Computational Modeling and Simulation of Nonlinear Dynamical System Stability in Applied Mathematics

Abstract

Nonlinear dynamical systems represent a fundamental area of study in applied mathematics due to their relevance across various disciplines, including physics, biology, and engineering. Their inherent complexity, characterized by phenomena such as bifurcation, chaos, and sensitivity to parameter variations, often limits the effectiveness of traditional manual analysis, particularly when addressing high-dimensional or computationally intensive models. This study aims to address these challenges by applying computational modeling and numerical simulation techniques to analyze the stability of nonlinear dynamical systems. The research employs analytical methods, including equilibrium point identification and linearization, which are then validated and extended through the fourth-order Runge-Kutta numerical method. Simulations were conducted to visualize equilibrium points, phase portraits, and parameter-driven bifurcation phenomena. The findings demonstrate a strong correspondence between analytical and numerical approaches, with minimal error margins (≤1%) observed in equilibrium point estimation, thus confirming the reliability of computational methods. Moreover, the bifurcation analysis revealed critical transitions such as pitchfork and Hopf bifurcations, which indicate sudden shifts from stability to instability behaviors that are difficult to capture through manual calculations alone. The integration of computational approaches provides clear advantages, offering systematic exploration of parameter spaces and detailed visualizations of system dynamics, thereby expanding the scope of stability analysis. In conclusion, this study emphasizes that computational modeling is not only an effective complement to analytical methods but also a necessary strategy for advancing the understanding of nonlinear dynamical systems in applied mathematics.