A Non-Perturbative Framework in Analyzing Weakly Nonlinear Oscillators and Their Chaotic Dynamics

Weakly nonlinear oscillators display complex behavior that perturbation methods struggle to analyze, particularly near critical thresholds. The non-perturbation approach (NPA) offers a unified, parameter-agnostic approach that is effective in strongly resonant situations, accurately capturing global phase space structures, and directly addressing chaotic transitions, providing predictive insights where traditional methods fail. The NPA as a novel technique successfully converts the nonlinear weakly oscillator of the ordinary differential equation (ODE) into a linear issue. Theoretical findings are confirmed through a numerical comparison using Mathematica Software (MS). The results of the numerical solution (NS) show excellent agreement. It is commonly acknowledged that all conventional perturbation methods utilize Taylor expansion to augment restoring forces, hence optimizing the usual conditions. A comprehensive analysis of the issue’s stability is easily achievable via NPA. Accordingly, when evaluating NS estimates of weakly nonlinear oscillators, NPA occupations serve as a more useful form of responsibility. Additionally, the stability analysis is easily accomplished via NPA. The system’s dynamics are examined by chaotic analyses, incorporating bifurcation diagrams (BDs), Poincaré maps (PMs), and Lyapunov exponents (LEs). This analysis identifies transitions between regular and complicated behavior and thoroughly examines the system’s stability features. The results provide a comprehensive understanding of the fundamental nonlinear dynamics and offer significant insights for future research on analogous systems.