Phase-lag synchronization in networks of coupled chemical oscillators

Chemical oscillators with a broad frequency distribution are photochemically coupled in network topologies. Experiments and simulations show that the network synchronization occurs by phase-lag synchronization of clusters of oscillators with zero- or nearly zero-lag synchronization. Symmetry also plays a role in the synchronization, the extent of which is explored as a function of coupling strength, frequency distribution, and the highest frequency oscillator location. The phase-lag synchronization occurs through connected synchronized clusters, with the highest frequency node or nodes setting the frequency of the entire network. The synchronized clusters successively “fire,” with a constant phase difference between them. For low heterogeneity and high coupling strength, the synchronized clusters are made up of one or more clusters of nodes with the same permutation symmetries. As heterogeneity is increased or coupling strength decreased, the phase-lag synchronization occurs partially through clusters of nodes sharing the same permutation symmetries. As heterogeneity is further increased or coupling strength decreased, partial synchronization and, finally, independent unsynchronized oscillations are observed. The relationships between these classes of behavior are explored with numerical simulations, which agree well with the experimentally observed behavior.

Figure 1

Phase-lag synchronization in a network of Belousov-Zhabotinsky (BZ) chemical oscillators. (a) The bidirectional links show the connectivity in the network. Nodes that oscillate simultaneously, or nearly so, define a synchronization cluster and are the same color. (b) Nodes that have the same permutation symmetries define a symmetry cluster and are the same color. (c) A reduced representation of the symmetry clusters. (d) Experimentally recorded grayscale values of all oscillators, showing the wave pattern. The node index corresponds to the numbering in (a). Red lines separate the synchronization clusters, with the synchronization clusters and symmetry clusters color coded (at left) as in (a) and (b), respectively. (e) Kuramoto order parameter [2] for each synchronization cluster, color coded as in (a), and the complete network (black) as a function of time. The network connections are established at 200 s (dashed line) following initial conditions of frequency synchronization with global coupling. (f) A linear fit through the phase occurrence of each node of each synchronization cluster reveals a constant phase difference between synchronization clusters [color coding as in (a)]. Note that the synchronization cluster index is equivalent to the network distance from the pacemaker with an offset of 1. See the Supplemental Material [38] for experimental setup and parameters as well as five other network topologies. Average natural period and standard deviation $T_0=35.02\pm 1.10$ s; coupling strength $\sigma =2.0$.

Figure 2

Phase-lag synchronization that occurs only partially through symmetry clusters. (a) and (d) Because the pacemaker (node 1) is not a complete symmetry cluster, as in Fig. 1, the red and blue synchronization clusters do not correspond to the symmetry clusters in Fig. 1. (b) Kuramoto order parameter [2] for each synchronization cluster, color coded as in (a), and the complete network (black) as a function of time. (c) A linear fit through the phase occurrence of each node of each synchronization cluster [color coding as in (a)]. Average natural period and standard deviation $T_0=50.59\pm 3.81$ s; coupling strength $\sigma$ = 2.0. Other parameters as in Fig. 1.

Figure 3

Network dynamics without complete synchronization. (a) and (c) Frequency synchronization in two groups of oscillators: nodes 1–7 and 10 and nodes 8 and 9 (separated by horizontal lines). Average natural period and standard deviation $T_0=55.65\pm 8.62$ s; coupling strength $\sigma$ = 0.60. (b) and (d) No synchronization occurs at low coupling strength, and a broad distribution in period is exhibited. Average natural period and standard deviation $T_0=55.97\pm 2.04$ s; coupling strength $\sigma$ = 0.05. Other parameters as in Fig. 1.

Figure 4

Simulations of the network of heterogeneous, photosensitive BZ oscillators with a modified ZBKE model [37]. (a) Degree of synchronization according to the standard deviation of the mean period as a function of coupling strength and the standard deviation of the uncoupled oscillator period distribution ${\left(T_0\right)}_{\mathrm{std}}$. (b) The number of phase-lag synchronized oscillators, $n\left(\mathrm{sync}\right)$, as a function of coupling strength and oscillator heterogeneity defined by ${\left(T_0\right)}_{\mathrm{std}}$. The number of phase-lag synchronized oscillators is determined according to $0.5<\mathrm{\Delta }\varphi <1.35$ for the phase difference between synchronization clusters and $\mathrm{\Delta }\varphi \le 0.5$ for the phase difference within synchronization clusters. (c) and (d) Panels (left to right) showing period distribution (c) and the relative phase of each oscillator (d), corresponding to symbols in (a) and (b) for increasing coupling strength: (circle) unsynchronized, (triangle) partial synchronization, (square) full synchronization with five synchronization clusters, (diamond) full synchronization with three synchronization clusters. See the Supplemental Material [38] for modified ZBKE model and parameters.