Catastrophe models Control variables Potential function Bifurcation set Normalization formulas Folding catastrophe 1 $V\left(x\right)={x}^{3}+{u}_{1}x$ ${u}_{1}=-3{x}^{2}$ ${X}_{u1}=\sqrt{{u}_{1}}$ Cusp catastrophe 2 $V\left(x\right)=\frac{1}{4}{x}^{4}+\frac{1}{2}{u}_{1}{x}^{2}+{u}_{2}x$ ${u}_{1}=-6{x}^{2},\text{}{u}_{2}=8{x}^{3}$ ${X}_{u1}=\sqrt{{u}_{1}}$ , ${X}_{u2}=\sqrt[3]{{u}_{2}}$ Swallowtail catastrophe 3 $V\left(x\right)=\frac{1}{5}{x}^{5}+\frac{1}{3}{u}_{1}{x}^{3}+\frac{1}{2}{u}_{2}{x}^{2}+{u}_{3}x$ ${u}_{1}=-6{x}^{2}$ , ${u}_{2}=8{x}^{3}$ , ${u}_{3}=-3x$ ${X}_{u1}=\sqrt{{u}_{1}}$ , ${X}_{u2}=\sqrt[3]{{u}_{2}}$ , ${X}_{u3}=\sqrt[4]{{u}_{3}}$ Butterfly catastrophe 4 $\begin{array}{l}V\left(x\right)=\frac{1}{6}{x}^{6}+\frac{1}{4}{u}_{1}{x}^{4}+\frac{1}{3}{u}_{2}{x}^{3}\\ \text{}+\frac{1}{2}{u}_{3}{x}^{2}+{u}_{4}x\end{array}$ ${u}_{1}=-10{x}^{2}$ , ${u}_{2}=20{x}^{3}$ , ${u}_{3}=-15{x}^{4}$ , ${u}_{4}=4{x}^{5}$ ${X}_{u1}=\sqrt{{u}_{1}}$ , ${X}_{u2}=\sqrt[3]{{u}_{2}}$ , ${X}_{u3}=\sqrt[4]{{u}_{3}}$ , ${X}_{u4}=\sqrt[5]{{u}_{4}}$