An Innovative Approach in Inspecting a Damped Mathieu Cubic–Quintic Duffing Oscillator

Abstract Purpose The objective of the present study is to analyze a damped Mathieu–cubic quintic Duffing oscillator as a parametric nonlinear oscillatory dynamical system. This equation has multiple applications in diverse fields, including optics, quantum physics, and general relativity. There are multiple concerns related to periodic motion and the analysis of boundary-value problems with elliptic symmetries. The current effort aims to determine the frequency amplitude of parametric nonlinear issues. Method The non-perturbative approach (NPA) is employed to transform the nonlinear ordinary differential equation (ODE) into a linear equation. The derivation of the approximate solutions is achieved without relying on typical perturbation approaches, separate from the series expansion. Hence, the objective of this study is to depart from traditional perturbation methods and acquire approximated solutions for minor amplitude parametric components without imposing any limitations. Furthermore, the technique is extended to ascertain optimal solutions for the nonlinear large amplitude of fluctuation. Results The current approach allows for rapid estimation of the frequency-amplitude relationship in order to attain successive approximations of the solutions for parametric nonlinear fluctuations. A validation is obtained for the derived parametric equation, demonstrating a high level of agreement with the original equation. An analysis of stability behavior is conducted in multiple scenarios. In addition, the Floquet theory is used to examine the transition curves. Conclusion The current technique is characterized by its clear principles, making it practical, user-friendly, and capable of achieving exceptionally high numerical precision. The current approach is highly beneficial for addressing nonlinear parametric problems due to its ability to minimize algebraic complexity during implementation.