Forced Oscillations of the van der Pol Oscillator with Delayed Amplitude Limiting

Forced oscillations of the van der Pol oscillator with delayed amplitude limiting are analyzed using the derivative-expansion method. The forcing function is taken to be either sinusoidal or white noise. The oscillator's output depends on whether the excitation is "hard" [i.e., excitation amplitude is
O(1)
] or "soft" [i.e., excitation amplitude is
O(\mu)
,
\mu
is a small parameter], and whether the excitation is resonant (i.e., excitation frequency is near natural frequency) or not. Explicit first-order expressions are obtained for the output in the nonresonant case. If the excitation is soft, the steady-state output is a combination of terms having frequencies equal to those of the natural and excitation frequencies, with the forced response being dominated by the natural response. On the other hand, if the excitation is hard, the natural response fades away as time increases over a wide range of frequencies and excitation amplitudes. Consequently, the output is harmonic, having a frequency equal to the excitation frequency. In the resonant case, the steady-state output is synchronized at the excitation frequency, irrespective of whether the excitation is soft or hard. The frequency response equation is a family of curves that depends on the excitation amplitude and the delay time as parameters. The stability of these harmonic oscillations is determined. For the noise perturbed oscillator, the conditional probability distribution for the deviations from the stationary stable state is presented.