Stochastic bifurcations and coherencelike resonance in a self-sustained bistable noisy oscillator

We investigate the influence of additive Gaussian white noise on two different bistable self-sustained oscillators: Duffing–Van der Pol oscillator with hard excitation and a model of a synthetic genetic oscillator. In the deterministic case, both oscillators are characterized with a coexistence of a stable limit cycle and a stable equilibrium state. We find that under the influence of noise, their dynamics can be well characterized through the concept of stochastic bifurcation, consisting in a qualitative change of the stationary amplitude distribution. For the Duffing-Van der Pol oscillator analytical results, obtained for a quasiharmonic approach, are compared with the result of direct computer simulations. In particular, we show that the dynamics is different for isochronous and anisochronous systems. Moreover, we find that the increase of noise intensity in the isochronous regime leads to a narrowing of the spectral line. This effect is similar to coherence resonance. However, in the case of anisochronous systems, this effect breaks down and a new phenomenon, anisochronous-based stochastic bifurcation occurs.

Figure 1

Stochastic $P$-bifurcations in the Duffing–Van der Pol oscillator [Eqs. (1, 2)]. (a) Bifurcation diagram of the system [Eq. (2)] in the parameter plane $\left(\epsilon ,D\right)$. The tinted region represents the bimodal distribution; lines $l_1$ and $l_2$ correspond to the appearance and disappearance of one of the maxima of $p\left(a\right)$. The vertical dashed line stands for the bistability border of the deterministic oscillator (region 2 corresponds to bistability in the deterministic model). (b) Stationary amplitude distribution for $\epsilon =-0.13$ and different values of the noise intensity $D$. The circles represent numerical computation for the oscillator [Eq. (1)], whereas the solid lines denote the algebraic calculations using formula [Eq. (4)] (the normalization constant $N$ is defined numerically). For numerical integration of the stochastic equations we used the Heun scheme [19].

Figure 2

(Color online) Normalized power spectral density of oscillation in (Eq. (1)) for $\epsilon =-0.13$ and different values of noise intensity: (a)—in the isochronous case $\left(\beta =0\right)$ and (b)—in the anisochronous case $\left(\beta =0.5\right)$.

Figure 3

(Color online) Phase trajectories of the system [Eq. (6)]: (a)—in the deterministic case $\left(D=0\right)$ in the region of bistability for ${\gamma }_y\approx 0.03701$ ($F$—stable focus, $L$—stable limit cycle, $L^{\prime }$—unstable limit cycle); (b)—in the presence of noise with intensity $D=0.00002$ near the border of bistability for ${\gamma }_y\approx 0.03702$.

Figure 4

(Color online) (a) Stationary amplitude distributions $p_c\left(a\right)$ (under the condition $\text{cos}\left(\Phi \left(t\right)=0\pm 0.001\right)$. (b) Normalized power spectra in system [Eq. (6)] for ${\gamma }_y=0.03702$ and different values of noise intensity $D$.