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Synchronization of fractionalorder dynamical network via aperiodically intermittent pinning control
Advances in Difference Equations volumeÂ 2019, ArticleÂ number:Â 165 (2019)
Abstract
In this paper, synchronization of fractionalorder network is investigated. The aperiodically intermittent pinning control scheme is adopted to design effective controllers for achieving the synchronization. Noticeably, the topology is directed and only the first node is controlled. Based on the Lyapunov function method and mathematical analysis technique, some sufficient conditions are derived and demonstrated to be effective by a numerical example.
1 Introduction
Synchronization as a typical and important collective behavior of dynamical networks has been extensively investigated [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23]. For many networks, especially those coupled with large number of nodes, they cannot synchronize themselves or synchronize with desired goals without external controls. Therefore, different control schemes, such as feedback control, intermittent control, impulsive control, pinning control, and so on, have been developed.
Intermittent control is a kind of discontinuous control consisting of work time and rest time in a sequence of intervals. In virtue of its high efficiency and strong practicability, it has been adopted to design effective controllers in practical applications, and lots of results have been obtained. For instance, the signal will become weak in transmission, so an external control should be added to increase the strength of the signal. Zhou et al. [16] studied the exponential lag synchronization for neural networks with mixed delays via intermittent control. Zhang et al. [17] considered the intermittent control for clusterdelay synchronization in directed networks. He et al. [23] investigated the exponential synchronization of dynamical network with distributed delays via intermittent control. From practical point of view, periodically intermittent control is unreasonable in some cases. As we know, the generation of wind power is typically aperiodically intermittent [8]. The heating system of central airconditioning can run by the aperiodically intermittent operation mode (offpreheatoccupancyoff) [2]. Therefore, aperiodically intermittent control is adopted to study the synchronization of dynamical networks. Guan et al. [4] studied the cluster synchronization of coupled genetic regulatory networks with delays via aperiodically adaptive intermittent control. Liu et al. [6] studied the finitetime synchronization of delayed dynamical networks via aperiodically intermittent control. On the other hand, for largescale networks, it is impractical to add controllers onto all nodes. Therefore, intermittent control and pinning control are combined together to design effective controllers. That is, only a fraction of network nodes is controlled. Liu and Chen [8] studied the synchronization of nonlinear coupled networks via aperiodically intermittent pinning control. Cai et al. [12] studied the outer synchronization between two hybridcoupled delayed dynamical networks via aperiodically adaptive intermittent pinning control.
The above results mainly concentrated on integerorder dynamical networks. Compared with integerorder dynamical networks, fractionalorder dynamical networks can excellently describe the memory and hereditary properties of various models. In fact, fractionalorder systems, such as viscoelastic systems, dielectric polarization, electromagnetic waves, heat conduction, robotics, finance, and so on [24,25,26,27,28,29,30], are ubiquitous in real world. Recently, synchronization of fractionalorder complex networks has been investigated as well [31,32,33,34,35,36,37,38,39,40]. Particularly, Li et al. [38] studied the synchronization of fractionalorder dynamical networks via periodically intermittent pinning control. Zhou et al. [40] studied the cluster synchronization of fractionalorder directed networks via intermittent pinning control. Naturally, how to design aperiodically intermittent pinning controllers for achieving synchronization of fractionalorder networks is an important issue and deserves further investigations. The main contribution of this paper is the design of aperiodically intermittent pinning controllers and the derivation of the sufficient conditions for achieving synchronization of fractionalorder network. It is noted that the obtained results generalize some of the results in Refs. [8, 38]. For example, periodically intermittent pinning control of fractionalorder dynamical network is generalized to aperiodically intermittent pinning control.
The paper is organized as follows. In Sect. 2, some necessary preliminaries about fractional calculus and the model of fractionalorder networks are presented. In Sect. 3, aperiodically intermittent pinning controllers are designed and sufficient conditions for achieving synchronization are derived based on Lyapunov stability theory and mathematical analysis method. In Sect. 4, numerical simulation is performed to demonstrate the effectiveness of the obtained results. Finally, some conclusions are presented in Sect. 5.
