Nonlinearity **7** (1994) 1565-1591.
doi:10.1088/0951-7715/7/6/003
## Chaos in magnetoconvection

A.M.Rucklidge

Department of Applied Mathematics and Theoretical Physics,

University of Cambridge, Cambridge, CB3 9EW, UK

**Abstract.**
The partial differential equations (PDEs) for two-dimensional
incompressible convection in a strong vertical magnetic field have a
codimension-three bifurcation when the parameters are chosen so that the
bifurcations to steady and oscillatory convection coincide and the limit
of narrow rolls is taken. The third-order set of ordinary differential
equations (ODEs) that govern the behaviour of the PDEs near this
bifurcation are derived using perturbation theory. These ODEs are the
normal form of the codimension-three bifurcation; as such, they prove to
be an excellent predictor of the behaviour of the PDEs. This is the first
time that a detailed comparison has been made between the chaotic
behaviour of a set of PDEs and that of the corresponding set of model
ODEs, in a parameter regime where the ODEs are expected to provide
accurate approximations to solutions of the PDEs. Most significantly, the
transition from periodic orbits to a chaotic Lorenz attractor predicted
by the ODEs is recovered in the PDEs, making this one of the few
situations in which the nature of chaotic oscillations observed
numerically in PDEs can be established firmly. Including correction terms
obtained from the perturbation calculation enables the ODEs to track
accurately the bifurcations in the PDEs over an appreciable range of
parameter values. Numerical calculations suggest that the T-point (where
there are heteroclinic connections between a saddle point and a pair of
saddle-foci), which is associated with the transition from a Lorenz
attractor to a quasi-attractor in the normal form, is also found in the
PDEs. Further numerical simulations of the PDEs with square rolls confirm
the existence of chaotic oscillations associated with a heteroclinic
connection between a pair of saddle-foci.

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