Convection offers a different paradigm, as here the transitions to oscillations and chaos occur at lower Rayleigh number, before the transition to full convective turbulence. I've made efforts to analyse the chaotic trajectories at high Rayleigh and Prandtl number (cf. Lorenz above), and Simon Acomb's thesis dealt with this. It's a difficult problem though, and to be realistic needs three dimensions, which makes life horrendous.
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