Moreover, the slip surface is often assumed a priori, potentially deviating substantially from the actual failure surface, while inter-slice forces are simplified, leading to inaccuracies in stress representation and, consequently, in stability evaluations [18]. Similarly, empirical formula methods, derived from extensive engineering practice and statistical analysis, heavily rely on empirical knowledge and typically possess relatively weak theoretical underpinnings [19]. As technological capabilities advance and engineering experiences accumulate, such empirical models require continuous updating and refinement. Furthermore, the accuracy of numerical simulation results is highly sensitive to the precision of input parameters; inappropriate or erroneous parameter selection can significantly compromise the reliability of the analysis outcomes. [PERSON] et al. [20] verified the applicability of the energy dissipation catastrophe judgment criterion in slope systems through engineering examples. [PERSON] et al. [21] established a method for calculating the safety factor of the locked section of a slope based on the energy principle and analyzed the joint weakening and the locking effect of rock slopes. The resultsshowed that the primary factor leading to the gradual dynamic instability of rock slopes is the residual energy in the critical sliding state of the slope system. [PERSON] et al. [22] calculated the slope safety factor based on the principle of energy conservation and compared it with the results from limit analysis. They found that the safety factor obtained using the energy method was lower than that calculated using the limit analysis method. [PERSON] et al. [23] established a sharp-point catastrophe theory displacement model for predicting tunnel instability. [PERSON] et al. [24] used monitoring data from the CadiMalta in Italy and the Baiheliang landslide and analyzed the data by establishing an elliptical catastrophe model to determine the initial instability point of rock slopes and the criteria for slope failure. Catastrophe theory has found certain applications in slope stability research, particularly in predicting the sudden instability of slopes under external disturbances or internal changes, and is especially relevant in nonlinear analyses involving multiple factors. However, several limitations hinder its practical application. First, the assumption of abrupt failure inherent to catastrophe models may not align with the often gradual nature of slope instability. Second, parameter selection presents significant challenges, as the complexity of real-world environments makes it difficult to accurately obtain all necessary data. Third, most catastrophe models neglect the temporal evolution of slope failure, despite the fact that instability typically develops dynamically over time. Moreover, the simplification of the physical processes underlying real slope behavior further limits the applicability of catastrophe models in complex geological settings. Therefore, although catastrophe theory offers valuable potential, further development and refinement are required to improve its effectiveness and reliability in practical slope stability analysis. The cusp catastrophe theory, as a nonlinear scientific framework, is particularly suited for describing and analyzing sudden and discontinuous changes in complex systems induced by small perturbations. In the context of slope stability analysis, cusp catastrophe theory can effectively capture the subtle evolutions that occur as a slope approaches a critical state, as well as the mechanisms by which these evolutions trigger abrupt failure [25; 26]. By analyzing the mechanical behavior and dynamic evolution of slopes during the destabilization process, this approach enables more accurate prediction and assessment of slope stability. Energy, as a fundamental concept in physical analysis, adds a crucial dimension to the evaluation of slope stability.