Equilibrium and Stability in the Cusp Catastrophe Framework

Background: This paper explores equilibrium configurations and stability characteristics within the framework of the cusp catastrophe. Methodology: The study employs discriminant analysis of the governing cubic equilibrium equation to classify the number and nature of steady states as functions of control parameters. Stability conditions are derived using derivative-based criteria, and the bifurcation set is constructed to partition the parameter space into regions of distinct dynamical behavior. Methods: Equilibria are obtained analytically from the cubic equation, their stability assessed through local linearization, and the cusp boundary identified via vanishing discriminant conditions. Geometric visualization of the equilibrium surface is used to interpret mechanisms of sudden transitions. Results: The analysis reveals that the system admits either one or three equilibria, depending on parameter values, with bistability arising within the cusp region. Stable equilibria coexist with unstable separators, and transitions occur along fold lines where stability is lost. Discussion: These findings highlight the geometric role of the cusp catastrophe in explaining abrupt qualitative changes in nonlinear systems. The bifurcation structure provides a rigorous framework for understanding how small parameter variations can induce large-scale state shifts. Conclusion: The cusp catastrophe serves as a minimal yet powerful model for nonlinear systems exhibiting multi-stability and discontinuous change, offering valuable insight into the mechanisms driving sudden transitions across diverse domains.