Predefined-Time Synchronization for Variable-Order Fractional Chaotic Systems via Smooth Control

This brief proposes a predefined-time synchronization smooth controller for a class of variable-order fractional chaotic systems (VOFCS) with uncertainties and external disturbances. Firstly, a novel predefined-time strategy is designed, which will not only guarantee the upper-bound time function independent of initial values of the system but also be more explicit than the conventional upper-bound time function. Then, a smooth controller is proposed, which can effectively restrict the chattering phenomenon compared with conventional control methods. Furthermore, by using the generalized Lyapunov stability criterion, a condition for the stabilization of the VOFCS under the predefined-time smooth control is established. Finally, two different chaotic systems are taken to illustrate that the VOFCS can be synchronized in predefined-time based on smooth control, thus better avoiding chattering and enhancing the synchronization convergence rate of the VOFCS. The proposed control strategy can be applied to variable-order chaotic systems, commensurate and incommensurate systems.