f(x)= x^3 - mu x + alpha + beta mu
The points for which f(x)=0 are the equilibria for this vector field. The set of points for which f(x) = f'(x) = 0 are called points of saddle-node bifurcation. See Chapter 3 of J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, 1983.
Since this model has a one-dimensional domain, one or more of the parameters must be permuted into the domain before attempting to trace level sets.