Cross-frequency synchronization of oscillators with time-delayed coupling

We carry out theoretical and experimental studies of cross-frequency synchronization of two pulse oscillators with time-delayed coupling. In the theoretical part of the paper we utilize the concept of phase resetting curves and analyze the system dynamics in the case of weak coupling. We construct a Poincaré map and obtain the synchronization zones in the parameter space for $m:n$ synchronization. To challenge the theoretical results we designed an electronic circuit implementing the coupled oscillators and studied its dynamics experimentally. We show that the developed theory predicts dynamical properties of the realistic system, including location of the synchronization zones and bifurcations inside them.

Figure 1

(a) Dynamics of the system (1). When the phase reaches unity it is reset to zero and the oscillator produces a pulse. Small jumps of the phase correspond to the instants when the oscillator receives delayed pulses from the other oscillator. (b) The return functions of the map (5) for different values of the parameter $\omega$. When $\omega$ increases the curve shifts upwards (as depicted by the double arrow) and various fixed points emerge and vanish sequentially. Stable fixed points are plotted by solid circles; unstable ones are plotted by empty circles.

Figure 2

Synchronization zones for the sinusoid PRC $f\left(\phi \right)=-sin\left(2\pi \phi \right)$. The rotation numbers of the zones are given in the bars. (a) The plane $\mu \text{−}\tau$ for the fixed value of $\varepsilon =0.1$. (b) The plane $\mu \text{−}\varepsilon$ for the fixed value of $\tau =0.5$.

Figure 3

Experimental setup. (a) The circuitry of the FHN oscillator. (b) The phase plane of the FHN oscillator: the current $i$ versus the voltage $u$. The dashed line shows the threshold for the output pulse. (c) The circuitry of two oscillators coupled via two delay lines. (d) The Lissajous figure for $3:2$ synchronization of the oscillators: the output voltages $u_2$ versus $u_1$.

Figure 4

The experimentally obtained synchronization zones for two FHN oscillators. The rotation numbers of the zones are given in the bars. (a) The plane $T\text{−}D$ for the fixed value of $E=5$ V. (b) The plane $T\text{−}E$ for the fixed value of $D=2.5$ ms.

Figure 5

(a) Definition of the PRC: An external pulse is applied to the oscillator and the perturbed period is measured. (b) The normalized PRC of the FHN oscillator. (c) Linear dependency between the magnitude of the PRC and the pulse amplitude $E$ (for $T=5$ ms). (d) Inversely linear dependency between the magnitude of the PRC and the period of the oscillator (for $E=5$ V).

Figure 6

The borders of the synchronization zones: theoretical predictions (solid lines) and experimental results (dots). (a) The plane $T\text{−}D$ for the fixed value of $E=5$ V. (b) The plane $T\text{−}E$ for the fixed value of $D=2.5$ ms.

Figure 7

(a) Bistability of the system: the Lissajous figures for two different regimes of $2:1$ synchronization (top) and the delay intervals in which they exist (bottom). (b) The map (5) in the interval of bistability has two stable (solid circles) and two unstable (empty circles) fixed points.