General solutions and stability analysis of two-dimensional linear fuzzy differential systems in the space of asymmetric linearly correlated fuzzy numbers

Abstract

Based on the well-behaved algebraic structure and topological properties of the space $R_{F\left(A\right)}$ of asymmetric linearly correlated fuzzy numbers, this paper presents a unified methodology for addressing the construction of the general solution and the stability analysis of the equilibrium solution of two-dimensional (2D) linear fuzzy differential systems. Specifically, leveraging the structural advantages of $R_{F\left(A\right)}$, a 2D linear fuzzy differential system with a crisp coefficient matrix can be equivalently transformed into two mutually independent classical linear differential systems that share the same coefficient matrix. Thereafter, the stability of the equilibrium solution is determined and the general solution is constructed by utilizing the mature stability theory and solution methods for planar linear systems. For a 2D linear fuzzy differential system with a fuzzy coefficient matrix, by virtue of the structural features of $R_{F\left(A\right)}$ and the favorable properties of the distributive product (which unifies the A-cross product and complex product), the system can be equivalently converted into a coupled fourth-order real-valued linear differential system. The stability analysis of the equilibrium solution and the construction of the general solution are then implemented using the theory of real-valued linear differential systems. Meanwhile, a detailed procedure is proposed to clearly and efficiently guide the stability analysis and solution process of such linear fuzzy differential systems. Several illustrative examples are provided to verify the feasibility and effectiveness of the proposed methods, particularly by specializing the distributive product to the A-cross product and complex product, respectively. Finally, as an application, the temporal evolution behaviors of charge and current in an LC oscillatory circuit under different fuzzy disturbance environments (including fuzzy initial conditions, fuzzy inductance, and fuzzy capacitance) are analyzed, which demonstrates the practical applicability of the proposed theoretical framework.