Comprehensions in Nonlinear Mathieu Oscillator: Analytic and Chaotic Inspections

Abstract Purpose Scrutinizing the nonlinear Mathieu oscillator is significant in realizing intricate behavior in practical systems, offering insights into engineering, physics, and applied mathematics, such as mechanical vibrations and structural stability. The point of this study is to acquire the frequency amplitude of parametric non-linear problems. Methodology The basic methodology is established on the non-perturbative approach (NPA), which is constructed principally in He’s frequency formulation (HFF). The NPA converts a weakly nonlinear second-order oscillator of ordinary differential equation (ODE) into a linear one. The approach response results freely from the usual perturbation methodologies. Accordingly, the investigation aims to leave all regular perturbation techniques aside and estimate responses of small amplitude parametric elements devoid of constraints. Besides, the procedure is unlimited to establish optimum responses of the nonlinear high-amplitude oscillations. A rapid measure of the oscillation-amplitude connection is needed to derive succeeding estimations of responses to parametric non-linear oscillations. The Mathematica Software program (MS) is working to validate the derived parametric ODE, demonstrating the primary formulation. Results The stability performance is examined across several situations. The present procedure is based on pure principles, is appropriate, and produces exceptionally high numerical accuracy. The present technique decreases measured complexity; the construction is valuable for the mathematical implementation of nonlinear parametric problems. The nonlinear dynamics of the model are analyzed through bifurcation diagrams, identifying critical parameters that influence system behavior. The Poincaré map further reveals periodic and chaotic oscillations, shedding light on long-term stability and chaos emergence.