<p>Preliminary cutting experiments indicate that the chip flow angle (CFA) catastrophe in transverse turning can significantly reduce cutting forces and power, offering potential for energy savings. However, the underlying principles of this phenomenon require further exploration. To address this, a set of CFA observation experiments was designed based on the cutting power function previously established. These experiments were conducted during the turning of concave and convex cone workpieces using a 0° rake angle straight double-edged turning tool. Ten sets of coordinate data for abrupt critical control points were collected. To reveal the law of catastrophe, a bifurcation set equation was derived. This involved assuming differential homeomorphism transformation functions between actual and theoretical control point coordinates, with three coefficients to be determined. The experimental data for critical control points were then transformed into theoretical coordinates. The objective function for optimizing the bifurcation set equation’s coefficients was derived using the cusp catastrophe equation from catastrophe theory. The optimization was carried out using a genetic algorithm, resulting in three possible models for the CFA catastrophe bifurcation set equation. By comparing the objective function values, the experimental equation based on a composite polynomial transformation function was selected as the bifurcation set curve of the CFA catastrophe, and the verification experiment is carried out. This research provides valuable insights for the development of energy-saving applications in CFA catastrophe.</p>