Abstract
The principal resonance of a van der Pol–Duffing oscillator to the combined excitation of a deterministic harmonic component and a random component has been investigated. By introducing a new expansion parameter ε=ε( ε ̄ ,u 0) , the method of multiple scales is adapted for the strongly non-linear system. Then the method of multiple scales is used to determine the equations of modulation of response amplitude and phase. The behavior and the stability of steady-state response are studied by means of qualitative analysis. The effects of damping, detuning, bandwidth, and magnitudes of random excitations are analyzed. The theoretical analyses are verified by numerical results. Theoretical analyses and numerical simulations show that when the intensity of the random excitation increases, the non-trivial steady-state solution may change from a limit cycle to a diffused limit cycle. Under some conditions the system may have two steady-state solutions. Random jump may be observed under some conditions. The results obtained in the paper are adapted for a strongly non-linear oscillator that complement previous results in the literature for the weakly non-linear case.
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