A comprehensive study of stability analysis for nonlinear Mathieu equation without a perturbative technique

The present research focuses on the challenges engineers face in predicting the behavior of nonlinear vibration systems accurately. The nonperturbative method is highlighted as a solution that provides insights into chaos, bifurcation, resonance response, and stability attributes. Specifically, the study delves into the dynamic analysis of the nonlinear Mathieu equation. The research involves a complex and extensive analytical exploration, transitioning from a nonlinear state to a linear one through various stages. The introduced computational method aims to examine the resonance response of the nonlinear Mathieu equation and offer innovative solutions for the Mathieu–Duffing–type oscillator. The nonperturbative approach remains essential in gaining a deeper understanding of nonlinear vibration systems.