Impulsive synchronization of fractional-order complex-variable dynamical network

*Correspondence: zhywu@jxnu.edu.cn 1School of Mathematics and Statistics, Jiangxi Normal University, No.99 Ziyang Avenue, Nanchang, P.R. China Abstract The impulsive synchronization of a fractional-order complex-variable network is investigated. Firstly, static impulsive controllers are designed and the corresponding synchronization criteria are derived. From the criteria, the impulsive gains can be calculated. Secondly, adaptive impulsive controllers are designed. Noticeably, the impulsive gains can be adjusted to the needed values adaptively. Finally, numerical examples are provided to verify the results.


Introduction
In recent years, fractional-order differential systems have gained increasing attentions due to the fact that they can better describe the memory and hereditary properties, such as elastic systems, dielectric polarization, heat conduction, electromagnetic waves, and financial systems [1][2][3][4][5][6][7][8][9][10][11][12]. In [7][8][9], the authors designed some kinds of memristive hyperchaotic system and discussed their applications. In [11], the author studied a fractionalorder financial system. For those large-scale fractional-order systems, they usually contain large number of interactive individuals and are modeled by fractional-order dynamical network. The nodes denote the individuals and the edges denote the interactions among individuals. In [13], the authors considered fractional-order neural networks. In [14], the authors investigated a time-delay neural network.
In the real world, many complex systems cannot be controlled by continuous control and endure continuous disturbance. Impulsive control, as a typical discontinuous control scheme, has been widely adopted to design proper controllers, i.e., the controllers are applied to the systems only at certain moments. That is, the impulsive controllers have a relatively simple structure and are easy to implement and have low cost. Researchers have obtained many valuable results about impulsive control and synchronization in integer-order dynamical networks [21-26, 29, 30, 32]. From a practical point of view, a fractional-order network can describe some practical phenomena more accurate than an integer-order model. Therefore, impulsive control is adopted to study the synchronization of fractionalorder dynamical networks as well. In [27], the authors investigated impulsive stabilization and synchronization of fractional-order complex-valued neural networks. In [28], the authors investigated synchronization for a class of fractional-order linear complex networks via impulsive control. In [27,28], some useful synchronization conditions are obtained, from which the impulsive gains and intervals can be calculated for a given network. However, for different networks, it is necessary to calculate the required impulsive intervals and gains repetitively. Therefore, how to design the universal impulsive controllers deserves further studies.
In this paper, we introduce the fractional-order complex-variable dynamical network model and present some preliminaries in Sect. 2. In Sect. 3, we design static and adaptive impulsive controllers, respectively. For static impulsive controllers, we derive the sufficient conditions for achieving synchronization. For adaptive impulsive controllers, we provide the updating laws of the impulsive gains. We perform three numerical examples to verify the results in Sect. 4. In Sect. 5, we give the conclusions.

Model description and preliminaries
In this section, some definitions and lemma are recalled.
where m is a positive integer such that m -1 < α < m.
Consider a fractional-order complex-variable dynamical network, described by is the zerorow-sum outer coupling matrix representing the network topology, defined as: if node k is affected by node l (k = l), then a kl = 0; otherwise, a kl = 0. The time series ]. The controlled network with impulsive controllers is written as where Assumption 1 Suppose that there exists a positive constant L such that holds for any x(t), s(t) ∈ C n and t > 0.
Let e k (t) = x k (t)s(t), we have the following error system: a kl e l (t), t ∈ (t σ , t σ +1 ),

Theorem 1 Suppose that Assumption 1 holds. If there exists a constant ξ > 0 such that
hold, then network (2) achieves synchronization.
Remark 1 By simple calculations, we can estimate the positive constant θ in Theorem 1, and then calculate the impulsive gains from conditions (4). However, for different networks, we must repeatedly calculate the impulsive gains. Therefore, we design adaptive impulsive controllers to avoid this situation.
Theorem 2 Suppose that Assumption 1 holds. If there exists a constant ξ > 0 such that the following conditions: and ω > 0 is a positive constant, then the controlled network (2) achieves synchronization.
Proof Consider the following Lyapunov function: When t ∈ (t σ , t σ +1 ), the function V (e(t)) can be written as and the derivative of V (e(t)) can be calculated as e T l (t)a kl e k (t) + e T k (t)a kl e l (t) When t = t σ +1 , one has Therefore, similar to the proof of Theorem 1, the proof is completed.
Remark 2 When τ σ and ξ are fixed, we choose such that the conditions (7) is satisfied, where ε > 0 is an arbitrary constant.
Example 2 Consider the above network in Example 1 via the adaptive impulsive controllers. Choose ω = 0.01, ξ = 0.001 and θ (0) = 1. According to Remark 2, choose Figure 2 The orbits of the real and imaginary parts of x kl (t) and s l (t) with ε = 0.001. Figure 3 shows the orbits of the real and imaginary parts of x kl (t) and s l (t), k = 1, 2, . . . , 10, l = 1, 2, 3. Figure 4 shows the impulsive gains B(t σ ).
Choose τ σ = 0.2, ω = 0.001, ξ = 0.001 and θ(0) = 0.1. According to Remark 2, choose with ε = 0.001. Figure 5 shows the orbits of the real and imaginary parts of x kl (t) and s l (t), k = 1, 2, . . . , 10, l = 1, 2, 3. Figure 6 shows the impulsive gains B(t σ ). From Examples 2 and 3, the impulsive gains need not be calculated in advance for different networks. And they can adjust themselves to the required values according to the updating laws. That is, the adaptive impulsive controllers are universal to some extent.

Conclusions
Both static and adaptive impulsive controllers were designed. Two corresponding synchronization conditions were derived as well. Particularly, for the adaptive impulsive con- Figure 6 The impulsive gains B(t σ ) versus σ trollers, the updating law for the impulsive gains was provided. Examples 2 and 3 demonstrated the points well and implied that the adaptive impulsive controllers are universal for different networks.