Cluster Synchronization in Variable-Order Fractional Community Network via Intermittent Control

: In this paper, the cluster synchronization of a variable-order fractional community network with nonidentical dynamics is investigated. For achieving the cluster synchronization, intermittent controllers are designed, and the sufﬁcient conditions with respect to system parameters, intermittent control instants and control gains are derived based on stability theory of fractional-order system and linear matrix inequalities (LMIs). To avoid verifying the LMIs, a corresponding simple corollary is provided. Finally, a numerical example is performed to verify the derived result.


Introduction
Many large-scale physical systems consisting of a large number of interactive individuals are modeled by various dynamical networks, including integer-order and fractionalorder networks . The individuals and interactions are denoted by nodes and (undirected/directed) edges. A community structure is a typical topology structure in dynamical networks. The individuals in the same community usually have the same local dynamics and higher density of interactions. The individuals in different communities, however, have different local dynamics and lower density of interactions. Therefore, those in the same community tend to achieve the same goal and those in different communities achieve different goals, i.e., the community network achieves cluster synchronization [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20].
Dynamical networks, as we know, are difficult and even impossible to synchronize themselves with desired goals without external control. Many kinds of control schemes are adopted to design effective controllers to achieve cluster synchronization, such as intermittent control [1][2][3][4][5][6][7][8][9], impulsive control [10][11][12][13], pinning control [17][18][19], finite-time control [19,20], and so on. Intermittent control, as a typical discontinuous control scheme, has been widely used in real systems, such as heat systems, central air-conditioning, and so on. In [1], Zhou et al. investigated the cluster synchronization of a colored community integer-order network via intermittent pinning control. In [6], Liu et al. considered the cluster synchronization of a delayed integer-order network via intermittent pinning control.
On the other hand, a fractional-order system, in virtue of its memory and genetic characteristics, has been adopted to describe many physical systems, such as the fractional-order inductor [21], cohesive fracture model [22], quantum mechanics [36], and so on. Further, the synchronization and control of a fractional-order dynamical network have been widely investigated. In [32], the synchronization of fractional-order complex-variable dynamical networks is studied using an adaptive pinning control strategy based on a close center degree. In [33], the adaptive cluster synchronization of fractional-order complex networks with internal and coupling delays as well as time-varying disturbances is investigated via the fractional-order hybrid controllers. In [34], cluster synchronization for fractional-order complex network with non-delay and delay coupling is investigated through designing both static and adaptive feedback controllers. In [35], cluster synchronization of a fractional network with fixed order is investigated via periodically intermittent control. In the abovementioned networks, the fractional orders are fixed, i.e., the present states always depend on the states from initial to present state, and the networks have long memory. Long memory, however, heavily increases the computational cost and reduces the performance in practical applications, for example, fast image encryption [24]. On the other hand, many real systems have only short memory, i.e., some past information is not needed and memory starts from a new state rather than the initial state. Furthermore, the fractional-orders may be variable as well [24,25,30,[37][38][39][40][41][42][43][44][45]. In [40], the nutrient-phytoplankton-zooplankton model is extended using variable-order fractional derivatives. In [42], a triaxial creep model for salt rocks based on variable-order fractional derivatives is introduced. In [44], a three-cell population cancer model with variable-order fractional derivative with power, exponential and Mittag-Leffler memories is proposed. In [45], the constitutive relation for viscoelasticity is described by a variable-order fractional system with short memory. In [25], a new fractional variable-order creep model with short memory is introduced. In [24], new variable-order fractional chaotic systems are developed for fast image encryption. In [31], short memory fractional different equations are introduced for new memristor and neural network design. In [30], the Mittag-Leffler stability analysis of tempered fractional neural networks with short memory and variable-order is investigated. Up to now, the cluster synchronization of a variable-order fractional dynamical network is seldom studied via intermittent control.
Motivated by the above discussions, we consider the cluster synchronization of a variable-order fractional dynamical network by designing proper intermittent controllers. The main contributions are two-fold: (1) We generalize the results about intermittent control and cluster synchronization of integer-order networks to a fractional-order network. (2) We design effective aperiodically intermittent controllers and derive the synchronization conditions. In Section 2, we introduce the network model and some preliminaries. In Section 3, based on the stability theory of the fractional-order system and mathematical analysis technique, we derive the sufficient conditions for achieving cluster synchronization with respect to the system parameters, intermittent control instants and control gains. In Section 4, we perform a numerical example to illustrate the effectiveness of the derived results. In Section 5, we conclude this paper.

