There are three fixed points of the system
When x=y=0 the equations reduce to dz/dt=-αz, which has solution z(t)=Ae-αt where A is a constant of integration. Thus any solutions initially on the z-axis will remain on it (i.e. the z-axis is invariant under the flow), and tend toward the origin. Physically this means that we will need a seed magnetic field in order for dynamo action to occur, just as we would hope for a dynamo model.
Taking the divergence of the flow shows that the equations are dissipative:
We use a Lyapunov function to show that the system has a trapping region into which all solutions eventually enter but never leave. Consider the function
Since σ and α are positive, the first three terms on the right hand side of this expression are negative. Defining λ=min{1,σ,α}>0 shows
The Lyapunov function increases as R→∞ and dE/dt<0 for R>D, so all solutions of the equations enter the sphere of radius D and cannot leave. Hence x, y, and z are bounded and the phase space is Cartesian with axes x, y, and z.
For all parameter choices the equations are invariant under the symmetry (x,y,z) → (-x,-y,z), which corresponds to reversal of the field
We use the Jacobian matrix at the fixed point 0 to determine its stability to small perturbations:
At b = 1 stability is transferred to the steady states C±. Subsequent behaviour depends on our choices of α, σ, and b, so determining their sizes becomes important.