Decentralized adaptive coupling synchronization of fractional-order complex-variable dynamical networks

• The synchronization of fractional-order complex-variable dynamical networks is investigated.
• A new lemma is proposed for estimating the Caputo fractional derivatives of Hermitian quadrtic Lyapunov functions.
• Fractional-order decentralized adaptive strategies are designed to tune the coupling gains.
• By applying the new lemma and proposed adaptive laws, two sufficient conditions are established for synchronization.
• The counterintuitive result is obtained, which is not consistent with that of integer-order case.

In this paper, we combine decentralized adaptive control with the fractional-order techniques to investigate the synchronization of fractional-order complex-variable dynamical networks. A new lemma is proposed for estimating the Caputo fractional derivatives of Hermitian quadrtic Lyapunov functions. Based on local information among neighboring nodes, an effective fractional-order decentralized adaptive strategy to tune the coupling gains among network nodes is designed. This analysis is further extended to the case where only a small fraction of coupling gains are choosen to be adjusted. By constructing suitable Lyapunov functions and utilizing the proposed lemma, two sufficient criteria are derived to guarantee the network synchronization by using the proposed adaptive laws. Finally, numerical examples are given to validate the theoretical results.