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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The complex Lorenz equations. Physica
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of the Lorenz attractor at high Prandtl number.
Physica
**5**D, 149-182. - Booty, M., J.D. Gibbon and A.C. Fowler 1982 A study
of the effect of mode truncation on an exact periodic
solution of an infinite set of Lorenz equations Phys. Letts.
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the Lorenz equations. Phys. Letts.
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period doubling and intermittency at high Prandtl
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for large
*r*. Stud. Appl. Math.**70**, 215-233. - Fowler, A.C. 1984 The use of the method of averaging
in predicting chaotic
motion. Phys. Letts.
**100**A, 1-6. - Fowler, A.C. 1986 Analytic methods for predicting chaos. In: Nonlinear Phenomena and Chaos, ed. S. Sarkar, Adam Hilger, Bristol: pp. 284-302.
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*n*dimensions. Stud. Appl. Math.**83**, 193-209. - Fowler, A.C. 1990 Homoclinic bifurcations for partial
differential equations
in unbounded domains. Stud. Appl. Math.
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Sparrow 1991 Bifocal homoclinic orbits in four dimensions (with C.
Sparrow). Nonlinearity
**4**, 1159-1182. - Kember, G. and A.C. Fowler 1992 Random sampling and
the Grassberger-Procaccia algorithm. Phys. Letts. A
**161**, 429-432. - Fowler, A.C. 1992 Convection and chaos. In: Chaotic processes in the geological sciences, ed. David A. Yuen, Springer-Verlag, pp. 43-69.
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chaotic time series. Phys. Letts.
A
**175**, 402-408. - Kember, G. and A.C. Fowler 1993 A correlation
function for choosing time delay in phase portrait
reconstructions. Phys. Letts. A
**179**, 72-80. - Kember, G., A.C. Fowler and J. Holubeshen 1993
Forecasting river flows that show low-dimensional
dynamics. Stoch. Hydraul. Hydrol.
**7**, 205-212. - Fowler, A.C. and G.
Kember 1995 A nonlinear filtering technique for multi-oscillator
systems. Comput. Maths. Applics.
**29**, 55-67 (1995). - Fowler, A.C. and G. Kember 1996
On the Lorenz-Krishnamurthy slow manifold. J. Atmos.
Sci.
**53**, 1433-1437.a name="91">

- Kember, G., A.C. Fowler and H.B. Evans 1997
Local nonlinear filtering. J. Nonlin.
Sci.
**7**, 411-425. - Fowler, A.C. and G. Kember 1998 Singular systems
analysis as a moving window spectral
method.
Euro. J. Appl. Math.
**9**, 55-79. - O'Brien, S.B.G, E.G. Gath, A.C. Fowler and
G. Kember 1998 Asymptotics with small exponent in a model for
ice-sheet surging. Proc. Roy. Ir. Acad.
**98A**, 67-80. - Fowler, A.C., G. Kember and S.G.B. O'Brien 2000
Small exponent asymptotics. IMA J. Appl. Math.
**64**, 23-38. - Kember, G.C., J.D. Evans, A.C. Fowler and
S.G.B. O'Brien 2000 Exponential asymptotics with a small exponent.
Quart. Appl. Math.
**58**, 561-576. - Fowler, A.C. and M.C. Mackey 2002
Relaxation oscillations in a class of
delay-differential equations. SIAM J. Appl. Math.
**63**(1), 299-323. - Fowler, A.C. and P.D. Howell 2003
Intermittency in the transition to turbulence.
SIAM J. Appl. Math.
**63**(4), 1184-1207.