Abstract
Nonlinear dynamics is a fundamental aspect of complex engineering systems where linear approximations fail to capture critical behaviors such as chaos, bifurcations, and multi-stability. This article reviews mathematical approaches utilized to model nonlinear dynamic systems across various engineering disciplines. Techniques covered include analytical methods, perturbation techniques, numerical simulations, and modern computational algorithms. The study emphasizes the importance of understanding nonlinearities for accurate system prediction, control, and optimization. A comparative analysis illustrates the effectiveness of these methods in capturing complex dynamic phenomena. The article also includes a graphical demonstration of bifurcation behavior in a nonlinear oscillator model. Finally, it concludes with current challenges and future research directions in nonlinear dynamic modeling
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