9 Problem 03.06.06 - Strogatz (Patterns in fluids). Statement for problem 03.06.06. (Patterns in fluids). G. Ahlers (1989) gives a fascinating review of experiments on one-dimensional patterns in fluid systems. In many cases, the patterns first emerge via supercritical or subcritical pitchfork bifurcations from a spatially uniform state. Near the bifurcation, the dynamics of the amplitude of the patterns are given approximately by d.A T = EA - 943 dt in the supercritical case, (9.0.1) or d.A T dt = EA - g43 - KA in the subcritical case. (9.0.2) Here A = A(t) is the amplitude of the pattern, 7 > 0 is a typical time scale, and c is a small dimensionless pa- rameter that measures the distance from the bifurcation. The parameter ? is positive in the supercritical case, whereas g 0 in the subcritical case. (In this context, the equation TA = EA - 94 is often called the Landau equation.) a) Dubois and Berge (1978) studied the supercritical bifurcation that arises in Rayleigh Benard convec- tion, and showed experimentally that the steady state amplitude depends on c according to the power law A* o e", where B = 0.50 + 0.01. What does the Landau equation predict