Stabilization of Nonlinear Coupled Parametric Oscillators of Mathieu’s Type in Fractal Space

In this work, the Renormalization Method (RM) is used to analyze the dynamics of a nonlinear two-degree-of-freedom (2DOF) system under parametric excitation, with a focus on fractal vibration behavior. This procedure comprises transforming the system into a comparable form. An equivalent linearized model is produced by isolating the system’s nonlinear interactions using a two-scale formulation and mean-square analysis. The non-autonomous fractal equations are transformed into an autonomous representation using the RM, and then the system is described in traditional derivative form using El-Dib’s fractal transformation. The fractal-coupled Mathieu system’s stability behavior can be effectively identified using this framework. An agreement with the analytical solutions is shown by numerical results. All things considered, the integrated RM-based approach provides a reliable tool for forecasting and managing intricate nonlinear fractal systems.