We first look at the bifurcation structure for d = 0. When this is the case the system can be written in cylindrical polar coordinates as
Kirk (1991) finds the bifurcation structure shown in the sketch below:
Continuing to examine the case when d = 0, we label these regions and examine the types of solution we expect to find in each.
In regions 1, 2, 7, 8, 11 and 12 the field free steady states are stable, so any seed magnetic field decays away. The initial conditions of the system will determine which of the three solutions to 0 = μ - z2 + cz3 the system reaches.
There are two ways in which periodic solutions are obtained. At the primary Hopf bifurcation a limit cycle bifurcates from the fixed points, giving rise to periodic orbits (for suitable initial conditions.) At the saddle-node bifurcation a pair of periodic orbits are created, one of which is stable. Hence periodic solutions are observed in regions 3, 4, 9, 10, and 13, an example of which is shown in below.

At the secondary Hopf bifurcation a torus bifurcates from the periodic orbit. This gives rise to quasiperiodic behaviour in regions 5 and 6, and example of which is illustrated below.