Notations. For real matrix \(A\in \mathbb{R}^{N\times N}\), \(A^{s}=(A+A^{T})/2\) denotes the symmetric part. If all the eigenvalues of A are real, let \(\lambda _{\max }(A)\) and \(\lambda _{\min }(A)\) be its largest and smallest eigenvalues, respectively. The symbol âŠ— denotes the Kronecker product.
2 Model description and preliminaries
In this section, some definitions, lemmas, and wellknown results about fractional differential equations are recalled. In addition, the mathematic model of fractional complex network is introduced.
2.1 Caputo fractional operator and MittagLeffler function
Caputo fractional operator plays an important role in the fractional systems, since the initial conditions for fractional differential equations with Caputo derivatives take the same form as for integerorder differential, which have wellunderstood physical meanings [24]. Thus, we use Caputo derivatives as a main tool in this paper. The formula of the Caputo fractional derivative is defined as follows.
Definition 1
([24])
The Caputo fractional derivative of function \(x(t)\) is defined as
where \(m1<\alpha <m, m\in Z^{+}\). Let \(m=1,0<\alpha <1\), then
For simplicity, denote \(D_{t}^{\alpha }x(t)\) as \({}_{0}^{C}D_{t}^{ \alpha }x(t)\). The following properties of Caputo operators are specially provided.
Lemma 1
([24])
If \(w(t),u(t)\in C^{1}[t_{0},b]\), and \(\alpha >0,\beta >0\), then

(1)
\(D_{t}^{\alpha }D^{\beta } w(t)=D_{t}^{\alpha \beta } w(t)\),

(2)
\(D_{t}^{\alpha }(w(t)\pm u(t))=D_{t}^{\alpha }w(t)\pm D_{t}^{ \alpha }u(t)\).
The MittagLeffler function is defined by
where \(\alpha >0,\beta >0\), and \(\varGamma (\cdot )\) is the gamma function. For short, \(E_{\alpha }(z):=E_{\alpha ,1}(z)\). The following properties of MittagLeffler function will be used below.
Lemma 2
Let \(V(t)\) be a continuous function on \([t_{0},+\infty )\) and satisfy
where \(0<\alpha <1\), Î¸ is a constant and \(t_{0}\) is the initial time, then
2.2 Model description
Consider a fractionalorder network consisting of N individuals described by
where \(0<\alpha <1 \), \(x_{i}(t)=(x_{i1}(t),x_{i2}(t),\ldots ,x_{in}(t))^{T} \in \mathbb{R}^{n}\) is the state variable of the ith node, \(f:\mathbb{R}^{n} \rightarrow \mathbb{R}^{n}\) is a continuously vectorvalued function, \(b>0\) is the coupling strength, and \(\varGamma =\operatorname{diag}(\gamma _{1},\ldots ,\gamma _{n}) \in \mathbb{R}^{n\times n}\) is the inner coupling matrix, \(C=(c_{ij}) \in \mathbb{R}^{N \times N}\) is the zerorowsum outer coupling matrix determining the topology and coupling strength of the network, which are defined as follows: if there exists a connection from node j to node \(i\ (i\neq j)\), then \(c_{ij}>0\); otherwise, \(c_{ij}=0\).
The objective here is to synchronize network (1) to the desired orbit \(\eta (t)\) by designing aperiodically intermittent pinning controllers, where \(\eta (t)\) is a solution of an isolated node satisfying \(D^{\alpha }_{t} \eta (t)=f(\eta (t))\). For simplicity, only the first node is controlled and the controlled network is written as follows:
where \(p=1,2,3,\ldots , k>0\) is the control gain, \(0=t_{1}< s_{1}< t _{2}< s_{2}<\cdots <t_{p}<s_{p}<\cdots\)â€‰. Then \(s_{p}t_{p}\) and \(t_{p+1}s_{p}\) denote the pth control and rest width respectively.
Let \(e_{i}(t)=x_{i}(t)\eta (t)\) be the synchronization errors. Then the error system is
Let \(\mathscr{E}(t)=(e_{1}(t)^{T},\ldots ,e_{N}(t)^{T})^{T}\) and
the error system can be rewritten as
where \(\overline{C}=C\operatorname{diag}(k,0,\ldots ,0)\).
