Nonlinear electrical systems are fundamental components of modern engineering applications, including power electronics, switching circuits, nonlinear semiconductor devices, permissive systems, and intelligent electrical networks [1]-[6]. Unlike linear circuits, nonlinear systems frequently exhibit complex dynamic behaviors such as multiple equilibrium states, hysteresis, sudden jumps, and bifurcation phenomena [7]-[15]. These behaviors cannot always be explained adequately using classical linear circuit theory, thereby motivating the development of mathematical models capable of describing abrupt qualitative changes in system behavior [16]-[21].

Catastrophe theory, introduced by René Thom, provides a rigorous mathematical framework for studying discontinuous transitions in systems whose governing parameters change continuously. Among the seven elementary catastrophes, the cusp catastrophe is one of the most widely applied because it describes systems controlled by two independent parameters and naturally explains bistability, hysteresis, and sudden state transitions. Consequently, cusp catastrophe theory has found applications in mechanics, biology, economics, engineering, psychology, epidemiology, and numerous other scientific disciplines where nonlinear phenomena arise [22][23].

In electrical engineering, nonlinear devices often possess multiple operating points under identical external conditions. Small variations in applied voltage, resistance, or other circuit parameters may produce abrupt switching between distinct equilibrium states. Such behavior is characteristic of tunnel diodes, memristors, nonlinear semiconductor devices, and other nonlinear electronic components [24][25].

This paper develops a cusp catastrophe model for investigating equilibrium current dynamics in nonlinear electrical systems. The model is formulated through the quartic potential function

where I represents the electrical current, V denotes the applied voltage, and R represents the electrical resistance. The equilibrium currents correspond to the stationary points of the potential function, obtained by solving the cubic equation I 3 + V I + R = 0 . The discriminant of the cubic equation provides a complete characterization of the number of equilibrium states and identifies the critical conditions under which fold bifurcations occur. Furthermore, the cusp boundary 4 V 3 + 27 R 2 = 0 partitions the parameter space into regions exhibiting unique and multiple equilibrium currents, thereby offering a geometric interpretation of nonlinear electrical behavior.

The proposed model extends the application of catastrophe theory to equilibrium current analysis by interpreting the control parameters as physically meaningful electrical quantities. Rather than relying solely on numerical simulations, the catastrophe-theoretic framework reveals the qualitative mechanisms responsible for sudden current switching, bistability, and hysteresis. This perspective provides valuable insight into the stability and control of nonlinear electrical systems and complements traditional methods of nonlinear circuit analysis [26]-[28].

The main contributions of this paper are summarized as follows:

1) A new cusp catastrophe formulation for equilibrium current dynamics in nonlinear electrical systems.

2) A physical interpretation of the catastrophe control parameters in terms of applied voltage and electrical resistance.

3) An analytical characterization of equilibrium currents using the discriminant of the governing cubic equation.

4) A geometric description of fold bifurcations and cusp boundaries that separate different operating regimes.

5) A qualitative framework for explaining bistability, hysteresis, and abrupt current transitions in nonlinear electrical devices.

Consider the potential function U ( x ) = x 4 4 + a 2 x 2 + b x , where x is the state variable, a and b are control parameters. The equilibrium points satisfy d U d x = x 3 + a x + b = 0 . Thus, equilibria are the roots of the cubic equation x 3 + a x + b = 0 [9]-[14]. See Figure 1Figure 1for the canonical cusp catastrophe.

Consider an electrical system in which the state variable I represents the electric current, while the control parameters V and R represent the applied voltage and the electrical resistance, respectively. The potential function can model the system, see Figure 2Figure 2.

where:

I is the electric current flowing through the circuit (state variable), V is the applied voltage, acting as the primary control parameter, R is the electricalresistance, serving as the secondary control parameter.

The potential function represents the energy landscape of the electrical system.

Figure 2.Cusp catastrophe model for equilibrium current dynamics in nonlinear electrical systems.

Stable operating conditions correspond to local minima of this potential, while unstable conditions correspond to local maxima.

Interpreting the cusp catastrophe model in terms of current, voltage, and resistance provides a qualitative description of nonlinear electrical behavior. The applied voltage ( V ) drives the system, while the electrical resistance ( R ) influences its operating characteristics. Their interaction determines whether the circuit has a single stable operating current or multiple possible current states. Near the cusp boundary, gradual variations in voltage or resistance may result in abrupt current switching, hysteresis, and bistable operation—phenomena commonly observed in nonlinear electrical devices such as switching circuits, tunnel diodes, memristive systems, and other nonlinear electronic components.

