= Five-mode rotating-convection truncation
A <Galerkin method> retaining one <stream function> mode, two temperature modes and two transverse-velocity modes produces five coupled amplitudes. A useful normalization gives $a'=-\sigma a+\sigma rb-\sigma^2qd$, $b'=a-b-ac$, $c'=h(-c+ab)$, $d'=a-\sigma d-ae$, $e'=-(4-h)\sigma e+had$. Here $\sigma>0$ is the <Prandtl number>, $0<h<4$, and $q$ measures squared rotation. The trivial <equilibrium point> has characteristic polynomial $(s+\sigma)^2(s+1)-\sigma r(s+\sigma)+\sigma^2q(s+1)$ in its three coupled linear modes; the remaining two modes decay. <Stationary bifurcations> and oscillatory onset meet at $q=(1+\sigma)/(1-\sigma)$, $r=2/(1-\sigma)$ when $\sigma<1$.
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