A Novel Methodology for Scrutinizing Periodic Solutions of Some Physical Highly Nonlinear Oscillators

The study offers a comprehensive investigation of periodic solutions in highly nonlinear oscillator systems, employing advanced analytical and numerical techniques. The motivation stems from the urgent need to understand complex dynamical behaviors in physics and engineering, where traditional linear approximations fall short. This work precisely applies He’s Frequency Formula (HFF) to provide theoretical insights into certain classes of strongly nonlinear oscillators, as illustrated through five broad examples drawn from various scientific and engineering disciplines. Additionally, the novelty of the present work lies in reducing the required time compared to the classical perturbation techniques that are widely employed in this field. The proposed non-perturbative approach (NPA) effectively converts nonlinear ordinary differential equations (ODEs) into linear ones, equivalent to simple harmonic motion. This method yields a new frequency approximation that aligns closely with the numerical results, often outperforming existing approximation techniques in terms of accuracy. To aid readers, the NPA is thoroughly explained, and its theoretical predictions are validated through numerical simulations using Mathematica Software (MS). An excellent agreement between the theoretical and numerical responses highlights the robustness of this method. Furthermore, the NPA enables a detailed stability analysis, an area where traditional methods frequently underperform. Due to its flexibility and effectiveness, the NPA presents a powerful and efficient tool for analyzing highly nonlinear oscillators across various fields of engineering and applied science.