Experiments on oscillator ensembles with global nonlinear coupling

We experimentally analyze collective dynamics of a population of 20 electronic Wien-bridge limit-cycle oscillators with a nonlinear phase-shifting unit in the global feedback loop. With an increase in the coupling strength we first observe formation and then destruction of a synchronous cluster, so that the dependence of the order parameter on the coupling strength is not monotonic. After destruction of the cluster the ensemble remains nevertheless coherent, i.e., it exhibits an oscillatory collective mode (mean field). We show that the system is now in a self-organized quasiperiodic state, predicted in Rosenblum and Pikovsky [Phys. Rev. Lett. 98, 064101 (2007)]. In this state, frequencies of all oscillators are smaller than the frequency of the mean field, so that the oscillators are not locked to the mean field they create and their dynamics is quasiperiodic. Without a nonlinear phase-shifting unit, the system exhibits a standard Kuramoto-like transition to a fully synchronous state. We demonstrate a good correspondence between the experiment and previously developed theory. We also propose a simple measure which characterizes the macroscopic incoherence-coherence transition in a finite-size ensemble.

Figure 1

Scheme of the experimental setup of $N=20$ globally coupled oscillators. All oscillators have an identical structure and therefore only the first one is shown in detail. The global coupling is organized via the common load $R_c$. A fraction of the voltage across $R_c$ is fed back to each oscillator via the feedback loop, consisting of a linear (standard $RC$ circuit) and nonlinear phase-shifting units.

Figure 2

Characteristic of the nonlinear phase-shifting unit: phase shift between the output and input vs the amplitude of the input.

Figure 3

Results of the experiment with the linear phase-shifting unit. Order parameter $R$ (a) and minimal mean-field amplitude $A_{\min }$ [(b), blue circles] reveal the synchronization transition at the coupling strength $\varepsilon \approx 0.17$. This is also confirmed by the plot of $\eta$ [(b), red squares]: This quantity shows that for $\varepsilon \gtrsim 0.17$, the instantaneous frequency of the mean field is always positive, as expected for a coherent, oscillatory mean field. The transition can be also very good seen from the frequency plot in (c): At $\varepsilon \approx 0.17$ several oscillators form a synchronous cluster and for $\varepsilon \gtrsim 0.2$ full frequency locking is observed, with $R$ close to 1. Here the circles show the frequencies of oscillators $f_i$ and the bold blue line shows the mean-field frequency $f_{\text{mf}}$. Notice that for subthreshold coupling $R$ is not small due to the finite-size effect; here $A_{\text{mf}}$ is more efficient for the determination of the threshold.

Figure 4

The same as in Fig. 3 but for the experiment with the linear and nonlinear phase-shifting units. At $\varepsilon \approx 0.12$ we observe the transition to a partially synchronous state, where the fastest oscillators lock onto each other and to the mean field. Between $\varepsilon \approx 0.43$ and $\varepsilon \approx 0.72$ the oscillators leave the cluster and for $\varepsilon \gtrsim 0.72$ the SOQ state is observed: The mean field is faster than all oscillators. Although the values of the order parameter in the asynchronous ($\varepsilon \lesssim 0.12$) and SOQ states are almost the same, these states are qualitatively different (see Fig. 5), and can be easily distinguished by the quantities $A_{\min }$ and $\eta$ (see text).

Figure 5

Phase portraits for the mean field. (a) Asynchronous state, $\varepsilon =0.05$. (b) SOQ state, $\varepsilon =0.85$ (cf. Fig. 4); ${\tilde{V}}_L$ is the Hilbert transform of the mean field $V_L$. The pattern in (a) is typical for a fluctuation process with the amplitude dropping to zero, whereas the portrait in (b) clearly demonstrates coherent, oscillatory mean-field dynamics with the well-defined phase and frequency. (c) Power spectra of one oscillator $S_o$ (circles) and of the mean field $S_f$ (solid line), for $\varepsilon =0.85$.