Dynamics of Continuous, Discrete and Impulsive Systems 5 (1999) 327-340.
Pascal Chossat, Martin Krupa, Ian Melbourne and Arnd Scheel
In 1972, Childress and Soward proposed the existence of a self-sustained magnetic dynamo through secondary bifurcation from a purely convective state in rotating Bénard convection.
It was only in the late '70s that the nature of the primary convective states in rotating convection was investigated. The standard rolls solution is stable for low rotation rates. At higher rates of rotation, rolls are unstable but there is a robust homoclinic cycle, the Busse-Heikes cycle, connecting rolls that are mutually oriented at 60o.
In this paper, we consider the full magnetohydrodynamic equations and analyze secondary bifurcations both from stable rolls and from the Busse-Heikes cycle. In particular, we show how the proposed solution of Childress and Soward to the linear stability (kinematic dynamo) problem, together with established techniques from equivariant dynamical systems, yields solutions to the full nonlinear dynamo problem. We obtain steady, periodic, quasiperiodic and intermittent weak-field dynamos.