Global synchronization in finite time for fractional-order coupling complex dynamical networks with discontinuous dynamic nodes

Synchronization Rather Than Finite-Time Synchronization Results of Fractional-Order Multi-Weighted Complex Networks

This article investigates the synchronization of fractional-order multi-weighted complex networks (FMWCNs) with order α ∈ (0,1) . A useful fractional-order inequality _t0^C D_t^α V(x(t)) ≤ -μV(x(t)) is extended to a more general form _t0^C D_t^α V(x(t)) ≤ -μV^γ(x(t)),γ ∈ (0,1] , which plays a pivotal role in studies of synchronization for FMWCNs. However, the inequality _t0^C D_t^α V(x(t)) ≤ -μV^γ(x(t)),γ ∈ (0,1) has been applied to achieve the finite-time synchronization for fractional-order systems in the absence of rigorous mathematical proofs. Based on reduction to absurdity in this article, we prove that it cannot be used to obtain finite-time synchronization results under bounded nonzero initial value conditions. Moreover, by using feedback control strategy and Lyapunov direct approach, some sufficient conditions are presented in the forms of linear matrix inequalities (LMIs) to ensure the synchronization for FMWCNs in the sense of a widely accepted definition of synchronization. Meanwhile, these proposed sufficient results cannot guarantee the finite-time synchronization of FMWCNs. Finally, two chaotic systems are given to verify the feasibility of the theoretical results.