Stability, bifurcation prediction and optimal control of a delayed integer-order small-world network based on the fractional-order PD control policy of variable order

Abstract In this paper, a fractional-order Proportional-Derivative (PD) control policy of variable order is initially designed in solving the problems of Hopf bifurcation and optimal control on the integer-order small-world network. Firstly, the time delay of the network as a key parameter is selected as the bifurcation parameter. Meanwhile, the impact of it is considered on dynamics of the controlled network. In order to analyse the dynamics of the controlled network, two methods are used to make the equivalent transformation for it and the analysis shows that transformed networks have similar linear features near the equilibrium point when variable order is a rational number. Then, through implementing the eigenvalue analysis for the characteristic equations of transformed networks, we are able to acquire some control conditions which can enable the controlled network to be stable or occur Hopf bifurcation. The finding represents that the bifurcation dynamics of the controlled network are effectively optimized by tuning the control parameters. Finally, numerical simulation results are illustrated to prove the availability of the designed control strategy and provide some diagrams between the bifurcation threshold and the control parameters. The diagrams indicate that there is an optimal order to maximize the bifurcation threshold of the controlled network under the fixed control gains.