Delayed Feedback as a Means of Control of Noise-Induced Motion

Time-delayed feedback is exploited for controlling noise-induced motion in coherence resonance oscillators. Namely, under the proper choice of time delay, one can either increase or decrease the regularity of motion. It is shown that in an excitable system, delayed feedback can stabilize the frequency of oscillations against variation of noise strength. Also, for fixed noise intensity, the phenomenon of entrainment of the basic oscillation period by the delayed feedback occurs. This allows one to steer the time scales of noise-induced motion by changing the time delay.

Figure 1

Phase portraits of noise-induced motion: (a),(b) Van der Pol oscillator at $D=0.003$; (c),(d) FitzHugh-Nagumo system at $D=0.09$ (the dashed lines denote the null isoclines), (a),(c) $K=0$; (b),(d) $K=0.2$, $\tau =T_0$.

Figure 2

Correlation time $t_{\mathrm{c}\mathrm{o}\mathrm{r}}$ vs noise intensity $D$ for (a) the Van der Pol oscillator and (b) the FitzHugh-Nagumo system. Grey (green online) lines: $K=0$; black lines: $K=0.2$, $\tau =T_0$. (b) Grey (green online) circles: $T_0$ for $K=0$; black circles: $T_1$ for $K=0.2$, $\tau =T_0$.

Figure 3

Fourier power spectra of noise-induced oscillations in dependence on $\tau$ for (a) the Van der Pol oscillator at $D=0.003$ and (b) the FitzHugh-Nagumo system at $D=0.09$, $K=0.2$. The spectrum is computed from $y$.

Figure 4

Spectral peaks, coherence, and eigenvalues vs $\tau$ at $K=0.2$. (a),(b): Van der Pol oscillator at $D=0.003$; (c),(d): FitzHugh-Nagumo system at $D=0.09$. (a),(c) Crosses (blue online): $T$, circles (yellow online): $T_1$, and black dots: $T^e$. (b),(d) Solid line (green online): $t_{\mathrm{c}\mathrm{o}\mathrm{r}}$, (b) black dots: seven largest $p$, and circles (yellow online): $p_1$.