Here the x-coordinate represents the toroidal field, and the y-coordinate the poloidal field. The hydrodynamic activity has been collapsed onto the z-axis, so that this coordinate represents the velocity.
The z-axis is invariant, as required, since a seed magnetic field is needed for dynamo action to set in. The term x2+y2 in equation 1(a) models the effect of the Lorenz force on the velocity and the equations have been normalised to set the coefficient of this term to 1.
In the absence of the magnetic field the hydrodynamics are described by the equation
It is shown later that the constants a and c have the effect of distorting the bifurcation structure, but do not alter it qualitatively (as long as their sign remains the same.)
Setting z=0 in equations 1(b) and 1(c) gives the simple system