Chaotic and dynamic vibration analysis of a time-delayed nonlinear mathieu oscillator via non-perturbative approach

The study compares van der Pol and Rayleigh oscillators with time-delayed effects, providing insights into stability, bifurcations, and organization in nonlinear systems. It emphasizes unique dynamical properties and resonance conditions, strengthening the theoretical basis of the design of delay-controlled oscillatory systems. The aim is to adopt the non-perturbative approach, which transforms a weakly nonlinear oscillator of an ordinary differential equation into a linear one. Computed through a refined series approximation, the response manages both small and large oscillatory amplitudes without constraining assumptions, avoiding reliance on small parameter expansions. Validation outcomes reveal a strong agreement between parametric clarifications and the original nonlinear model, confirming the credibility of the proposed framework. Furthermore, a comprehensive assessment of the system’s stability is conducted in diverse situations. Two different instances of nonlinear Mathieu oscillators are inspected. A comprehensive stability analysis is performed for two nonlinear Mathieu-type oscillators with van der Pol and Rayleigh damping. Numerical simulations are carried out employing time histories, phase portraits, Poincaré maps, bifurcation diagrams, and Lyapunov exponents. These simulations ensure the accuracy and reliability of the analytical predictions. The outcomes expound distinct and contrasting stability performances for the van der Pol and Rayleigh oscillators. They demonstrate that nonlinear damping, excitation amplitude, natural frequency, and time delay play vital roles in governing the system’s dynamic response. It is observed that stability generally decreases with increasing natural and excitation frequencies, while it enhances with higher damping and excitation amplitudes. Moreover, the van der Pol and Rayleigh oscillators exhibit opposite impacts of nonlinear damping and natural frequency on stability. The novelty of this study lies in employing a non-perturbative approach to time-delayed Mathieu-type oscillators. The obtained findings provide valuable physical insight and practical guidance for the analysis and design of delay-controlled oscillatory systems in mechanical, structural, and engineering implementations.