Here's the cusp catastrophe. Think about being on the upper fold of the surface and moving to the left. Eventually you fall off and end up on the bottom fold. The points on top where you fall off are the bifurcation points. A 2-D plot of those points is the bufurcation points (red diagram in your figure).
The top surface represents "stable states" like a vase on the top of a table that you move about on the table. Nothing much happens. However, if you move the vase to the very edge of the table. It's now on it's bifurcation curve. Moving it ever so slightly and it will "traject" abruptly and qualitatively change states from being a stable vase on a table to a broken one on the floor.
In the language of Rene' Thom, the table surface is a basin of attraction for the stable state of the vase on the table. Pushing it past it's bifurcation point, and it moves into the basin of attraction of the floor attractor and undergoes "morphogenesis" in passing through the bifurcation point to the new stable state (analogous to the bottom fold of the cusp).
Yea, I know what you're thinking, "nevermind Salty, anyone else up there?".