Characterizations and criteria for synchronization of heterogeneous networks to linear subspaces

Quanyi Liang, Zhikun She, Lei Wang, Michael Z.Q. Chen, Qing Guo Wang

Research output: Contribution to journalArticlepeer-review

30 Citations (Scopus)


In this paper, we investigate synchronization issues in heterogeneous complex networks. We start with a necessary condition for the existence of synchronous trajectories by analyzing the vector fields of isolated nodes. Then, by geometrical methods of differential equations, a closed invariant set is introduced to characterize the synchronous trajectories. Especially, the ω-limit sets of all the synchronous trajectories are contained in this closed invariant set. Afterward, assuming that the vector fields of all the isolated nodes are identical in a linear subspace, an equiv- alent network is found such that the equivalent network can be composed of two simpler networks with lower dimensions. Based on this composition, global synchronization criteria for the equivalent network are developed by combining the corresponding Lyapunov-like functions of quadratic form (and even beyond quadratic form) for these two simpler networks. Furthermore, we alternatively use different functions for different nodes to construct Lyapunov functions, arriving at an extended criterion which can be applicable for more networks. Finally, three examples are given to illustrate the validity and advantages of our theoretical results.

Original languageEnglish
Pages (from-to)4048-4071
Number of pages24
JournalSIAM Journal on Control and Optimization
Issue number6
Publication statusPublished - 2017


  • Heterogeneous networks
  • Invariant set
  • Lyapunov functions
  • Synchronization
  • ω-limit set

ASJC Scopus subject areas

  • Control and Optimization
  • Applied Mathematics


Dive into the research topics of 'Characterizations and criteria for synchronization of heterogeneous networks to linear subspaces'. Together they form a unique fingerprint.

Cite this