Equilibrium Current Dynamics in Nonlinear Electrical Systems Using a Cusp Catastrophe Model

The behavior of nonlinear electrical systems is often characterized by abrupt transitions, multiple operating states, and hysteresis phenomena that conventional linear models cannot adequately describe. This paper presents a cusp catastrophe model for analyzing the equilibrium dynamics of electrical current in nonlinear electrical systems. The proposed model is based on the quartic potential function, $U\left(I\right)=\frac{I^4}{4}+\frac{V}{2}{I}^2+RI$ where the state variable $I$ denotes the electric current, while the control parameters $V$ and $R$ represent the applied voltage and electrical resistance, respectively. The equilibrium states are obtained from the stationary condition of the potential function, leading to the cubic equation $I^3+VI+R=0$. The discriminant of this equation is employed to determine the number and stability of equilibrium currents, revealing three distinct operating regimes corresponding to one equilibrium, multiple equilibria, and fold bifurcations. The cusp boundary $4V^3+27R^2=0$ separates regions of unique and multiple steady-state solutions in the control-parameter space. The proposed catastrophe-theoretic framework provides a qualitative understanding of sudden current transitions, bistability, and hysteresis arising from variations in voltage and resistance. The results demonstrate that cusp catastrophe theory offers a powerful mathematical framework for studying nonlinear electrical behavior and may contribute to the design, stability analysis, and control of switching circuits, nonlinear electronic devices, and emerging intelligent electrical systems.