Nonlinear Analysis of Periodic/Quasi-Periodic Dynamics of a Forced Duffing Oscillator Using the Nonperturbative Approach

The study investigates nonlinear stability and dynamical behavior of a mathematical model of a forced Duffing oscillator using the nonperturbative approach that is based on He’s frequency formula. The method constructs an amplitude-dependent equivalent representation of a nonlinear system. It does not use small-parameter assumptions or perturbation expansions. It retains the full effects of damping, external forcing, and cubic stiffness. The main objective is to describe nonlinear resonance and stability transitions in moderately nonlinear regimes. The formulation provides closed-form expressions of effective frequency and damping. These issues depend on the oscillation amplitude and system parameters. The analytical results are validated using Mathematica Software. The numerical solutions show an excellent agreement in both transient and steady-state responses. This confirms the validity of the proposed formulation. The stability inspection for the linear ordinary differential equation is conducted. Regarding the original nonlinear variant, the stability analysis is performed via Lyapunov exponents, Poincaré maps, phase portraits, and bifurcation diagrams. The system exhibits stable periodic and quasi-periodic responses. The transitions between different response types depend on damping, cubic stiffness, and excitation amplitude. Time-history results explain energy exchange and amplitude modulation. The method is computationally efficient compared to direct numerical integration. It is suitable for parametric studies in various engineering applications.