Numerical stability analysis of nonlinear differential equations in engineering systems

Nonlinear differential equations describe the dynamic behaviours of many engineering systems, from mechanical oscillators with nonlinear springs to electrical circuits containing diodes. The safe and accurate simulation of such systems requires understanding both the qualitative stability of equilibrium solutions and the numerical stability of discretization schemes. This paper focuses on a mathematical stability analysis of a damped nonlinear pendulum, a canonical example in mechanical engineering. Challenges arise from the nonlinearity of the governing equations, the presence of multiple equilibrium points and the sensitivity of numerical solutions to step size. The proposed methodology involves deriving equilibrium points, linearizing the system, computing eigenvalues of the Jacobian to assess local stability, and analyzing the stability regions of numerical integration schemes. Results show how damping influences the eigenvalues and thus the asymptotic behaviours of the pendulum, and how explicit time stepping methods impose strict step size restrictions for stable simulation. Outcomes demonstrate that proper selection of integration schemes and parameters ensures accurate long term predictions of nonlinear dynamics.