(untitled) (European Journal of Pure and Applied Mathematics, 2025)
The Mathieu–cubic-quintic Duffing oscillator (MDO) exemplifies a fundamental model for a parametric stimulated system characterized by nonlinearities. This issue is essential for realizing intricate, nonlinear dynamical behavior, including bifurcations, chaos, and resonance phenomena, pertinent to engineering and physics for systems influenced by periodic external stimuli. A universal two-degree-of-freedom (TDOF) of MDO is examined. This study utilizes an adaptation of the non-perturbative approach (NPA), to encompass the coupled system, to inspect a parametric nonlinear oscillatory system and evaluate its efficacy. The NPA is based mainly on the He’s frequency formula (HFF). The main aim of the NPA is to convert any nonlinear ordinary differential equation (ODE) into a linear one. The approximate solutions are derived independently of conventional perturbation methods, excluding the series expansion. Consequently, the objective of this study is to depart from traditional perturbation methods and derive estimated solutions for tiny amplitude parametric components without restrictions. Furthermore, the technique is extended to ascertain optimal responses for the nonlinear large amplitude fluctuations. The capacity to swiftly assess the frequency-amplitude correlation is essential for deriving successive approximations of the solutions to parametric nonlinear fluctuations. The Mathematica Software (MS) is employed to verify the agreement parametric equation, which demonstrates significant concordance with the original equation. The stability behavior is examined in multiple instances. The existing methodology is characterized by clear principles, appropriateness, user-friendliness, and exceptional numerical precision. The current method reduces mathematical complexity, making it advantageous for addressing nonlinear parametric problems. It also discussed the influence of multiple coefficients on solution behavior, ensuring stability through time-dependent plots and PolarPlots, and detecting periodic behavior with parameter variations.