Abstract
This paper discusses the out lag synchronization of fractional order complex networks (FOCN) including both internal delay and coupling delay and with the employment of pinning control scheme. Using comparison theorem and constructing the auxiliary function, several synchronization criterions by linear feedback pinning control are presented. The model and the obtained results in this work are more general than the previous works. Correctness and effectiveness of the theoretical results are validated through numerical simulations.
1. Introduction
Complex networks (CN) have attracted great attentions due to their tremendous potentials in deferent fields, such as social organizations, World Wide Web, power grids, communication networks, and biological networks [1–4]. Among all the dynamical behaviours of CN, synchronization is an important and interesting behavior, which can explain many problems in the real world. Several control methods were employed to help synchronizing the CN, including intermittent control [5], impulsive control [6], state feedback control [7], adaptive control [8], sampled-data control [9], and pinning control [10, 11]. However, the existing results are mainly achieved on the case of “inner synchronization” where all nodes can gain a coherent dynamical behavior within a network. Unlike inner synchronization, synchronization can be performed between two or more CN, and this is referred to as “out synchronization.” Some excellent conclusions about out synchronization in complex dynamical networks have been presented in the existing literatures [12–14]. In [12], authors discussed out synchronization between two uncertain general coupled CN. Ref. [13] studied out synchronization between master-slave dynamical networks by adaptive impulsive pinning control. Through the implementation of aperiodically adaptive intermittent pinning control, out synchronization in two hybrid-coupled delayed CN was investigated in [14].
However, integer order networks were considered in all previous works. Fractional calculus, which was first given by Leibniz in the 17th century, can act with differential and integrals of any arbitrary order. Compared with classical ordinary differential networks, fractional calculus provides an excellent instrument to depict memory and hereditary properties in processing. Taking these factors into account, many scientists have introduced fractional differential and integrals to complex dynamical networks, forming fractional-order complex networks (FOCN). Recently, the stability and synchronization analysis of FOCN have attracted considerable interest. Many interesting results were derived for FOCN such as hybrid synchronization of coupled FOCN that was studied in [15]. Ref. [16] investigated the pinning synchronization in delayed FOCN with nondelayed and delayed couplings. In [17], the outer synchronization between FOCN was considered using a nonfragile observer-based control scheme. Using centralized and decentralized data-sampling principles and the theory of fractional differential systems, out synchronization in FOCN was considered in [18].
Time delay is inevitable in real networks due to finite information transmission and processing speeds among the units. Therefore, both internal delay and coupling delay should be considered in dynamical networks. Synchronization of integer order CN with both internal delay and coupling delay was studied in [19, 20]. In general, many types of synchronization were considered in the literature, including lag synchronization [21], complete synchronization [22], projective synchronization [23], and robust synchronization [24]. Among all these types of synchronizations, lag synchronization, which requires the response system to synchronize the drive system with a propagation delay , has been extended to various fields, such as electronic circuits, lasers, and neural systems [25]. Lay synchronization has become an important concept in both theory and application. However, up to now, the out lag synchronization in FOCN was totally ignored in the literature.
Inspired by the above discussions, the goal of this work is to consider out lag synchronization of FOCN with both node delay and coupling delay through the use of pinning control. The main contributions of this paper are listed as follows. In order to form a general model, node delay and coupling delay are incorporated into the fractional order model. The theory of fractional order differential equations and some new analysis techniques are employed to obtain novel sufficient conditions. Out lag synchronization scheme is adopted to perform synchronization between two complex networks.
2. Preliminaries
In this paper, the definition of Caputo fractional derivative is used.
Definition 1 (see [26]). The Caputo fractional derivative of order for a function is given as where , , , and is the Gamma function.
Consider the following FOCN consisting of coupled identical nodes with linearly diffusive couplings, in which each node having an dimensional fractional order delayed dynamical system acts as a drive system; the whole network is described as follows:where , is the state variable of the th node, are the coupling strength, is the delay at each node, is the coupling delay between nodes, represent the inner matrix connecting the coupled variables, and and are the topological structure of the drive networks at time and , respectively. The parameters are given by the following: if there is a connection from node to node , then ; otherwise . The diagonal elements of are defined as , , respectively. is a smooth nonlinear vector function. The initial values of system (2) are defined as .
Remark 2. In model (2), the symmetric and irreducible condition of coupling matrix is not necessary. Furthermore, the node delay is different from the coupling delay .
Remark 3. Synchronizations of FOCN were discussed in previous works [15, 17, 18], but without considering the coupling delay in models. Model (2) is more practical in this paper.
Corresponding, the response system is given:where denotes the response state variable of the th node and are the controllers.
To proceed further, we give assumption and lemmas in the following.
