Bipartite Complete Synchronization of Fractional Heterogeneous Networks via Quantized Control Without Gauge Transformation

Recently, gauge transformation-based bipartite synchronization has received much interest, but the method of gauge transformation alters the original signed topological structure and the competition or cooperation among individuals is obscured. In addition, the heterogeneity of nodes brings great difficulty and challenge for heterogeneous networks to achieve complete synchronization like homogeneous networks. In this article, without converting signed graph into corresponding unsigned structure via the gauge transformation, the bipartite complete synchronization of heterogeneous fractional networks is explored. Above all, a mathematic model of fractional networks with signed topology and heterogeneous nodes’ dynamics is introduced, in which the topological graph possesses both negative and positive edges to illustrate the competition and cooperation between individuals, and the desired synchronized state is an arbitrarily specified smooth orbit and not necessarily the decoupled state. Additionally, two innovative control schemes with logarithmic quantizer are developed, and several conditions are obtained to reach bipartite complete synchronization of fractional heterogeneous networks just by virtue of the Laplacian matrix of the original signed graph rather than the traditional technique of gauge transformation. The theoretical analysis is eventually confirmed by several numerical results.