Cluster synchronization in a complex dynamical network with two nonidentical clusters.

*(English)*Zbl 1172.93002Summary: This paper further investigates cluster synchronization in a complex dynamical network with two clusters. Each cluster contains a number of identical dynamical systems, however, the sub-systems composing the two clusters can be different, i.e., the individual dynamical system in one cluster can differ from that in the other cluster. Complete synchronization within each cluster is possible only if each node from one cluster receives the same input from nodes in the other cluster. In this case, the stability condition of one-cluster synchronization is known to contain two terms: the first accounts for the contribution of the inner-cluster coupling structure while the second is simply an extra linear term, which can be deduced by the “same-input” condition. Applying the connection graph stability method, the authors obtain an upper bound of input strength for one cluster if the first account is known, by which the synchronizability of cluster can be scaled. For different clusters, there are different upper bounds of input strength by virtue of different dynamics and the corresponding cluster structure. Moreover, two illustrative examples are presented and the numerical simulations coincide with the theoretical analysis.

##### MSC:

93A30 | Mathematical modelling of systems (MSC2010) |

94C15 | Applications of graph theory to circuits and networks |

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\textit{L. Chen} and \textit{J. Lu}, J. Syst. Sci. Complex. 21, No. 1, 20--33 (2008; Zbl 1172.93002)

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