Response of a strongly non-linear oscillator to narrowband random excitations

The principal resonance of a van der Pol–Duffing oscillator subject to narrowband random excitations has been studied. By introducing a new expansion parameter ε=ε(ε̄,u0) the method of multiple scales is adapted for the strongly non-linear system. The behavior of steady state responses, together with their stability, and the effects of system damping and the detuning, and magnitude of the random excitation on steady state responses are analyzed in detail. Theoretical analyses are verified by some numerical results. It is found that when the random noise intensity increases, the steady state solution may change form a limit cycle to a diffused limit cycle, and the system may have two different stable steady state solutions for the same excitation under certain conditions. The results obtained for the strongly non-linear oscillator complement previous results in the literature for weakly non-linear systems.