Singular response of bistable systems driven by telegraph noise

We show that weak periodic driving can exponentially strongly change the rate of escape from a potential well of a system driven by telegraph noise. The analysis refers to an overdamped system, where escape requires that the noise amplitude $\theta$ exceed a critical value ${\theta }_c$. For $\theta$ close to ${\theta }_c$, the exponent of the escape rate displays a nonanalytic dependence on the amplitude of an additional low-frequency modulation. This leads to giant nonlinearity of the response of a bistable system to periodic modulation. Also studied is the linear response to periodic modulation far from ${\theta }_c$. We analyze the scaling of the logarithm of the escape rate with the distance to the saddle-node and pitchfork bifurcation points. The analytical results are in excellent agreement with numerical simulations.

Figure 1

The scaling functions $R_{\mathrm{sn}}\left(z\right)$ and $R_{\mathrm{pf}}\left(z\right)$ for the escape exponent near the saddle-node and pitchfork bifurcation points, Eq. (13). The vertical lines show the values of $z$ where the corresponding functions diverge. The squares and circles are the results of simulations of the escape rate, with $R$ obtained from Eqs. (11) and (13) with $\eta =1$. The agreement between theory and simulations improves with increasing $\nu$, which corresponds to the increasing exponent of the escape rate ${\Psi }_{\mathrm{e}}$, as seen from Eq. (13).

Figure 2

Strong square-wave-like signal induced by a comparatively weak sinusoidal force $A\cos \omega t$. The results refer to the potential $U\left(q\right)=-q^2/2+q^4/4$ and a symmetric telegraph noise, ${\nu }_{\uparrow \downarrow }={\nu }_{\downarrow \uparrow }=\nu /2$. The noise amplitude $\theta =0.384$ is close to the maximal slope of $U\left(q\right)$ between the minima, ${\theta }_c=\left|U^{\prime }\left(q_{\max }^{(1,2)}\right)\right|\approx 0.385$. The noise switching rate is $\nu =2$ and the signal amplitude is $A=0.1$. The panels refer to $\omega ={10}^{-3}$ and ${10}^{-2}$. They show how the shape of the response changes and the amplitude decreases with increasing frequency.

Figure 3

A sketch of a metastable potential well. The average potential $\tilde{U}\left(q\right)=U\left(q\right)+\theta q{\nu }_{-}/\nu$ is shown by the solid line. The long- and short-dashed lines show the potentials $U\left(q\right)+\theta q$ and $U\left(q\right)-\theta q$; they correspond to the instantaneous values of the noise $\mp \theta$. The potential is switching between these two values. Escape from the potential well occurs once the system goes sufficiently far beyond the local maximum of $\tilde{U}\left(q\right)$ (point ${\tilde{q}}_{\mathcal{S}}$), from where the probability of returning to the interior of the well is very small. The plots refer to $U\left(q\right)=q-q^3/3,\theta =1.5,{\nu }_{-}/\nu =0.2$.