I was working on a problem for my Dynamical Systems class (essentially, using Lyapunov functions to prove that some square is in the basin of attraction of an equilibrium point) when I found myself trying to show some bound on the real roots of the function $f(x) = -\delta-3x+x^3$, where $0 < \delta < 2$. This devolved into a whole digression on the behavior of the roots of this function, which are really just the equilibria of the system $u' = -\delta-3u+u^3$ (the real and imaginary parts of which are plotted below as a function of $\delta$; ignore the colors a bit, as Mathematica doesn't really do strict path-following in calculating the values of Root[] objects):
At $\delta = \pm 2$, we have these pitchfork bifurcations (at least, I think they're pitchfork bifurcations). They represent values of $\delta$ where the discriminant of $-\delta-3x+x^3$ in $x$ crosses $0$. Firstly, how would I go about proving that these are pitchfork bifurcations using the normal form of the pitchfork bifurcation? Also, are there any good references on the dynamics of higher-degree one-dimensional polynomials (essentially, polynomial systems that don't have nice closed-form solutions like this one)? Thanks!
Edit: Thanks to Moo, I see how to show that these bifurcations are actually saddle-node bifurcations (which makes sense from the plots I provided... which I had been misreading. Doh!). I'd still love to have a good reference for the dynamics of one-dimensional high-degree polynomial systems, though!