Assumption 1
([1])
Suppose that there exist two positive definite diagonal matrices \(P=\operatorname{diag}(p_{1},p_{2},\ldots ,p_{n})\) and \(\Delta =\operatorname{diag}(\delta _{1},\delta _{2},\ldots ,\delta _{n})\) such that
for any \(u,v\in \mathbb{R}^{n}\).
Lemma 3
([1])
Suppose that C is an irreducible zerorowsum matrix with nonnegative offdiagonal elements and \(\mu <0\). Then there exists a positive definite diagonal matrix \(\varPhi =\operatorname{diag}(\phi _{1},\ldots ,\phi _{N})\) such that \(C_{\mu }=C+\operatorname{diag}( \mu ,0,\ldots ,0)\) is Lyapunov stable, i.e., \(\varPhi C_{\mu }+C^{T}_{ \mu }\varPhi <0\).
Assumption 2
([2])
For the aperiodically intermittent control strategy, there exist two positive scalars Ïƒ and Ïˆ such that
Assumption 3
Suppose that the outer coupling matrix C is irreducible and the inner coupling matrix Î“ is positive definite.
3 Main results
In this section, some sufficient conditions for achieving synchronization of the controlled network (2) are provided.
Theorem 1
Suppose that Assumptions 1â€“3 hold. The controlled network (2) with the intermittent pinning control can achieve synchronization if there exist positive constants \(a_{1}\), \(a_{2}\) and \(0<\varpi <1\) such that the following conditions hold:
where P and Î” are defined in (5), \(\widetilde{C}_{1}=( \varPhi \overline{C})^{s},\widetilde{C}_{2}=(\varPhi C)^{s}\), and Î¦ is defined according to Lemma 3, i.e., \(\widetilde{C}_{1}<0\).
Proof
Consider the following Lyapunov function:
Then the derivative of \(V(t)\) along the trajectories of (4) satisfies the following:
When \(t\in [t_{p},s_{p})\),
Therefore, from Lemma 2,
Similarly, when \(t\in [s_{p},t_{p+1})\),
which implies
For \(0=t_{1}\leq t< s_{1}\), from (7),
For \(s_{1}\leq t< t_{2}\), from (8) and (9),
For \(t_{2}\leq t< s_{2}\), from (7) and (10),
For \(s_{2}\leq t< t_{3}\), from (8) and (11),
By mathematical induction, for \(t_{l}\leq t< s_{l}\),
and for \(s_{l}\leq t< t_{l+1}\),
Assume that inequalities (13) and (14) hold when \(l\leq k\). Then
Now, for \(t_{k+1}\leq t< s_{k+1}\),
and for \(s_{k+1}\leq t< t_{k+2}\),
that is, inequalities (13) and (14) hold for \(l=k+1\). Thus, for any l,
i.e., \(V(t_{l})\to 0\) as \(l\to \infty \). And for \(t_{l}\leq t< s_{l}\),
which implies that \(V(t)\to 0\) as \(t\to \infty \). Similarly, one can show that \(V(t)\to 0\) as \(t\to \infty \) for \(s_{l}\leq t< t_{l+1}\). Then, the synchronization is achieved and the proof is completed.â€ƒâ–¡
Let \(\kappa ^{*}=\lambda _{\max }(\varPhi \otimes (P\Delta ))\), \(\vartheta ^{*}=\lambda _{\max }(\varPhi \otimes P)\), \(\vartheta _{*}=\lambda _{\min }( \varPhi \otimes P)\), \(c^{*}_{1}=\lambda _{\max }(\widetilde{C}_{1}\otimes (P\varGamma ))\), and \(c^{*}_{2}=\lambda _{\max }(\widetilde{C}_{2}\otimes (P\varGamma ))\). We have the following corollary.
Corollary 1
Suppose that Assumptions 1â€“3 hold. The controlled network (2) with the intermittent pinning control can achieve synchronization if there exist positive constants \(a_{1}\), \(a_{2}\), and \(0<\varpi <1\) such that the following conditions hold:
Remark 1
If we choose \(t_{p+1}t_{p}=T>0\) and \(s_{p}t_{p}= \kappa T\) with \(0<\kappa <1\), then the intermittent control is periodic. And condition (iii) in Theorem 1 (or Corollary 1) is rewritten as \(E_{\alpha }(a_{1}(\kappa T)^{\alpha })E_{\alpha }(a_{2}((1\kappa ) T)^{\alpha })<\varpi <1\), which is similar to condition (iii) in Theorem 1 of Ref. [38]. That is, the obtained results generalize the results in Ref. [38] from periodic control to aperiodic control.
