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

Area of Science:

• Numerical Analysis
• Differential Equations

Background:

• Nonlinear delay differential-algebraic equations (DDAs) with multiple delays present significant challenges in stability analysis and numerical computation.
• Understanding how numerical methods affect the stability of these complex systems is crucial for reliable simulations.

Purpose of the Study:

• To develop a stability analysis framework for nonlinear DDAs with two delays.
• To investigate sufficient conditions for numerical methods to preserve the stability and asymptotic stability of the continuous nonlinear system.
• To validate the effectiveness of proposed numerical approaches through simulations.

Main Methods:

• Stability analysis using direct linearization based on asymptotic stability considerations.
• Derivation of stability preservation conditions for the implicit Euler method and the 2-step backward differentiation formula.
• Numerical experiments to demonstrate the efficacy of the methods.

Main Results:

• A stability analysis system based on local linearization was established.
• Sufficient conditions were derived for the implicit Euler and 2-step BDF methods to preserve stability.
• Numerical results confirmed that the proposed methods effectively maintain the stability of nonlinear DDAs.

Conclusions:

• The developed stability analysis framework provides insight into discretization effects on stability.
• The implicit Euler and 2-step BDF methods are shown to be effective in preserving stability for nonlinear DDAs.
• The findings support the practical application of these numerical methods for solving complex delay differential-algebraic equations.