<p>Chip-splitting, a catastrophic phenomenon occurring during double-edged cutting, significantly reduces both cutting forces and power consumption, thereby presenting substantial potential for energy efficiency improvements. Establishing the equilibrium surface equation of the catastrophic phenomenon is the basis for its control and utilization. However, prior research failed to provide a state variable to describe the chip-splitting catastrophe and establish an equilibrium surface equation for chip-splitting catastrophe. Building upon the regularized cusp catastrophe model, this paper proposes a method to establish the chip-splitting cusp catastrophe equilibrium surface equation by using the specific main cutting force as the actual state variable, and using the blade angle and rake angle of the straight double-edged turning tool as the actual control variables. The form of a set of diffeomorphism transformation functions between the actual state/control variables and the theoretical state/control variables is determined by the topological analysis method. And coefficients of the transformation function are fitted through experimental data. Then the cusp catastrophe equilibrium surface equation and bifurcation set equation for chip-splitting are established. Verification experiments confirm 100% accuracy in predicting chip-splitting catastrophe using the equilibrium surface equation, with a maximum absolute error of 22.07% in specific main cutting force prediction. The experiment identifies a set of structural parameters for double-edged turning tools that can save up to 23.07% cutting energy by utilizing chip-splitting catastrophe.</p> Graphical abstract <p></p>