Based on the Zeeman catastrophe machine theory, the critical equation of potential function of Pearcey beams is solved. It is found that the number of roots of the critical equation determines the optical morphogenesis of Pearcey beams. This critical equation has at most three real roots, which correspond to three stable points and three diffraction lines, respectively. By choosing suitable control variables of Pearcey beams, if two of the three real roots are equal, the two corresponding diffraction lines overlap and the caustic line of Pearcey beams is formed. If three real roots are all equal, the optical cusp of Pearcey beams is formed. The study not only clarifies the mathematical mechanism of the generation of caustics and optical cusps of Pearcey beams, but also shows the caustic line equation and the position of the optical cusp.