K. Fujimura: Centre manifold reduction and the Stuart-Landau equation for fluid motions, Proc. R. Soc. Lond. A 453 (1997) 181-203.
Abstract
The centre manifold reduction to derive the Stuart-Landau equation is examined. A double expansion in terms of the Fourier series and linear eigenfunctions is introduced in hydrodynamic equations. A centre manifold reduction scheme is then applied to reduce the resultant system of ordinary differential equations to the Stuart-Landau equation. Through a formal expansion in linear eigenfunctions, the latter equation is shown to be equivalent with the one derived by the method of multiple scales. Numerical coefficients involved in the quintic Stuart-Landau equation are evaluated for plane Poiseuille flow, convection in a vertical slot, and Rayleigh-Benard convection. In all the cases, the coefficients converge as a dimension of the ODE system increases and approach the numerical values obtained by the method of multiple scales.
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