Insightful Examination of Some Nonlinear Classifications Linked with Mathieu Oscillators

Abstract Purpose The aim of the current study is to comparehend an analytical description of different classes of nonlinear Mathieu oscillators. The van der Pol-Duffing-Mathieu oscillator, and the generalized van der Pol-Duffing-Mathieu oscillator are examined. Additionally, the hybrid Rayleigh-van der Pol-Duffing-Mathieu oscillators, as well as the nonlinear Mathieu oscillator, are scrutinized. Method The non-perturbative approach (NPA) is utilized to convert the nonlinear ordinary differential equations (ODEs), of the illustrated dynamical systems, into linear ones. The approximate solutions are derived independently in the series expansion and without the use of conventional perturbation techniques. Therefore, the goal is to deviate from conventional perturbation techniques and get approximations of small amplitude parametric components without imposing any restrictions. The method is also expanded to determine the best solutions for the nonlinear immense amplitude of fluctuation. Results The current method offers successive approximations of the solutions of parametric nonlinear fluctuations may be obtained by quickly estimating the frequency-amplitude relationship. The resulting parametric equations are validated, showing high degree of agreement with the original equation. Stability behavior is analyzed under various circumstances. The transition curves, bifurcation diagram, Poincaré map, and phase portrait are also examined using the Floquet theory. Conclusion The stability regions are found to be diminishing with the rise of the natural frequency, and the excited frequency. Moreover, the achieved regions are found to be growing with the rise of the damping coefficient and the excitation amplitude. The stability settings have been examined by considering the effects of various factors in both the damped and un-damped phases for each situation. In the un-damped state, PolarPlots are examined of the transition curves of the two corresponding solutions, namely Cos- and Sin-oscillations. The conclusions of the acquired results suggested that the approach presented here is highly efficient, robust, founded on solid premises, and remarkably intuitive.