Control of tipping in a small-world network model via a novel dynamic delayed feedback scheme

Small-world networks have been of increasing interest because of their high similarity with complex networks in reality. Achievements about controlling the propagation and evolution of small world networks have been highlighted, but little is known about the tipping mechanism and control of such network. In this paper, a novel dynamic delayed feedback scheme is put forward and applied to successfully control the tipping phenomenon in small-world networks. Firstly, the linear characteristic equation of the controlled system is analyzed and the conditions for stability and the occurrence of bifurcation-induced tipping are given. Then, the direction of Hopf bifurcation, which reveals the further evolution mechanism of tipping, is analyzed by using the normal form theory and center manifold reduction. Finally, the numerical simulations are provided to support the theoretical analysis and to verify the effectiveness of the proposed dynamic delayed feedback controller for the small-world network.