Experimental Observations of Group Synchrony in a System of Chaotic Optoelectronic Oscillators

We experimentally demonstrate group synchrony in a network of four nonlinear optoelectronic oscillators with time-delayed coupling. We divide the nodes into two groups of two each, by giving each group different parameters and by enabling only intergroup coupling. When coupled in this fashion, the two groups display different dynamics, with no isochronal synchrony between them, but the nodes in a single group are isochronally synchronized, even though there is no intragroup coupling. We compare experimental behavior with theoretical and numerical results.

Figure 1

(a) Schematic of four nodes separated into two groups, A (red, solid line) and B (blue, dashed line). (b) Experimental setup of a single node, showing coupling to the other nodes according to the configuration in (a).

Figure 2

Maximum Lyapunov exponent ${\lambda }_{\max }$ as a function of ${\beta }^{(A)}$ and ${\beta }^{(B)}$: (a) in the longitudinal directions $\gamma =\pm 1$ and (b) in the transverse direction $\gamma =0$. White areas correspond to ${\lambda }_{\max }=0$. The black dot indicates values of ${\beta }^{(A)}$ and ${\beta }^{(B)}$ used in this experiment.

Figure 3

(a) Simulated motion in the synchronization manifold, obtained by numerically solving Eqs. (6, 7), showing the predicted group-synchronous state. (b)–(d) Experimentally measured time traces from three different network configurations (indicated by the coupling scheme and the coupling matrices $\mathbf{K}$) that achieve group synchrony. All three networks have the same eigenspectra, but the configuration in (b) is symmetric while those in (c) and (d) are not.

Figure 4

Correlation functions of $3.9\mathrm{s}$ of experimental data. (a) Autocorrelation functions for the dynamics of group A (left) and group B (right), with no coupling in the system. (b) Cross-correlation functions between the two nodes in group A (left) and group B (right), for coupled nodes. (c) Cross-correlation functions between one node in group A and one node in group B for the uncoupled system (left) and coupled system (right).