Synchronization of directed uniform hypergraphs via adaptive pinning control

Abstract

In recent years, higher-order networks have been gradually investigated, and uniform hypergraph is one of their typical expressions. In this paper, in addition to undirected uniform hypergraphs, directed uniform hypergraphs with newly defined hyperedges are proposed by equipping each node with identical chaotic oscillator. In order to realize global synchronization of proposed uniform hypergraphs, two kinds of adaptive pinning control schemes with and without desired synchronized states are designed. By newly defining Laplacian tensor of the uniform hypergraphs, several sufficient conditions of globally stable synchronization are obtained by applying the Lyapunov stability theory. The Laplacian tensor is supersymmetric when considering undirected structures and asymmetric when considering directed structures. Based on the stability condition, an algorithm for determining the pinned nodes is developed. The result further shows that the directionality of uniform hypergraphs can restrain the realization of synchronization. Numerical examples are presented to illustrate the theoretical results.