Resonance analysis based on the Galerkin technique and coupled nonlinear Mathieu equations that have been renormalized

The stability evaluation of damping-coupled Mathieu-type equations necessitates robust analytical methodologies. To conquer the limitations of small-parameter assumptions inherent in perturbative strategies, the non-autonomous system is transformed into an equivalent autonomous system, simplifying the analysis. The Galerkin technique provides a powerful foundation for investigating these complex systems by carefully including oscillator coupling to capture problematic electricity transfer and interaction consequences that are frequently overlooked in traditional approaches. The coupled Galerkin approach enables a comprehensive study of balance behavior, offering significant insights into the effects of oscillator coupling on typical balance. The method highlights the effects of coupling on resonance stability transitions, such as internal and harmonic resonances, by determining stability barriers via essential parameter evaluation. This method provides a radical insight into the stability dynamics of coupled Mathieu oscillators. It is a critical tool for engineering and science applications that require precise control over complicated oscillatory systems.