The exploration of nonlinear damped oscillators is driven by their common occurrence in real systems where damping and nonlinearity dictate stability and energy dissipation. This study is innovative due to the complex phenomena they display, including amplitude-dependent frequencies, bifurcations, and chaos, which are not represented by linear frameworks. Therefore, this issue addresses distinct five nonlinear dynamical systems, focusing in understanding and improving dynamic behavior of mechanical systems like vibration isolation, energy harvesting, and precision control. The goal of the study is to implement He’s frequency formula (HFF) in order to realize analytical justifications of extremely weakly nonlinear oscillators. The novel methodology, which is effectively converting a nonlinear ordinary differential equation (ODE) to a linear one, is referred to as the non-perturbative approach (NPA). It is well known that all conventional perturbation methods rely Taylor expansions to approximate restoring forces when present, which often introduces limitations and reduces the accuracy of the resulting solutions. The results are validated throughout a graphical comparison as well as Tabular comparison with the Mathematica Software (MS). Additionally, some problems are validated with modified algebraic method (MAGM). Moreover, the NPA allows in investigation of the matters’ stability analysis, which was difficult in the previous approaches. The application of NPA of science and technology and applied research is therefore useful in studying approximations of abundant nonlinear dynamical systems as well as fluid dynamics.
Ismail et al. (2025) studied this question.