Chaos, Solitons and Fractals 121 (2019) 108–118 Contents lists available at ScienceDirect Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos Graph-theoretic approach to synchronization of fractional-order coupled systems with time-varying delays via periodically intermittent control Yao Xu, Yanzhen Li, Wenxue Li ∗ Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai, 264209, PR China a r t i c l e i n f o Article history: Received 5 September 2018 Revised 25 January 2019 Accepted 31 January 2019 Keywords: Synchronization Fractional-order coupled systems Time-varying delays Periodically intermittent control Graph-theoretic approach a b s t r a c t This paper deals with synchronization problem of fractional-order coupled systems (FOCSs) with time- varying delays via periodically intermittent control. Here, nonlinear coupling, time-varying internal delay and time-varying coupling delay are considered when modeling, which makes our model more general in comparison with the most existing fractional-order models. It is the first time that periodically inter- mittent control is applied to synchronizing FOCSs with time-varying delays. Combining Lyapunov method with graph-theoretic approach, some synchronization criteria are obtained. Moreover, the synchroniza- tion criteria we derive depend on the fractional order α, control gain, control rate and control period. Besides, the synchronization issues of fractional-order coupled chaotic systems with time-varying delays and fractional-order coupled Hindmarsh–Rose neuron systems with time-varying delays are also inves- tigated as applications of our theoretical results, and relevant sufficient conditions are derived. Finally, numerical simulations with two examples are provided in order to demonstrate the effectiveness of the theoretical results and the feasibility of control strategy. © 2019 Published by Elsevier Ltd. 1. Introduction In the last decades, coupled systems have received increasing interest due to their impressive applications in various areas. Many of these applications heavily depend on the dynamical behaviors of coupled systems, such as stability [1] and synchronization [2,3]. It is well known that Lyapunov method plays an important role in investigating the dynamical behaviors of coupled systems, and lots of excellent results have been obtained by taking advantage of this method, see [4–6] for example. Nevertheless, in fact, con- structing a suitable Lyapunov function is quite difficult. Fortunately, in [7], Li and Shuai proposed a new method (combination of Lya- punov method and graph-theoretic approach) to construct a global Lyapunov function, i.e., the weighted sum of Lyapunov function of each subsystem, which can also make the topological structure of networks be well reflected. Inspired by their work, some scholars have made full use of this method to investigate the dynamical be- haviors of coupled systems, and a number of results have been re- ported [8–10]. It is not difficult to find that most investigations for dynami- cal behaviors of coupled systems are based on the integer-order ∗ Corresponding author. E-mail address: wenxuetg@hitwh.edu.cn (W. Li). dynamical models in aforementioned references. However, in com- parison with classical integer-order derivative, a more effective in- strument for the description of various materials and processes is provided by fractional-order derivative. And the incorporation of the fractional-order derivative into dynamical systems is an extreme improvement since fractional-order systems possess the characters of non-locality, memory and history-dependent. For in- stance, the memory of fractional-order systems leads to modeling a fractional-order HIV-immune system in [11]. As many scholars have focused on the dynamical behaviors of fractional-order cou- pled systems (FOCSs), the synchronization problem of FOCSs has gradually become a hot research point, and a variety of interesting results have been derived, see [12–14] and references therein. It is noted that time delays sometimes are not taken into consideration in FOCSs, see [15–17] for example. This may be attributed to the fact that it is difficult to extend Lyapunov– Krasovskii functional method, which is frequently used in integer- order coupled systems, to FOCSs directly. However, while work- ing with real phenomena, whether coupling time delay or internal time delay cannot be neglected since they may facilitate the pro- duction of undesirable dynamical behaviors such as instability and oscillations [18]. For some FOCSs, generally, constant fixed time delays can serve as good approximation to a certain extent, see [12,19]. Even so, we still cannot ignore the truth that time delays are variable in most situations. Thus, time-varying delays were https://doi.org/10.1016/j.chaos.2019.01.038 0960-0779/© 2019 Published by Elsevier Ltd.