One of the most important physical implications of the model is the prediction of bistability. When the discriminant satisfies Δ = − 4 V 3 − 27 R 2 > 0 , the system possesses three equilibrium currents. In practical electrical systems, this typically corresponds to two stable operating currents separated by an unstable equilibrium. Depending on the initial operating conditions or the previous state of the system, the electrical circuit may settle into either of the stable current levels. Such behavior is frequently observed in nonlinear switching devices and memory-based electronic components.

The cusp catastrophe model also explains the occurrence of hysteresis. As the applied voltage or the resistance changes gradually, the current does not necessarily follow a unique path. Instead, the system may remain in one stable equilibrium until a critical point is reached, after which it suddenly jumps to another equilibrium state. When the control parameter is reversed, the return transition occurs at a different critical value, forming a hysteresis loop. This phenomenon is common in nonlinear semiconductor devices, magnetic circuits, permissive systems, and switching electronics.

Another important implication is the existence of critical transition points. The cusp boundary, 4 V 3 + 27 R 2 = 0 , represents the set of parameter values at which two equilibrium currents merge througha fold bifurcation. At these critical conditions, extremely small variations in voltage or resistance may produce disproportionately large changes in the current. Such sensitivity is particularly relevant in the design of electrical protection systems, fault detection algorithms, and adaptive control strategies.

The model further illustrates how voltage and resistance jointly determine the stability of electrical operation. An increase in the applied voltage may drive the system toward regions where multiple equilibria exist, while variations in resistance modify the location of the cusp boundary and the accessibility of stable operating states. Consequently, voltage and resistance should not always be considered independently when analyzing nonlinear electrical behavior, since their interaction governs the overall stability structure.

This paper has presented a cusp catastrophe model for investigating equilibrium current dynamics in nonlinear electrical systems. By representing the electric current as the state variable and the applied voltage and electrical resistance as control parameters, the proposed model establishes a direct relationship between nonlinear circuit behavior and catastrophe theory.

The cubic equation determines the equilibrium currents

whose discriminant completely characterizes the number of possible operating states. The analysis demonstrates that the electrical system may exhibit one equilibrium, multiple equilibria, or fold bifurcations depending on the values of the voltage and resistance. Furthermore, the cusp boundary

separates regions of unique and multiple equilibrium currents, providing a clear geometric interpretation of nonlinear electrical behavior.

The proposed framework successfully explains several important physical phenomena observed in nonlinear electrical systems, including bistability, hysteresis, critical transitions, and sudden current switching. Unlike purely numerical approaches, the cusp catastrophe model provides analytical insight into the qualitative changes in system behavior resulting from continuous parameter variations. Consequently, the model contributes to a deeper understanding of stability mechanisms in nonlinear electrical circuits and offers an effective mathematical tool for predicting catastrophic operating transitions.

From an engineering standpoint, the model may assist in the design and analysis of nonlinear electronic devices, switching circuits, intelligent electrical networks, and emerging energy systems where abrupt state changes are either undesirable or intentionally utilized.

Although the present study focuses on equilibrium analysis, several promising research directions remain open.

Extending the model to time-dependent nonlinear electrical systems described by ordinary or partial differential equations.Incorporating capacitive and inductive elements to investigate transient and oscillatory electrical behavior.Developing fractional-order cusp catastrophe models for electrical systems exhibiting memory effects and anomalous dynamics.Studying stochastic catastrophe models that account for measurement noise, parameter uncertainty, and environmental disturbances.Applying the proposed framework to permissive circuits, nonlinear semiconductor devices, tunnel diodes, Josephson junctions, and smart-grid technologies.Investigating multi-parameter catastrophe models, such as butterfly or swallowtail catastrophes, to describe more complex electrical phenomena involving additional control variables.Designing catastrophe-based control strategies capable of preventing undesirable switching or exploiting bistability in intelligent electrical systems.Validating the theoretical predictions through numerical simulations and experimental investigations on practical nonlinear electrical circuits.

Overall, the cusp catastrophe model provides a mathematically rigorous and physically meaningful framework for understanding nonlinear electrical phenomena. Its ability to relate equilibrium structure, bifurcation behavior, and catastrophic transitions makes it a promising analytical tool for future research in nonlinear circuit theory, electrical engineering, and intelligent dynamical systems.