Bifurcation threshold of the delayed van der Pol oscillator under stochastic modulation

We obtain the location of the Hopf bifurcation threshold for a modified van der Pol oscillator, parametrically driven by a stochastic source and including delayed feedback in both position and velocity. We introduce a multiple scale expansion near threshold, and we solve the resulting Fokker-Planck equation associated with the evolution at the slowest time scale. The analytical results are compared with a direct numerical integration of the model equation. Delay modifies the Hopf frequency at threshold and leads to a stochastic bifurcation that is shifted relative to the deterministic limit by an amount that depends on the delay time, the amplitude of the feedback terms, and the intensity of the noise. Interestingly, stochasticity generally increases the region of stability of the limit cycle.

Figure 1

Phase portrait of the velocity of the oscillator $y=\dot{x}$ as a function of its position $x$ of the deterministic $(D=0)$ van der Pol oscillator with delayed feedback for different values of the damping parameter $\beta$. The parameters are fixed at ${\omega }_0=1$, $b=1$, $\eta =1$, $\kappa =1$, and $\tau =1$.

Figure 2

Power spectrum $P\left(\omega \right)=\langle |\widehat{X}\left(\omega \right){|}^2\rangle$ as a function of the frequency $\omega$ for different values of the intensity of the damping $\beta$. The parameters are fixed at ${\omega }_0=1$, $\eta =1$, $\kappa =1$, $b=1$, $D=0.1$, and $\tau =0.01$ so that according to Eq. (30) the critical damping is ${\beta }_c\approx -1.06$. The peak of the power spectrum is located at ${\omega }_{\text{max}}\approx 1.407$, in agreement with Eq. (8).

Figure 3

Frequency of oscillation $\omega$ as a function of the time delay $\tau$ for different values of the amplitude of the delayed velocity term $\kappa$. The other parameters are fixed at ${\omega }_0=1$, $\eta =1$, $\beta =-\kappa$, $b=1$, and $D=0.1$. The symbols are the numerically determined frequencies whereas the solid curves show the numerical solution of Eq. (8).

Figure 4

Numerically obtained stationary probability distribution function $p\left(x\right)$ for the van der Pol oscillator with delayed feedback. We show results for several upper bounds of the time window used to calculate the probability density as a test of stationarity (denoted by $t$ in this figure). The parameters are fixed at ${\omega }_0=1$, $\eta =1$, $b=1$, $\kappa =1$, $\tau =0.025$, and $D=0.1$. The bifurcation is located at ${\beta }_c\simeq -1.08$ for this set of parameters. The probability density is not stationary below threshold, as shown in (a) $(\beta =-1.1)$. However, the distribution is stationary above threshold, as shown in (b) and (c), where $\alpha \simeq -0.39$ ($\beta =-1.04$) and $\alpha \simeq 0.02$ ($\beta =-1$), respectively. The solid line denotes the domain of $x$ used for the fit to determine the exponent $\alpha$.

Figure 5

Exponent of the power law $\alpha$ as a function of the damping parameter $\beta$ calculated from the stationary probability distribution function $p\left(x\right)$ for the stochastic van der Pol oscillator with delayed feedback. The parameters are fixed at ${\omega }_0=1$, $\eta =1$, $b=1$, $\kappa =1$, and $D=0.1$. The calculations are done for three values of the time delay $\tau$. The bifurcation point is located at $\alpha =-1$, the point at which the distribution function becomes non-normalizable. The symbols are the results of the numerical simulations whereas the solid curves are the numerical determination of $\alpha$ [Eq. (29)] from Eqs. (8) and (30).

Figure 6

Bifurcation diagram of Eq. (1). We show the critical value ${\beta }_c$ as a function of the time delay $\tau$. The other parameters are fixed at ${\omega }_0=1$, $\eta =1$, $b=1$, and $\kappa =1$. The symbols ($◯$) and ($\square$) correspond, respectively, to $D=0$ and $D=0.1$. They are the numerically determined threshold calculated as the point for which the exponent of the power law of the stationary probability distribution function is $-1$, whereas the solid curves are given by Eqs. (8) and (30).