STABILITY ANALYSIS OF NONLINEAR DYNAMICAL SYSTEMS USING DIFFERENTIAL EQUATIONS

Abstract

Stability analysis is fundamental to understanding and predicting the long-term behavior of nonlinear dynamical systems, which are ubiquitous in engineering, physics, biology, and many other scientific domains. Unlike linear systems, nonlinear systems can exhibit complex phenomena such as multiple equilibria, limit cycles, bifurcations, and chaos, making their stability properties far more intricate to characterize. This paper presents a comprehensive overview of stability theory for nonlinear systems governed by ordinary differential equations. It traces the historical development of the field from the early contributions of Torricelli, Maxwell, Routh, and Hurwitz to the groundbreaking work of Lyapunov and Poincaré, and further to modern frameworks including LaSalle’s Invariance Principle, Input-to-State Stability (ISS), and Center Manifold Theory. Special emphasis is placed on Lyapunov’s Direct Method, which employs energy-like Lyapunov functions to establish stability without explicitly solving the differential equations. The paper also discusses LaSalle’s Invariance Principle for handling semi-definite Lyapunov derivatives, linearization techniques via the Hartman-Grobman theorem, and advanced topics such as bifurcation theory and the transition to chaotic behavior. Applications across diverse fields including biological systems (Lotka–Volterra, SIR models), fluid dynamics, electrical circuits, and power systems are explored to illustrate the practical relevance of these theoretical tools. The survey further examines contemporary computational approaches such as Sum-of-squares optimization and emerging frontiers including Neural Lyapunov certificates and stability analysis of non-smooth and high-dimensional systems. By bridging classical theory with modern techniques, this work provides researchers and engineers with a solid foundation for analyzing, designing, and ensuring the robustness and reliability of nonlinear dynamical systems.