Suppression of deterministic and stochastic extreme desynchronization events using anticipated synchronization

We numerically show that extreme events induced by parameter mismatches or noise in coupled oscillatory systems can be anticipated and suppressed before they actually occur. We show this in a main system unidirectionally coupled to an auxiliary system subject to a negative delayed feedback. Each system consists of two electronic oscillators coupled in a master-slave configuration. Extreme events are observed in this coupled system as large and sporadic desynchronization events. Under certain conditions, the auxiliary system can predict the dynamics of the main system. We use this to efficiently suppress the extreme events by applying a direct corrective reset to the main system.

Figure 1

Coupling scheme used for anticipated synchronization. Each subsystem consist of two unidirectionally coupled oscillators in a master-slave configuration. The main system is unidirectionally coupled to the auxiliary system according to Eqs. (1) and (2).

Figure 2

(a) Time trace of $x_{\perp }$ for the main system (black solid line) and $y_{\perp }$ for the desynchronization events predicted by the auxiliary system (dashed red line). (b) A magnification of an extreme event in the main system (solid black line) and the event predicted by the auxiliary system (dashed red line). (c) A magnification of the small desynchronization events in the main system (solid black line) and the events predicted by the auxiliary system (dashed red line). $\tau =1$, $C=0.75$, and the other parameters are listed in Table 1. This and other figures have been obtained by a numerical integration of the governing equations of the system using a Runge-Kutta method of second order with a small enough time step ($dt={10}^{-4}$) to ensure the convergence of the method. Nevertheless, we have also verified that the results obtained are statistically independent for larger time steps.

Figure 3

Boundary of stable anticipated synchronization with an accuracy, $\rho$, larger than 0.9 in the plane ($C$, $\tau$). The parameter mismatch between the main and the auxiliary systems is $\sim 2\%$ (solid black line), $\sim 5\%$ (dashed red line), and $\sim 10\%$ (dash-dotted blue line). The boundaries were averaged using 4 different sets of parameters.

Figure 4

(a) Time trace for $x_{\perp }$ (solid black line) and $y_{\perp }$ (dashed red line). The control is triggered when $y_{\perp }$ exceeds the threshold ${\mathcal{T}}_2=0.5$. (b)–(d) Time trace of each of the variables of the main system $M_1$ (solid black line) and $S_1$ (dashed blue line). In (b) we also show a magnification of the control [solid red (gray) line]. (e) Time trace for the $x_{\perp }$ setting where the control turns on when $x_{\perp }$ exceeds the threshold ${\mathcal{T}}_1=0.5$. (f)–(h) Time trace of each of the variables of the main system $M_1$ (solid black line) and $S_1$ (dashed blue line) for a threshold ${\mathcal{T}}_1=0.5$. In (f) we also show a magnification of the control [solid red (gray) line]. $|V_c|=$ 0.27.

Figure 5

Fraction of detected DKs for a reset amplitude $V_c$ divided by the number of DKs when the control is off. The reset is triggered when $x_{\perp }={\mathcal{T}}_1$ (dashed lines) or when $y_{\perp }={\mathcal{T}}_2$ (solid lines). The values of the thresholds are from left to right: ${\mathcal{T}}_2=0.1$ [solid blue (dark gray) line], ${\mathcal{T}}_1=0.1$ [dashed blue (dark gray) line], ${\mathcal{T}}_2=0.5$ (solid black line), ${\mathcal{T}}_1=0.5$ (dashed black line), ${\mathcal{T}}_2=1$ [solid red (light gray) line], and ${\mathcal{T}}_1=1$ [dashed red (light gray) line].

Figure 6

Number of control resets as a function of $V_c$. The reset is triggered when $x_{\perp }={\mathcal{T}}_1$ (dashed lines) or when $y_{\perp }={\mathcal{T}}_2$ (solid lines) where the values of the thresholds are from top to bottom ${\mathcal{T}}_1=0.1$ [dashed blue (dark gray) line], ${\mathcal{T}}_2=0.1$ [solid blue (dark gray) line], ${\mathcal{T}}_1=0.5$ (dashed black line), ${\mathcal{T}}_2=0.5$ (solid black line), ${\mathcal{T}}_1=1$ [dashed red (light gray) line], and ${\mathcal{T}}_2=1$ [solid red (light gray) line].

Figure 7

Fraction of detected DKs for a reset amplitude $V_c$ divided by the number of DKs when the control is off. The reset is triggered when $x_{\perp }={\mathcal{T}}_1$ (dashed lines) or when $y_{\perp }={\mathcal{T}}_2$ (solid lines). The values of the thresholds are from left to right ${\mathcal{T}}_2=0.1$ [solid blue (dark gray) line], ${\mathcal{T}}_1=0.1$ [dashed blue (dark gray) line], ${\mathcal{T}}_2=0.5$ (solid black line), ${\mathcal{T}}_1=0.5$ (dashed black line), ${\mathcal{T}}_2=1$ [solid red (light gray) line], and ${\mathcal{T}}_1=1$ [dashed red (light gray) line]. $D=3\times {10}^{-3}$.

Figure 8

Fraction of detected DKs when the control is on divided by the number of DKs when the control is off as a function of the logarithm of the noise strength $D$. The reset is triggered when $x_{\perp }={\mathcal{T}}_1$ (squares) or when $y_{\perp }={\mathcal{T}}_2$ (circles). ${\mathcal{T}}_1=0.2$, $V_c=0.1$ (open squares); ${\mathcal{T}}_1=0.5$, $V_c=0.28$ (filled squares); ${\mathcal{T}}_2=0.2$, $V_c=0.1$ (open circles); and ${\mathcal{T}}_2=0.5$, $V_c=0.28$ (filled circles).