Synchronization of stochastic bistable systems by biperiodic signals

We study the nonlinear response of a noisy bistable system to a biperiodic signal through experiments with an electronic circuit (Schmitt trigger). The signal we use is a biharmonic one, i.e., a superposition of low and high frequency harmonic components. It is shown that the mean switching frequency (MSF) of the system can be locked at both low and high frequencies. Moreover, the phenomenon of MSF locking at the lower frequency can be induced and enhanced by the higher frequency excitation. Thus high frequency bias can control synchronization at the low frequency.

Figure 1

A circuit diagram of Schmitt trigger (a) and its (idealized) input-output hysteresis characteristic (b).

Figure 2

(Color online) Experimental results obtained from the circuit model shown in Fig. 1. (a) The mean switching frequency (MSF) $⟨f⟩$ is plotted as a function of noise amplitude $\sigma$ (dimensionless units) for different combinations of amplitudes of the two-frequency signal with $\mathrm{\Delta }U=1$. The full curves correspond to $A_L=0.2$ and $A_H=0$ (line 1), $A_H=0.43$ (line 2), $A_H=0.64$ (line 3), $A_H=0.7$ (line 4); the dashed curve corresponds to $A_L=0$, $A_H=0.7$. (b) Plots of the MSF $⟨f⟩$ as a function of $\sigma$ for different values of the threshold: $\mathrm{\Delta }U=1$ (line 1), $\mathrm{\Delta }U=0.9$ (line 2), $\mathrm{\Delta }U=0.34$ (line 3), $\mathrm{\Delta }U=0.25$ (line 4). The HF component is absent $A_H=0$ and the amplitude of LF component $A_L=0.2$.

Figure 3

(Color online) Comparison of experimentally measured (data points) and calculated (full curves) MSFs $⟨f⟩$ as functions of noise amplitude $\sigma$ (dimensionless units). The dashed lines connecting data points are guides to the eye. (a) The influence of the HF component on MSF locking at the low-frequency $f_L$. The data points $\times$ correspond to the parameter set $\mathrm{\Delta }U=1$, $A_L=0.2$, and $A_H=0$; $엯$ to $\mathrm{\Delta }U=1$, $A_L=0.2$, and $A_H=0.7$; $◻$ to $\mathrm{\Delta }U=0.25$, $A_L=0.2$, and $A_H=0$. (b) The influence of the LF component on MSF locking on the high-frequency $f_H$. The data points $엯$ correspond to the parameter set: $\mathrm{\Delta }U=1$, $A_L=0$, and $A_H=0.7$; $◻$ to $\mathrm{\Delta }U=1$, $A_L=0.2$, and $A_H=0.7$.

Figure 4

(Color online) The distributions $p\left(\tau \right)$ and $p\left(\phi \right)$ in the region of synchronization at $f_L$ (a) $⟨f⟩<f_L$, (b) $⟨f⟩=f_L$, (c) $⟨f⟩>f_L$. Time is normalized by the period of the LF component $T_L=1∕f_L$ in the plots of $p\left(\tau \right)$.

Figure 5

(Color online) The distributions $p\left(\tau \right)$ and $p\left(\phi \right)$ in the region of synchronization at $f_H$ (a) $⟨f⟩<f_H$, (b) $⟨f⟩=f_H$, (c) $⟨f⟩>f_H$. Time is normalized by the period of the HF component $T_H=1∕f_H$ in the plots of $p\left(\tau \right)$.