Periodic Solutions of Strongly Nonlinear Oscillators Using He’s Frequency Formulation

Abstract

In this paper, we address several scientific and technological challenges with a novel non-perturbative approach (NPA), simplifying processing time compared to traditional methods. The proposed approach transforms nonlinear ordinary differential equations into linear ones, akin to simple harmonic motion, producing a new frequency. This method yields highly accurate outcomes, surpassing well-known approximate methodologies, as validated through numerical comparisons in mathematical software. The congruence between numerical solution tests and theoretical predictions further supports our findings. While classical perturbation methods rely on Taylor expansions to simplify restoring forces, NPA also enables stability analysis. Thus, for analyzing approximations of highly nonlinear oscillators in mathematical software, NPA serves as a more reliable tool.