Non-Perturbative Approach in Scrutinizing Nonlinear Time-Delay of Van der Pol-Duffing Oscillator

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. Examining this oscillator facilitates the investigation of complex dynamics, including chaos, bifurcations, and stability alterations, making it essential for disciplines like control theory, engineering, and neuroscience. The current oscillator is analyzed using using the non-perturbative approach (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. The analytical solution (AS) with the associated numerical data that reveals high nonlinearity, and 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 the 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.