Stability and Computational Analysis of Nonlinear Dynamical Systems Using Lyapunov-Based Methods

Abstract

The paper introduces an analytical and computational framework of the stability analysis of nonlinear dynamical systems based on Lyapunov methods. Complex behaviors involving bifurcations, limit cycles and chaotic dynamics characterize nonlinear systems and the stability assessment is challenging and necessary to confident system design. The given research integrates the classical Lyapunov stability theory with numerical simulations to offer a formal and confirmable method of stability analysis. We adopt a generalized approach to constructing Lyapunov functions to analyze the stability of equilibrium, and then we test it with benchmark nonlinear systems, such as the Van der Pol oscillator, Duffing oscillator, and Lorenz system. The phase-space analysis and time-domain simulations with different initial conditions and parameter settings support the analytical results. The results indicate that the Lyapunov-based framework is a good predictor of the local and asymptotic stability properties, whereas numerical simulations allow one to learn more about transient and long-term behavior of a system. The suggested methodology contributes to the credibility of stability analysis because it moves the theoretical rigor to the computational validation. The work adds some form of structure that can be further extended to complex engineering situations, such as control systems, power networks, and nonlinear oscillatory models to enhance not only the quality of analytical results but also their practical value.