Advanced non-perturbative and galerkin scrutinizes of forced van der pol–duffing oscillator

Abstract The investigation of the forced van der Pol–Duffing oscillator (VDO) with cubic and quintic nonlinearities offers an innovative framework in modeling complex dynamical behavior such as chaos and bifurcations, which are fundamental in engineering, physics, and biological systems. Analyzing and managing nonlinear oscillatory processes in mechanical structures, electrical circuits, and vibration systems provides a driving force. For these fields, precise resonance prediction, energy harvesting, and stability improvement are essential. The study focuses on a robust non-perturbative approach (NPA) to derive periodic solutions for both damped and conservative coupled systems. The method is based on He’s frequency formula (HFF) and stands apart from any traditional perturbation techniques by avoiding Taylor-series expansions. NPA is particularly suitable for weakly nonlinear oscillator ordinary differential equations (ODEs), offering a valid approximation even under strong nonlinearity. The resulting parametric expressions are validated via numerical solution (NS) and employing Mathematica Software (MS), showing excellent agreement and confirming the reliability of the method. NPA demonstrates to be simple, powerful, and flexible, with applications extending to interconnected dynamical systems and fluid dynamics. While equivalent linearization techniques can yield solvable models, they often fail to capture resonance accurately. To address this, an enhanced procedure incorporating the Galerkin method is used to investigate resonance behavior and establish stability criteria. Quantitative comparisons confirm that the derived analytical outcomes align with advanced numerical findings. Furthermore, a comprehensive bifurcation analysis, employing bifurcation diagrams, largest Lyapunov exponent (LLE), phase portraits, and Poincaré maps, is conducted to classify the system's dynamic responses. This enables the identification of transitions between periodic, quasiperiodic, and chaotic motions, shedding light on the rich dynamics of VDO.