On cluster synchronization of heterogeneous systems using contraction analysis

In this paper, we employ contraction theory to solve cluster synchronization problem under directed topology. According to the cluster structure, we develop an invariant subspace. Then we take the advantage of the complementary space to prove that the whole systems will synchronize to this invariant subspace, leading to cluster synchronization. We first deal with the linear systems which are linearly coupled under the framework of directed topology. Some sufficient conditions are given to guarantee that the coupled linear systems can achieve cluster synchronization. Moreover, the case of linearly coupled non-linear systems is also considered. Two simulation examples are given to verify our theoretical findings.