Assumption 4. The function satisfies the uniform semi-Lipschitz condition with Lipschitz constants; i.e., there exist positive constants such thatfor all .
It has to be noted that Assumption 4 is very necessary and important. Many typical systems, such as the Lorenz system, L system, the unified chaotic system, and delayed Hopfield neural network, satisfy Assumption 4. The results in this paper are achieved, while satisfying Assumption 4.
Lemma 5 (see [27]). Assume is a continuous and derivable function. Then, we can get
Lemma 6 (see [28]). Let . Thenfor any .
Lemma 7 (see [29]). Consider the following fractional differential inequality with multiple delaysand the linear fractional differential equation with multiple delayswhere are continuous in , and is continuous in . If , and for all , then
3. Out Lag Synchronization
This section investigates the out lag synchronization by employing the linear feedback pinning control. Without loss of generality, let the first nodes be controlled. The controllers are chosen as follows:where the control gains .
The error vector is defined aswhere is the propagation delay.
Therefore, one gets the error dynamical system:
Let be the minimum eigenvalue and the maximum eigenvalue of the matrix, respectively. denotes the Kronecker product. is the dimensional identity matrix. Note that , . , .
Theorem 8. Suppose that Assumption 4 and condition (9) hold; if there exists such thatthen system (2) can synchronize system (3), where
Proof. Constructing the auxiliary function:where
From Lemma 5 and Assumption 4, we can obtainThen we can obtainBased on Lemma 6, we have the estimations:Substituting (15), (16) into (14) givesNext, we consider the linear fractional order systems:and giving the initial values is the same as in system (17).
According to Corollary 3 in Ref. [30], when the characteristic equation of (18)has no purely imaginary roots and , then we can get that the zero solution of (18) is asymptotically stable.
In case of having a pure imaginary root of (19), in which is a real number. When , , while , We can lead toEquivalently,that is,Since and , if , (21) has no real roots. According to Lemma 7, one can obtain , indicating that the zero solution of (17) is asymptotically stable. Therefore, the system (2) can synchronize the system (3).
Remark 9. Refs. [31, 32] discussed the out synchronization in CN with both internal delay and coupling delay, while considering integer order. In comparison with integer order networks, our proposed system can be applied to some CN in the real world. Moreover, the lag synchronization is considered, which is more general in Refs. [31, 32].
Remark 10. Unlike LMI method [33] for discussing the synchronization of coupled-delay FOCN, we here apply the comparison theorem via pinning control to establish the synchronization criteria for the FOCN with delayed couplings.
If the coupling matrix , then model (2) can be rewritten as follows:For this case, we can get the result below.
Corollary 11. Let Assumption 4 and condition (9) hold; if there exists such thatthen system (2) synchronizes system (3), where
If the coupling matrix , then the fractional order complex networks only contain inner delay; i.e., coupling delay is not considered here, and accordingly model (2) will change to
For this case, we can get the result below.
Corollary 12. Let Assumption 4 and condition (9) hold; if there exists such thatthen system (2) synchronizes system (3), where
Remark 13. In complex dynamical systems, the challenging problems lie on the types of pinning nodes and their possible minimum number. In [34], it is reported that nodes whose out-degrees are bigger than their in-degrees should be pinned candidates. Here, according to the difference between out-degrees and in-degrees, we could rearrange nodes in the light of descending sort and select the first nodes as pinned candidates which satisfy Theorem 8.
4. Numerical Example
The following 2-D CN model is considered as a drive system:where , , , The coupling strengths are , the inner connecting matrix is , and coupling time delay is The weight configuration coupling matrices are taken as
The response system is given bywhere , are the control functions.
By simple computing, one gets In the simulation, we choose , and the conditions of Theorem 8 are satisfied. Initial conditions are chosen as Based on the pinned node selection strategy in Remark 13 and Theorem 8, the nodes 1, 2, 3 are chosen as pinned candidates. The synchronization errors and are shown in Figures 1 and 2.
5. Conclusions
Recently, synchronization of CN has been studied extensively. However, for fractional order dynamical networks, it is more appropriate to choose fractional order stability theory to realise the synchronization. In this paper, the stability theory of fractional order systems and comparison theorem are implemented. Synchronization criterions are achieved by pinning control strategy. A numerical example is elaborated to verify the feasibility of the obtained results.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors would like to acknowledge the project supported by the National Natural Science Foundation of China (61573096), the Natural Science Foundation of Anhui Province (1908085MA01), the Natural Science Foundation of the Higher Education Institutions of Anhui Province (KJ2018A0365, KJ2019A0573), and the Special Foundation for Young Scientists of Anhui Province (gxyq2019048).