Role of delay in the mechanism of cluster formation

We study the role of delay in phase synchronization and phenomena responsible for cluster formation in delayed coupled maps on various networks. Using numerical simulations, we demonstrate that the presence of delay may change the mechanism of the unit to unit interaction. At weak coupling values, the same parity delays are associated with the same phenomenon of cluster formation and exhibit similar dynamical evolution. Intermediate coupling values yield rich delay-induced driven cluster patterns. A Lyapunov function analysis sheds light on the robustness of the driven clusters observed for delayed bipartite networks. Our results reveal that delay may lead to a completely different relation, between dynamical and structural clusters, than that observed for the undelayed case.

Figure 1

Phase diagram of phase synchronization patterns in system (1) for a 1D lattice with $N=50$, $\langle k\rangle =4$. Grayscale encoding represents values of (a) $f_{\text{inter}}$ and (b) $f_{\text{intra}}$. The local dynamics is governed by a logistic map $f\left(x\right)=4x(1-x)$ and a coupling function $g\left(x\right)=f\left(x\right)$. The figure is obtained by averaging over 20 random initial conditions. The regions, which are black in both (a) and (b), correspond to states of no cluster formation. Both subfigures with gray shades correspond to clusters having both inter- and intracouplings. The regions in (a), which are lighter as compared to the corresponding $\varepsilon$ and $\tau$ values in (b), refer to dominant D phase synchronized clusters, and the reverse refer to dominant SO phase synchronized clusters. White regions in (a) and (b) refer to ideal D or ideal SO clusters, respectively. The regions, which are dark gray in (a) and black in (b) or vice versa, correspond to states where many less clusters are formed. (c) and (d) are for scale-free and (e) and (f) are for bipartite networks and demonstrate the same as (a) and (b), respectively.

Figure 2

A typical behavior of coupled dynamics illustrating D patterns observed with changing $\tau$. Squares represent clusters, diagonal dots represent freely evolved nodes, while off-diagonal dots imply that the two corresponding nodes are coupled (i.e., $A_{ij}=1$). In each case the node numbers are reorganized so that the nodes belonging to the same cluster are numbered consecutively. The example presents a scale-free network with $N=50$ and $\varepsilon =0.6$. For $\tau =0$, very few nodes are forming a cluster. For $\tau =1$, 3, and 5, nodes form dominant D clusters, whereas $\tau =2$ and 4 yield very few nodes forming clusters of an ideal D type.

Figure 3

Three-node schematic diagram illustrating the impact of delay. Arrows depict the direction of information flow as governed by Eq. (1). The dashed lines show the flow of information from the $(t-2)\text{th}$ time step. For $\tau =0$, the evolution of all nodes (solid orange circles) receives information from the second node (left panel), whereas in the presence of delay, the evolution of the connected nodes at a particular time does not involve any common term (right panel). For both panels, the first and third nodes are connected with the second one, leading to the construction of the smallest possible bipartite network.

Figure 4

Phase synchronized patterns for coupled circle maps on scale-free networks with $N=50$, $\langle k\rangle =2$, $g\left(x\right)=x$, and $\varepsilon =0.24$.