Model and Preliminaries
In this section, some basic definitions, lemmas and notations about the Caputo fractional-order calculus are presented, which are useful throughout this paper. In addition, the mathematical model of a variable-order fractional complex network is proposed.

Lemma 3 ([48])
. Let x(t) ∈ R be a continuous and derivable function. Then, for any time instant Then, when x(t) ∈ R n is continuous and derivable, it implies where Q ∈ R n×n is a positive definite matrix.
Consider a variable-order fractional community network consisting of N nodes and m communities with 2 ≤ m N. The set of the nodes can be divided into {1, · · · , N} = C 1 C 2 · · · C m , where The node dynamics of the pth community is chosen as the following variable-order fractional system with short memory and lim k→+∞ t k = +∞. Figure 1 shows an example of a variable-order fractional system with short memory. The community network is described by c > 0 is the coupling strengths, Γ = diag(γ 1 , γ 2 , · · · , γ n ) > 0 is the inner coupling matrix, G = (g ij ) N×N is the outer coupling matrix. If node i is affected by node j (j = i), then g ij = 0; otherwise, g ij = 0, and the the diagonal elements of G are defined as g ii = − ∑ N j=1,j =i g ij , i = 1, · · · , N. The matrix G can be written as and G is irreducible, denoted as G ∈ G 1 .
Let s p (t) be a trajectory, which is satisfied with If lim t→∞ x i (t) − s p (t) = 0 and lim t→∞ s p (t) − s l (t) = 0, l = p, where i ∈ C p , · is the Euclidean norm, then the network is said to achieve cluster synchronization. For achieving the cluster synchronization of the Community Network (2), we add feedback controllers u i (t) to the network. The controllers work when t ∈ [t 2η , t 2η+1 ] and rest when t ∈ [t 2η+1 , t 2η+2 ], η = 0, 1, · · · . As we know, the control scheme is called intermittent control, which is a typical discontinuous control scheme. We write the network with intermittent controllers as where i ∈ C p , p = 1, 2, · · · , m, η = 0, 1, · · · , u i (t) = −ch i Γ(x i (t) − s p (t)), h i > is feedback gain.

Remark 1.
We call u i (t) continuous feedback controllers if they work for t ∈ [t k , t k+1 ] and impulsive controllers if they only work for t = t k , k = 0, 1, 2, · · · . If t 2η+2 − t 2η and t 2η+1 − t 2η are constants for η = 0, 1, · · · , we call u i (t) periodically intermittent controllers. That is, the designed controllers in this paper are aperiodically intermittent controllers.

Main Result
In this section, we derive the sufficient conditions for achieving cluster synchronization with respect to the system parameters, intermittent control instants and control gains. Theorem 1. Suppose Assumptions 1 and 2 hold and G ∈ G 3 . Controlled Network (4) can achieve cluster synchronization if there exist positive constants a 1 , a 2 and 0 < < 1 such that the following inequalities hold: where G S = G+G T 2 .

Corollary 1. Suppose all Assumptions in Theorem 1 hold. Controlled Network
Remark 2. For any given Network (4), by simple calculations, we can choose proper a 2 satisfying condition (ii) in Theorem 1, and then choose a proper a 1 satisfying condition (iii). Finally, we can choose feedback gains h i such that condition (i) holds.

Remark 3.
In this paper, the variable-order function is a piecewise constant function. When α k = α for all k = 0, 1, · · · , the variable-order fractional network degenerates to a conventional fractional-order network in [49]. Furthermore, condition (iii) in Theorem 1 (or Corollary 1) is rewritten as E α (−a 1 δ α )E α (a 2 (T − δ) α ) < , which is similar to condition (iii) in Theorem 1 of Ref. [49]. That is, the obtained results generalize the results in Ref. [49] from constant order to variable order.
That is, we generalize the results in Ref. [8] from integer-order network to fractional-order network.

Simulation Results
Consider a variable-order fractional community network consisting of 19 nodes and 3 communities. The topology of the network is shown in Figure 2.

Conclusions
The cluster synchronization of a variable-order fractional community network is studied via intermittent control. The local dynamics of different communities are nonidentical. The synchronization conditions are derived according to the stability theory of a fractionalorder system and LMIs. Furthermore, the obtained result is verified to be correct by a numerical example.
Author Contributions: Y.W. wrote the original draft preparation. Z.W. reviewed and edited the whole paper. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement:
The data can be found in the manuscript.

Conflicts of Interest:
The authors declare no conflict of interest.