Bifurcation, Stability, and Nonlinear Vibration Analysis of a Harmonically Excited Duffing Oscillator Coupled with a Two-Degree-of-Freedom Nonlinear Energy Sink

The study investigates the nonlinear dynamics of a harmonically excited Duffing oscillator coupled with an unforced two-degrees-of-freedom nonlinear energy sink. The external excitation is applied only to the primary oscillator; meanwhile, the NES response is induced through nonlinear internal coupling. The governing nonlinear ordinary differential equations are analyzed using the proposed non-perturbation approach, which does not rely on small-parameter assumptions or Taylor-series expansions. The formulation is used to obtain amplitude-dependent equivalent linear representations and analytical approximations of the coupled system. The analytical results are compared with direct numerical simulations, showing overall agreement with the full nonlinear model. The stability of the steady-state solutions is examined under variations of the main system parameters. The results indicate that the nonlinear coupling and stiffness parameters significantly affect the response amplitudes, stability characteristics, and overall dynamical behavior. Additional analyses using bifurcation diagrams, Lyapunov exponents, Poincaré maps, and basins of attraction reveal transitions between periodic, quasi-periodic, and chaotic regimes, as well as the presence of multi-stability and sensitivity to initial conditions. The proposed framework provides a useful analytical tool in studying the dynamics and stability of nonlinear oscillatory systems over a wide range of operating conditions.