Impact of heterogeneous delays on cluster synchronization

We investigate cluster synchronization in coupled map networks in the presence of heterogeneous delays. We find that while the parity of heterogeneous delays plays a crucial role in determining the mechanism of cluster formation, the cluster synchronizability of the network gets affected by the amount of heterogeneity. In addition, heterogeneity in delays induces a rich cluster pattern as compared to homogeneous delays. The complete bipartite network stands as an extreme example of this richness, where robust ideal driven clusters observed for the undelayed and homogeneously delayed cases dismantle, yielding versatile cluster patterns as heterogeneity in the delay is introduced. We provide arguments behind this behavior using a Lyapunov function analysis. Furthermore, the interplay between the number of connections in the network and the amount of heterogeneity plays an important role in deciding the cluster formation.

Figure 1

Schematic diagram depicting the (a) ideal D, (b) ideal SO, and (c) dominant SO clusters. The nodes (small solid circles) in the circular region represent that they are synchronized.

Figure 2

Phase diagrams (a) and (b) show different regions in the parameter space of ${\tau }_1,{\tau }_2$ ($\tau$ for homogeneous delays) and $\epsilon$ for $f\left(x\right)=4x(1-x)$. The gray (color) denotes different regions: turbulent (T) (stands for no cluster formation), ideal driven (D), dominant driven (DD), ideal self-organized (SO), dominant self-organized (DSO), and mixed (M). In these phase diagrams, the boundaries of the ideal D and ideal SO clusters do not depend on the threshold value, while the boundaries of the dominant D, SO, and mixed clusters depend on the threshold chosen. (c) and (d) show variations in the fraction of nodes forming clusters ($F_{\text{clus}}=N_{\text{clus}}/N$, where $N_{\text{clus}}=\text{total}$ number of nodes forming clusters) in the parameter space of ${\tau }_1,{\tau }_2$ ($\tau$ for homogeneous delays) and $\varepsilon$ for $f\left(x\right)=4x(1-x)$. The values on the y axis represent the delay values. Network parameters are $N=500$ and $\langle k\rangle =4$. The gray (color) coding represents the variation in the fraction of nodes forming clusters. (a), (c) correspond to the 1D lattice and (b), (d) correspond to the SF networks.

Figure 3

The ideal SO clusters for the (a) 1D lattice, (b) SW, and (c) random networks. Squares represent clusters, diagonal dots represent isolated 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 corresponds to networks with $N=50$, $\langle k\rangle =4$, and $\varepsilon =0.17$. All the graphs correspond to $f_{{\tau }_1}=f_{{\tau }_2}$, ${\tau }_1=1$, and ${\tau }_2=3$.

Figure 4

Phase diagrams (a) and (b) showing different regions in the parameter space of ${\tau }_1,{\tau }_2$ ($\tau$ for homogeneous delays) and $\epsilon$ for a complete bipartite network of $N=500$. The figure description is the same as for Figs. 2 and 2.

Figure 5

Time evolution of a few nodes in the complete bipartite network of $N=500$ [the coupling strength is chosen as 0.68 for which the network is shown to form two ideal D clusters for a homogeneous delay ($\tau =2$) for $t<t_0$]. At $t=t_0$ the heterogeneity in the delay is introduced by randomly making $50\%$ of the connections conducting with ${\tau }_2=4$ and the rest keep on conducting with ${\tau }_2=2$.

Figure 6

Schematic diagram representing two set of nodes, when a pair of nodes in set A receiving the same inputs are not directly connected (I) and when they are directly connected (II).

Figure 7

Variation of $f_{\text{inter}}$ (solid circles) and $f_{\text{intra}}$ (open circles) as a function of the amount of heterogeneity. (a) SF network with $N=500$ and ${\tau }_1=1,{\tau }_2=3$. (b) The complete bipartite networks with $N=200$ and ${\tau }_1=2,{\tau }_2=4$. Both graphs are for $f\left(x\right)=4x(1-x)$.

Figure 8

A typical behavior of coupled dynamics illustrating different cluster patterns for changes in the parity of heterogeneous delays. The figure description is the same as in Fig. 3. The example presents a scale-free network with $N=50$, $\langle k\rangle =4$, and $\varepsilon =0.02$. All the graphs correspond to $f_{{\tau }_1}=f_{{\tau }_2}$

Figure 9

Node vs node diagram demonstrating various cluster states for (a) and (b) for coupled circle maps on complete bipartite networks of $N=50$ at $\varepsilon =0.85$, and (c) and (d) for coupled logistic maps on a globally connected network of $N=200$ at $\varepsilon =1.0$. (a) and (c) $\tau =0/\tau =1$ indicate that exactly the same patterns are obtained for the undelayed ($\tau =0$) and the homogeneous delayed ($\tau =1$) cases. Circles and dots remain the same as in Fig. 3. All the graphs correspond to $f_{{\tau }_1}=f_{{\tau }_2}$.

Figure 10

Variation of $f_{\text{inter}}$ (solid) and $f_{\text{intra}}$ (open) circles as a function of $\varepsilon$ for SF (left) and complete bipartite (right) networks with $N=500$ and for Gaussian distributed delays with mean $\overline{\tau }=10$ and variance $c=9$.