Synchronization of Multi-Directed Hypergraphs Via Adaptive Pinning Control

This paper proposes a new complex dynamical network defined on multi-directed hypergraphs, and designs an adaptive pinning control scheme to regulate the network to achieve complete synchronization. By using the Lyapunov function method and analyzing the defined Laplacian matrices of multi-directed hypergraphs, we obtain the sufficient condition for global asymptotical stability of the synchronization of controlled dynamical network. Based on the obtained stability condition and renumbering theorem, we develop novel algorithms to determine pinned nodes under different directed higher-order structures. By introducing directed weighted hypergraph models, we further find that the directionality of multi-directed hypergraphs can either promote synchronization or suppress synchronization, depending on the weighting method and coupling strength. By utilizing the chaotic Chua’s circuit as nodes’ local dynamics, some numerical examples are presented to illustrate the theoretical results.