The Bifurcation Set of the Space of Smooth Real Functions

range

first derivative

second derivative

The diagram on the left shows the full bifurcation in the plane of the polynomials

5

x

-

3

x

+a

2

x

+bx

. It consists of two hypersurfaces. The first, shown in blue, is called the caustic and consists of polynomials with degenerate critical points (both first and second derivatives vanish at the same point). The second, shown in red, is called the Maxwell set, and consists of polynomials with coinciding critical values. The full bifurcation set divides the plane into seven components of equivalent polynomials. The diagram on the right shows the graph of the polynomial chosen by the locator in the diagram on the left and, optionally, the graph of its first and second derivatives.