Research ArticleNETWORK SCIENCE

Complete characterization of the stability of cluster synchronization in complex dynamical networks

See allHide authors and affiliations

Science Advances  22 Apr 2016:
Vol. 2, no. 4, e1501737
DOI: 10.1126/sciadv.1501737
  • Fig. 1 Examples of symmetries in networks.

    (A) A network of four identical oscillators coupled through three identical links. (B) The same network after a reflection operation. (C) The same network after a rotation operation. (D) An 11-node network showing three clusters (blue, green, and white).

  • Fig. 2 Patterns of clusters in a five-node network.

    Left: All possible patterns displayed when the network connectivity is given by the adjacency matrix (Eq. 5). Right: Additional patterns displayed when the network connectivity is given by the Laplacian matrix (Eq. 6).

  • Fig. 3 Reduction of the dimension of the three-dimensional synchronization manifold.

    This shows schematically how the merging of clusters (1,3) and (5) produces a new synchronization direction in the (1,3) and (5) plane of the synchronization manifold along with a new transverse direction orthogonal to the new synchronization direction.

  • Fig. 4 The figure shows the experimental synchronization error for each synchronization pattern as a function of the parameter σ for a five-node experimental system modeled after Fig. 2.

    Underneath the top portion, we plot the results of our stability analysis applied to each one of the CS patterns, where a colored dot labels the values of σ for which the corresponding pattern is stable.

  • Fig. 5 Experimental phase space plots with lines connecting successive iterates.

    (A) Three clusters (A3). (B) Two clusters (L3). (C) One cluster (L1).

  • Fig. 6 Experimental patterns of light intensity of different clusters in the five-node network.

    (A to C) Snapshots of the experimental dynamics for the cases of (A) three clusters (A3), (B) two clusters (L3), and (C) one cluster (L1).

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/2/4/e1501737/DC1

    movie S1. Video of light intensity dynamics of node areas for the five-node network analyzed in the text, displayed in Fig. 2, and whose equations are shown in Table 1 for different values of parameters β and σ as shown in Fig. 6.

    movie S2. Video rotating the view of state space trajectories of the synchronized node clusters for the five-node network analyzed in the text, displayed in Fig. 2, and whose equations are shown in Table 1 for different values of parameters β and σ as shown in Fig. 5.

  • Supplementary Materials

    This PDF file includes:

    • Legends for movies S1 and S2

    Download PDF

    Other Supplementary Material for this manuscript includes the following:

    • movie S1 (.mp4 format). Video of light intensity dynamics of node areas for the five-node network analyzed in the text, displayed in Fig. 2, and whose equations are shown in Table 1 for different values of parameters β and σ as shown in Fig. 6.
    • movie S2 (.mp4 format). Video rotating the view of state space trajectories of the synchronized node clusters for the five-node network analyzed in the text, displayed in Fig. 2, and whose equations are shown in Table 1 for different values of parameters β and σ; as shown in Fig. 5.

    Files in this Data Supplement:

Navigate This Article