FOLD CATASTROPHE (CODIMENSION 1)

V = x^3 / 3 + 0
dV / dx = x^2 + 0


CUSP CATASTROPHE (CODIMENSION 2)

V = x^4 / 4 x^2 / 2 + 0
dV / dx = x^3 x + 0


SWALLOW'S TAIL CATASTROPHE (CODIMENSION 3)

V = x^5 / 5 x^3 / 3 x^2 / 2 + 0
dV / dx = x^4 x^2 x + 0


BUTTERFLY CATASTROPHE (CODIMENSION 4)

V = x^6 / 6 x^4 / 4 x^3 / 3 x^2 / 2 + 0
dV / dx = x^5 x^3 x^2 x + 0


General Notes
1. Throughout his text, Thom seems to use "x" and "y" interchangeably.
    "x" is used throughout, here, for consistency.
2.