Experimental evidence of explosive synchronization in mercury beating-heart oscillators

We report experimental evidence of explosive synchronization in coupled chemo-mechanical systems, namely in mercury beating-heart (MBH) oscillators. Connecting four MBH oscillators in a star network configuration and setting natural frequencies of each oscillator in proportion to the number of its links, a gradual increase of the coupling strength results in an abrupt and irreversible (first-order-like) transition from the system's unordered to ordered phase. On its turn, such a transition indicates the emergence of a bistable regime wherein coexisting states can be experimentally revealed. Finally, we prove how such a regime allows an experimental implementation of magneticlike states of synchronization, by the use of an external signal.

Figure 1

The experimental setup. Schematic diagram of the four MBH oscillators, $S_1,S_2,S_3$, and $S_4$, which are connected in a star configuration with oscillator $S_1$ at the center. The oscillators $S_2,S_3$, and $S_4$ are connected to $S_1$ via variable resistors $R_2,R_3$ and $R_4$, respectively. The value of all the resistors is kept identical, i.e., $R_2=R_3=R_4=R$. The iron nails represented by Fe are connected to each other externally using the insulated copper wire. The potential of each of the MBH oscillators $(V_i,i=1,2,3,4)$ is measured with respect to the Fe nails which act as the common ground.

Figure 2

The system's unordered phase. Superimposed time series of redox potentials for the four MBH oscillators [$V_1$ dotted green (dotted gray), $V_2$ yellow (light gray), $V_3$ dashed yellow (dashed gray), and $V_4$ blue (dark gray) lines] at $R=100 \mathrm{\Omega }$ (i.e., in the regime of weak coupling strengths). This corresponds to the typical evolution of the systems in the regime of low values for the order parameter $r$. The time series of $V_1$ oscillates at its natural, higher, frequency.

Figure 3

The system's ordered phase. Superimposed time series of redox potentials for the four MBH oscillators [$V_1$ dotted green (dotted gray), $V_2$ yellow (light gray), $V_3$ dashed yellow (dashed gray), and $V_4$ blue (dark gray) lines] at $R=10 \mathrm{\Omega }$ (i.e., in the regime of high coupling strength). The systems' dynamics evolve in the ordered (synchronized) phase, and all oscillators lock to the high frequency of the central one.

Figure 4

The explosive transition. Order parameter $\left(r\right)$ versus the coupling strength $\left(1/R\right)$. In the experiment, $R$ was varied systematically in steps of $5 \mathrm{\Omega }$. The forward direction, shown in blue (gray) color, and the backward direction, shown in dotted blue (dotted gray) color, correspond to a gradual increase and decrease in the coupling strength. The experimentally observed hysteretic region extends from $1/65\text{mho}$ to $1/35\text{mho}$. Inset: The natural frequency of the central oscillator (of the leaves oscillators) is $10.90\pm 0.05$ Hz ($5.80\pm 0.08$ Hz). For the sake of comparison, the inset reports the same forward and backward transitions in a completely different experimental arrangement, where all oscillators are prepared to have quasi-identical natural frequencies. In the inset, $R$ varies between 300 and $5 \mathrm{\Omega }$, with a step size of $20 \mathrm{\Omega }$; the central oscillator has a frequency of 7.725 Hz; while the frequencies of the surrounding oscillators are, respectively, 7.563, 7.838, and 7.65 Hz. The volume of the Hg is kept at 2 ml, and the concentration of ${\text{K}}_2{\text{Cr}}_2{\text{O}}_7$ (in oscillator 2 M ${\text{H}}_2{\text{SO}}_4$) was maintained at 0.1 mM. It is evident that the transition root to the synchronized phase has, here, a smooth and fully reversible nature.

Figure 5

Magneticlike states of synchronization. Scattered plot of the order parameter $r$ versus the external signal frequency $f$ (simultaneously superimposed to all oscillators; see text for details on the procedure). The external signals are applied inside the hysteretic region of Fig. 4, and removed after a while. Dots and stars stand, in fact, for two different values of coupling strength (stars for 1/$70 \text{mho}$ and dots for 1/$40 \text{mho}$). Red (dark gray) symbols represent the initial state of the $r$ and yellow (light gray) symbols represent the final state of the $r$, before the application of the external signal and after the external signal is removed.