Uncertain differential equation is a type of differential equation driven by canonical Liu process. By applying some uncertain theories, the sufficient conditions of the exponential stability in mean square is obtained for nonlinear uncertain differential equations. At the same time, some new criteria ensuring the existence of the global attracting sets of considered equations are presented.

This paper studies the asymptotically mean-square stability of a Runge–Kutta type predictor–corrector numerical scheme for uncertain differential equations. The method consists of a fourth-order Runge–Kutta predictor coupled with a one-step implicit correction, which leads to a fully discrete scheme for the associated $\alpha$-path equations. For a linear test equation driven by a Liu process, the corresponding growth factor of the numerical solution is derived. By employing vectorization techniques and Kronecker product representations, a recursive relation is established for the second-moment matrix of the numerical solution. It is shown that the asymptotically mean-square stability of the proposed scheme is equivalent to a necessary and sufficient condition on the modulus of the growth factor. Numerical examples are provided to illustrate the theoretical results and to demonstrate the influence of the step size on the stability.