Insightful Examination of Some Nonlinear Classifications Linked with Mathieu Oscillators

• DOI: 10.1007/s42417-024-01688-3
• Publication Date: 2025-02-01T07:15:36Z
• Authors: Galal M. Moatimid; Mona A. A. Mohamed; Khaled Elagamy
• Keywords: Mathieu function; Parametric oscillator; Duffing equation; Floquet theory

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