The Aizawa attractor II

In the previous post, it was apparent (from numerical experiment) that the Aizawa attractor has three purely real equilibrium points along the z-axis, so long as parameter α is greater than a certain value, namely 0.932169751786... (with the other parameter values as listed in the previous post). If α is less than this value, then there is evidently one real equilibrium point, and two complex equilibrium points.

Here we derive an expression for α which gives the demarcation point between all-real and real and two complex equilibrium points.

The two (potentially) complex roots / equilibrium points are given by (see previous post):

Assuming that λ is not itself complex (see below), the above will roots will be purely real if λ² = α (since this forces the right hand side of the above to be zero). From the previous post we have

Hence we need to solve 

which, by inspection, has the solution

giving the value of 0.932169751786... which agrees with the value determined by numerical experiment.

Note that although the above gives the value for α for which the imaginary parts of the equilibrium points are exactly zero, it is not obvious that values of α greater than this will give purely real equilibrium points, since if α > α₀ then λ will itself be complex, because 9γ² - 4α³ will be negative, and the above argument / derivation of α₀ assumes λ to be real.