Synchronization Rather Than Finite-Time Synchronization Results of Fractional-Order Multi-Weighted Complex Networks
Xiangqian Yao, Yu Liu, Zhijun Zhang, Weiwei Wan
Abstract
This article investigates the synchronization of fractional-order multi-weighted complex networks (FMWCNs) with order <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha \in (0,1)$ </tex-math></inline-formula> . A useful fractional-order inequality <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${}_{t_{0}}^{C} D_{t}^{\alpha } V(x(t))\leq -\mu V(x(t))$ </tex-math></inline-formula> is extended to a more general form <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${}_{t_{0}}^{C} D_{t}^{\alpha } V(x(t))\leq -\mu V^{\gamma }(x(t)),\gamma \in (0,1]$ </tex-math></inline-formula> , which plays a pivotal role in studies of synchronization for FMWCNs. However, the inequality <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${}_{t_{0}}^{C} D_{t}^{\alpha } V(x(t))\leq -\mu V^{\gamma }(x(t)),\gamma \in (0,1)$ </tex-math></inline-formula> 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.