The nonlinear Mathieu-Duffing oscillator is a basic model of parametrically driven dynamical systems with nonlinear coefficients, and it serves as an important model for the study of such phenomena as bifurcation, chaotic transitions and resonance instabilities in engineering structures and physical systems subjected to periodic external excitations. In this work, a non-perturbative analytical approach is proposed to analyse a coupled parametric nonlinear oscillatory system and to determine its dynamical features with much more computing efficiency. The proposed methodology is based on He’s frequency formulation, which systematically linearises weakly nonlinear ordinary differential oscillators, avoiding the fundamental limitations of classical perturbation methods and being effective for large amplitude nonlinear oscillatory regimes. A prominent contribution of this work is the extension of the non-perturbative framework to the consideration of linked oscillator systems, which is a unique application domain for the method. The analytical conclusions are validated by symbolic computation in Mathematica. It is found that there is strong conformity with the original governing equations for different parametric configurations. The stability study is conducted under different operating situations, and it always shows the accuracy, analytical tractability, and numerical accuracy of the procedure. The global dynamical behavior of the system is also characterized by bifurcation diagrams, which identify critical parameter-dependent transitions, and the largest Lyapunov exponent, which quantitatively distinguishes between periodic and chaotic oscillatory regimes and defines the threshold conditions for the onset of chaos. The results agree with the conclusion that the non-perturbative technique is a powerful, and efficient for the analysis of parametrically stimulated nonlinear dynamical systems.
Almutlg et al. (2026) studied this question.