The time-delayed (TD) of velocity and position are employed throughout this investigation to lessen the nonlinear vibration of an exciting Van der Pol-Duffing oscillator (VdPD). The issue encompasses multiple real-world elements such as feedback lags, signal transmission delays, and delayed responses in mechanical, electrical, or biological systems. Actually, examining this oscillator facilitates the investigation of complex dynamics, including chaos, bifurcations, and stability alterations, rendering it essential for disciplines like control theory, engineering, and neuroscience. The current oscillator is analyzed using the NPA. This methodology is based mainly on the He’s frequency formula (HFF). Simply, this approach transforms the nonlinear ordinary differential equation (ODE) into a linear one. Accordingly, the stability standards are constructed, depicted, and sketched. Contrasting the analytical solution (AS) with the associated numerical data, which reveals high nonlinearity, the numerical estimation is validated via the Mathematica Software (MS). In contrast to other traditional perturbation methods, the NPA exhibits high convenience, accessibility, and great precision in analyzing the behavior of strong nonlinear oscillators. Subsequently, this technique enables the analysis of issues related to other oscillators in dynamical systems. It is an effective and promising method for addressing similar dynamic system challenges, providing a qualitative assessment of theoretical outcomes. The study describes time histories of solutions for different natural frequencies and TD parameters and discusses the main findings based on displayed curves. It also examines how various regulatory limits impact the vibrating system. The performance is applicable in engineering and other domains owing to its flexibility in various nonlinear systems. Consequently, the NPA can be considered significant, effective, and intriguing, with potential for use in more categories within the domain of coupled dynamical systems.

The goal of this article is to reduce the vibration of a hybrid oscillator with a cubic–quintic nonlinear term under internal and external forces in the worst resonance case. To eliminate the harmful vibration in the system, the following strategies are suggested: nonlinear derivative feedback control (NDF), linear negative velocity feedback control (LNVC), nonlinear integral positive position feedback (NIPPF), integral resonant control (IRC), negative velocity with time delay (TD), and positive position feedback (PPF). It is discovered that the PPF control suppresses vibration more effectively than typical controllers, which reduces the vibration to 0.0001 with an effectiveness of 99.92%. Moreover, the main advantages of the PPF controller are its low cost and the fast response. The multiple time scale perturbation technique (MSPT) is used to apply the theoretical methodology and obtain a perturbed response. In order to close the concurrent primary and internal resonance case, frequency response (FR) equations are also used to study and analyze the system’s stability. The MATLAB software is used to complete and clarify all numerical topics. The FR curves are examined to determine the amplitude’s subsequent impact from variations in the parameters’ values. Lastly, a comparison of the numerical and analytical solutions is performed utilizing time history. Along with comparing the impact of our PPF damper on the hybrid oscillator, earlier research is also provided.