Stability analysis of nonlinear algebraic-differential equations with 2-delays and numerical methods

This paper investigates stability analysis and numerical approaches for nonlinear delay differential-algebraic equations with 2-delays. Based on asymptotic stability considerations for Hessenberg type delay differential-algebraic equations, we develop a stability analysis approach using direct linearization. This analysis provides some insight into how stability properties can be preserved under discretization. The main contribution of this paper lies in establishing a stability analysis system based on local linearization, and consists in deriving sufficient conditions for the implicit Euler method and the 2-step backward differentiation formula method to preserve stability and asymptotic stability properties of the nonlinear continuous system. Numerical results demonstrate that the proposed numerical methods can effectively maintain the stability of the original systems, thereby supporting the practical solution of delay differential-algebraic equations.