrossRefGoogle Scholar
[7] Lorenz, E. N. and Krishnamurthy, V. “On the nonexistence of a slow manifold”, J. Atmos. Sci. 44 (1987) 2940–2949.2.0.CO;2>CrossRefGoogle Scholar
[8] Mercer, G. N. and Roberts, A. J. “The application of centre manifold theory to the dispersion of contaminant in channels with varying flow properties”, SIAM J. Appl. Math. (1990) to appear.CrossRefGoogle Scholar
[9] Mielke, A. “On Saint-Venant's problem and Saint-Venant's principle in nonlinear elasticity” in Trends in Appl. of Math, to Mech. eds. Besseling, J. F. & Eckhaus, W. (1988) 252–260.CrossRefGoogle Scholar
[10] Muncaster, R. G., “Invariant manifolds in mechanics I: the general construction of coarse theories from fine theories”, Arch. Rat. Mech. Anal. 84 (1983) 353–373.CrossRefGoogle Scholar
[11] Nicolaenko, B., Scheurer, B. and Temam, R. “Some global dynamical properties of the Kuramoto-Sivashinsky equations: nonlinear stability and attractors”, Physica D 16 (1985) 155–183.CrossRefGoogle Scholar
[12] Roberts, A. J. “Simple examples of the derivation of amplitude equations for systems of equations possessing bifurcations”, J. Austral. Math. Soc. Ser. B 27 (1985) 48–65.CrossRefGoogle Scholar
[13] Roberts, A. J., “Appropriate initial conditions for asymptotic descriptions of the long term evolution of dynamical systems”, J. Austral. Math. Soc. Ser. B 31 (1989) 48–75.CrossRefGoogle Scholar
[14] Roberts, A. J., “The utility of an invariant manifold description of the evolution of a dynamical system”, SIAM J. Math. Anal. 20 (1989) 1447–1458.CrossRefGoogle Scholar
[15] Roberts, A. J., “The invariant manifold of beam deformations. Part 1: the simple circular rod” preprint (1990).Google Scholar
[16] Smith, R. “Longitudinal dispersion coefficients for varying channels”, J. Fluid Mech. 130 (1983) 299–314.CrossRefGoogle Scholar