The cusp catastrophe corresponds to the equation
$$F(x,a,b)=x^4+ax^2+bx$$
where $a, b$ are the control parameters. The following diagram of cusp catastrophe shows the curves that satisfy $\frac{dF}{dx}=0$ for the parameters $a,b$ drawn for parameter $b$ continuously varied, for several values of parameter $a$.
The blue curve is the bifurcation curve which is the locus of the extremas of the cusp surface. In another diagram, the projection of the bifurcation set on the control surface spanned by $(a,b)$ is shown as
I had plotted the cusp surface but I couldn't understand how to plot the bifurcation curve (i.e. the boundary of the bifurcation set in the second figure). I think that the bifurcation curve satisfies an equation of the form $g(a,b)=0$ which can be plotted in the $(a,b)$-plane.
My questions are:
- How to find the equation $g(a,b)=0$?
- How to plot the projection of the bifurcation curve on the control plane?
Best Answer
As a visualization matter we can see below in light blue, the surface
$$ 4x^3+2a x+b = 0 $$
and in light orange the two leaves for
$$ \pm i\left(\frac 23 a\right)^{\frac 32}+b=0 $$