Advanced nonlinear dynamics and bifurcation structures in multi-coupled oscillators using a powerful non-perturbative framework

Abstract

Nonlinear oscillators with two degrees of freedom (2DOF) serve as fundamental models for describing complex dynamical behavior in engineering and applied mechanics. Accurate prediction of their responses is crucial for stability enhancement, vibration suppression, and optimal design of coupled mechanical systems. In this study, three distinct 2DOF coupled oscillator models are examined, encompassing both linear and strongly nonlinear restoring forces that govern free and damped vibration regimes. These models provide realistic frameworks for analyzing nonlinear interactions, resonance phenomena, and stability boundaries in coupled dynamical systems. The primary objective is to develop and apply a robust non-perturbative approach (NPA) for deriving periodic solutions of conservative and damped coupled oscillators. The proposed approach, rooted in He’s Frequency Formula (HFF), fundamentally differs from classical perturbation techniques as it avoids Taylor-series expansions, linearization assumptions, and small-parameter constraints. Instead, the nonlinear governing equations are transformed into analytically tractable linear forms, enabling efficient treatment of strongly nonlinear performance. The analytical solutions are validated through comprehensive numerical simulations implemented in Mathematica Software (MS), and are systematically compared with direct numerical integrations, demonstrating excellent accuracy and computational efficiency. Furthermore, bifurcation diagrams and Poincaré maps (PMs) are employed to characterize the qualitative dynamical transitions and classify the complex response patterns exhibited by each coupled model.