Remark 2
If we choose \(\alpha =1\), then network (2) is an integerorder network and condition (iii) in Theorem 1 (or Corollary 1) is rewritten as \(e^{a_{1}\sigma }e^{a_{2}\psi }<\varpi \). Further, we have \(a_{2}\psi a_{1}\sigma <\ln \varpi <0\), which is similar to the condition in Corollary 4 of Ref. [8]. That is, we generalize the results from integerorder network to fractionalorder network.
4 Numerical simulations
Consider a fractionalorder dynamical network consisting of 10 nodes. The node dynamics is chosen as the following fractionalorder Chua circuit [41]:
where \(i=1,2,\ldots ,10\), \(\alpha =0.99\), \(\varphi (x_{i1})=5/7x_{i1}3/14(x _{i1}+1x_{i1}1)\).
For simplicity, choose P as an identity matrix. Then one has
Let \(\rho =1.36\) and \(\beta =0.83\), we can choose \(\Delta =\operatorname{diag}(8.19,8.19,8.19)\) such that Assumption 1 holds.
In numerical simulations, we choose \(k=1\), \(b=100\), Î“ as an identity matrix and
The real parts of the eigenvalues of C are \(0, 0.7876, 2, 3.3482, 3.615, 3.615, 4.3919, 4.6587, 6.2918, 6.2918\). That is, the matrix C is irreducible. According to the discussions in [1], we choose
in Lemma 3. By simple calculations, we have \(\kappa ^{*}=3.4447\), \(\vartheta ^{*}=0.4206\), \(\vartheta _{*}=0.1253\), \(c^{*}_{1}=0.0857\), and \(c^{*}_{2}=0\). Then we choose \(a_{1}=12\) and \(a_{2}=27.5\) such that conditions (i) and (ii) in Corollary 1 hold. Further, we choose \(\sigma =0.05\), \(\psi =0.02\), and \(\varpi =0.96\). We have \(E _{\alpha }(12\sigma ^{\alpha }) E_{\alpha }(27.5\psi ^{\alpha })=0.9581< \varpi \), i.e., condition (iii) in Corollary 1 holds. Specially, we choose the first 7 elements of sequences \(\{t_{p}\}\) and \(\{s_{p}\}\) as \((0,0.06,0.14,0.21,0.3,0.36,0.42)\) and \((0.05,0.12,0.19,0.28,0.35,0.41,0.5)\). Figure 1 shows the orbits of the synchronization errors \(e_{ij}(t)\), \(i=1,2,\ldots ,10, j=1,2,3\).
5 Conclusions
In this paper, the synchronization of complex network coupled with fractionalorder dynamical systems is studied. Aperiodically intermittent control scheme is adopted to design effective controllers combining with pinning strategy. That is, only the first node is controlled. Sufficient conditions for achieving the synchronization are derived and verified by numerical example. Obviously, the obtained results for networks with only one controller are also valid for networks with more controllers. Therefore, in practical applications, especially for largescale networks, more nodes are usually controlled to reduce the coupling strength and/or control gain.
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The authors thank the editor and anonymous referees for their valuable suggestions and comments, which improved the presentation of this paper.
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This work is jointly supported by the NSFC under Grant No. 61463022, the NSF for Distinguished Young Scholar of Jiangxi Province of China under Grant 20171BCB23031, the program of China Scholarships Council under Grant No. 201708360078, the Graduate Domestic Visiting Project of Jiangxi Normal University, and the Graduate Innovation Project of Jiangxi Normal University under Grant No. YJS2017060.
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Zhou, J., Yan, J. & Wu, Z. Synchronization of fractionalorder dynamical network via aperiodically intermittent pinning control. Adv Differ Equ 2019, 165 (2019). https://doi.org/10.1186/s1366201921091
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DOI: https://doi.org/10.1186/s1366201921091