Limiting the stroke of a Schmitt trigger with multiplicative noise

We have devised an experiment whereby a bistable system is confined away from its deterministic attractors by means of multiplicative noise. Together with previous numerical results, our experimental results validate the hypothesis that the higher the slope of the noise's multiplicative factor, the more it shifts the stationary states.

Figure 1

Shape of the SPDF $P_{\mathrm{st}}\left(u\right)$ in three instances. Constant-$u$ lines indicate the boundaries of the confinement region $u_l=\pm 0.7$. For additive noise ($\lambda =\sqrt{.02}$), the SPDF has its maxima outside the bounds. For $\lambda =5$ and $b=20$ instead, it becomes confined (inside the bounds: thin line, analytical calculation; thick line, numerical calculation) [31].

Figure 2

Schematic block diagram of the electronic device. $u$ is the output voltage of the feedback system; the noise's multiplicative factor ${\mathrm{\Gamma }}_e^{1/2}\left(u\right)$ is the input-output function of the exponential amplifier; $Q_e\left(u\right)$ is the input-output function of the Schmitt trigger, which mimics the cubic nonlinearity; $\eta \left(t\right)$ is the external Gaussian noise; and $F\left(u\right)=u+Q_e\left(u\right)+{\mathrm{\Gamma }}_e^{1/2}\left(u\right)\eta \left(t\right)$.

Figure 3

Time window on the 2-ms scale of $u\left(t\right)$—the system's output voltage and Schmitt trigger's feedback input—without multiplicative noise ($\lambda =0$ V). The transitions are due to the internal (additive) electronic noise, which is always present in these devices. Constant-$u$ lines indicate the boundaries of a confinement region for $u_l=\pm 3$ V.

Figure 4

Scope screen shots of the electronically synthesized input-output function ${\mathrm{\Gamma }}_e^{1/2}\left(u\right)$ of the exponential amplifier, for (a) $u_l=\pm 2$ V and (b) $u_l=\pm 3$ V. Oscilloscope use mode, XY; channel A, 1.00 V; channel B, 2.00 V; and sampling frequency, 1 Ms/s.

Figure 5

Time window on the 10-ms scale of the output voltage $u\left(t\right)$ for $u_l=\pm 3$ V (constant-$u$ lines) and $\lambda =3.18$ V, illustrating switchings between two basically stationary solutions that lie inside the confinement region. The detailed dynamics of the Schmitt trigger is such that internal (additive) electronic noise succeeds in driving the system outside the confinement region during very short periods before each switching.

Figure 6

Time windows on the 1-ms scale of the output voltage $u\left(t\right)$ between transitions, for the values of $u_l$ and $\lambda$ in Table 1: (a) $u_l=\pm 2$ V and (b) $u_l=\pm 3$ V. As $\lambda$ increases, the statistically stationary (before runoff) states shift into the confinement region.

Figure 7

Experimental SPDF for the cases in Fig. 6. Constant-$u$ lines indicate the boundaries of the confinement region: (a) $u_l=\pm 2$ V and (b) $u_l=\pm 3$ V. As $\lambda$ increases, the experimental SPDF changes from peaking outside to peaking inside the confinement region.