×
Rucklidge, A. M.

Chaos in models of double convection. (English) Zbl 0747.76089

J. Fluid Mech. 237, 209-229 (1992).
In certain parameter regimes, it is possible to derive third-order sets of ordinary differential equations that are asymptotically exact descriptions of weakly nonlinear double convection and that exhibit chaotic behaviour. This paper presents a unified approach to deriving such models for two-dimensional convection in a horizontal layer on Boussinesq fluid with lateral constraints. Four situations are considered: thermosolutal convection, convection in an imposed vertical or horizontal magnetic field, and convection in a fluid layer rotating uniformly about a vertical axis. Thermosolutal convection and convection in an imposed horizontal magnetic field are shown here to be governed by the same sets of model equations, which exhibit the period-doubling cascades and chaotic solutions that are associated with the Shil’nikov bifurcation.

Cited in 2 Reviews
Cited in 45 Documents

MSC:

76R10 Free convection
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37G99 Local and nonlocal bifurcation theory for dynamical systems

Keywords:

ordinary differential equations; Boussinesq fluid; thermosolutal convection; magnetic field; period-doubling cascades; chaotic solutions; Shil’nikov bifurcation
PDF BibTeX XML Cite
\textit{A. M. Rucklidge}, J. Fluid Mech. 237, 209--229 (1992; Zbl 0747.76089)
Full Text: DOI

References:

[1] DOI: 10.1007/BF01307443
[2] DOI: 10.1017/S0022112081002127 · Zbl 0485.76066
[3] DOI: 10.1017/S0022112081002115 · Zbl 0485.76065
[4] DOI: 10.1017/S0022112081000918 · Zbl 0475.76047
[5] DOI: 10.1017/S0022112084001968 · Zbl 0547.76093
[6] DOI: 10.1137/0143052 · Zbl 0534.35004
[7] DOI: 10.1017/S0022112066000818
[8] Coullet, C.R. Acad. Sci. Paris 299I pp 253– (1984)
[9] Veronis, J. Mar. Res 23 pp 1– (1965)
[10] Arter, Geophys. Astrophys. Fluid Dyn. 25 pp 259– (1983)
[11] DOI: 10.1007/BF01081886 · Zbl 0411.58013
[12] Shimizu, Phys. Lett 6 pp 163– (1980)
[13] ArneAodo, Phys. Lett. 109A pp 367– (1985)
[14] Shil’nikov, Sov. Maths Dokl 6 pp 163– (1965)
[15] ArneAodo, Geophys. Astrophys. Fluid Dyn. 31 pp 1– (1985)
[16] ArneAodo, Phys. Lett. 92A pp 369– (1982)
[17] Rucklidge, Geophys. Astrophys. Fluid Dyn 3 pp 619– (1992)
[18] Rucklidge, Physica 3 pp 619– (1992)
[19] DOI: 10.1088/0951-7715/3/3/005 · Zbl 0717.58040
[20] DOI: 10.1017/S0022112073002600 · Zbl 0261.76028
[21] DOI: 10.1038/261459a0 · Zbl 1369.37088
[22] Lyubimov, Physica 20 pp 130– (1983)
[23] DOI: 10.1175/1520-0469(1963)020 2.0.CO;2
[24] DOI: 10.1017/S0022112081003443 · Zbl 0513.76078
[25] Knobloch, Geophys. Astrophys. Fluid Dyn 51 pp 195– (1990)
[26] DOI: 10.1017/S0022112081002139 · Zbl 0478.76050
[27] DOI: 10.1017/S0022112081002139 · Zbl 0478.76050
[28] DOI: 10.1017/S0022112086000216 · Zbl 0612.76047
[29] Knobloch, Contemp. Math 56 pp 56– (1986)
[30] Knobloch, Geophys. Astrophys. Fluid Dyn 36 pp 161– (1986)
[31] DOI: 10.1017/S0022112076002759 · Zbl 0353.76028
[32] Guckenheimer, Geophys. Astrophys. Fluid Dyn 23 pp 247– (1983)
[33] DOI: 10.1007/BF01020649 · Zbl 0635.58031
[34] DOI: 10.1007/BF01010828 · Zbl 0588.58041
[36] Dangelmayr, Phys. Lett. 117A pp 394– (1986)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.