Vanadium dioxide (VO \(_2\) ) devices exploit a thermally driven metal–insulator transition accompanied by strong hysteresis, making them promising candidates for compact memristive elements and neuromorphic functionalities. Here we present a fully explicit electrothermal LLP–VO \(_2\) model aimed at enabling practical implementation of LLP hysteresis operators in VO \(_2\) and related materials. The formulation couples a single thermal balance equation to a discrete LLP operator that governs the metallic fraction while capturing key Preisach-like features, including nested hysteresis loops and return-point memory. The resulting resistance expression combines an Arrhenius-type conduction term with a metallic offset. A systematic comparison between explicit Euler and fourth-order Runge–Kutta (RK4) time integration schemes examines numerical accuracy and solver-dependent stiffness effects across the transition region. The results indicate that LLP-based hysteresis can be simulated efficiently without introducing additional differential states, while maintaining numerical reproducibility. The proposed model may serve as a lightweight and implementation-ready framework for applications requiring controlled VO \(_2\) hysteresis behavior, and is designed to be compatible with SPICE-class solvers for future integration and benchmarking against HP- and Chua-type compact models.
