Synchronization in Tempered Fractional Complex Networks via Auxiliary System Approach

Ma, Weiyuan; Li, Changpin; Deng, Jingwei

doi:https://doi.org/10.1155/2019/6071412

Complexity

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Abstract Introduction Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2019 | Article ID 6071412 | https://doi.org/10.1155/2019/6071412

Synchronization in Tempered Fractional Complex Networks via Auxiliary System Approach

Weiyuan Ma,¹Changpin Li,²and Jingwei Deng¹

Academic Editor: Thierry Floquet

Received29 Oct 2018

Revised25 Jan 2019

Accepted26 Sept 2019

Published25 Nov 2019

Abstract

In the famous continuous time random walk (CTRW) model, because of the finite lifetime of biological particles, it is sometimes necessary to temper the power law measure such that the waiting time measure has a convergent first moment. The CTRW model with tempered waiting time measure is the so-called tempered fractional derivative. In this article, we introduce the tempered fractional derivative into complex networks to describe the finite life span or bounded physical space of nodes. Some properties of the tempered fractional derivative and tempered fractional systems are discussed. Generalized synchronization in two-layer tempered fractional complex networks via pinning control is addressed based on the auxiliary system approach. The results of the proposed theory are used to derive a sufficient condition for achieving generalized synchronization of tempered fractional networks. Numerical simulations are presented to illustrate the effectiveness of the methods.

1. Introduction

In the past few decades, the study of fractional calculus has attracted substantial attention. Fractional calculus is a powerful mathematical tool for modeling systems in the fields of secure communication, biological science, chemical reactors, laser systems, and so on [1–5]. Recently, tempered fractional derivatives have drawn the wide interests of researchers. A tempered fractional derivative is closer to reality in the sense of the finite life span or bounded physical space of particles. As a generalization of fractional calculus, tempered fractional calculus does not simply have the properties of fractional calculus but can describe some other complex dynamics [6, 7]. Tempered fractional derivatives and the corresponding tempered fractional differential equations have played key roles in poroelasticity [8], finance [9], ground-water hydrology [10], geophysical flows [11], and so on. Therefore, complex network models with tempered fractional dynamics can become more realistic.

Synchronization, as one of the vital dynamical phenomena, exists not only in integer systems but also in fractional systems. Its applications range from electrology to biology, from physics to engineering, and even from economics to nervous system, communication system, and control processing [12–16]. Many studies have focused on the outer synchronization between two fractional complex networks [17–19]. Yu et al. investigated α-synchronization for fractional neural networks [20]. Wu et al. constructed outer synchronization for two different fractional complex networks with nonlinear controllers [21]. Yang and Jiang considered adaptive synchronization for fractional complex networks with uncertain parameters [22]. Chai et al. proposed a pinning synchronization for general fractional complex networks [23]. Dadras et al. studied fractional integral sliding-mode control for uncertain fractional nonlinear systems [24]. Shao et al. discussed adaptive sliding-mode synchronization for a class of fractional chaotic systems [25]. However, in most of this work, all individuals in two networks are assumed to have completely identical dynamics, which is not the case in real practice, for example, predator-prey interactions in ecological communities. Clearly, groups of predators and prey have their own dynamical behavior. Therefore, outer synchronization between two different tempered fractional complex networks is a practical and significant problem worth investigating.

The scenario in which two networks reach harmonious coexistence is regarded as generalized synchronization, which is weaker than complete synchronization [26]. Generalized synchronization exists widely in nature and society. For example, predators and preys influence each other’s behaviors. Predators cannot live without preys, and too many predators would bring the preys into extinction. The complex systems of predators and preys will finally reach harmonious coexistence without man-made sabotage. The auxiliary system approach [27] was proposed to realize generalized synchronization between two complex networks. In addition to the drive and response systems, this method constructs an auxiliary system that has an identical dynamical system to that of the response system, as described bywhere are the states of drive, response, and auxiliary system, respectively. is the tempered fractional Caputo derivative operator which is defined later on. If the response system and auxiliary system reach complete synchronization, that is, for any initial conditions , then generalized synchronization between the drive system and response systems is achieved. Figure 1 gives the schematic diagram of generalized synchronization based on the auxiliary system approach. Specifically, layer I is the drive system, and layer II is the response system which is driven by signals from layer I. Layer III is the auxiliary system, which is an identical duplication of layer II driven by the same signals from the drive layer. This technique is a very effective method to realize outer synchronization between two complex networks [29]. However, as far as we know, no one has discussed the auxiliary system approach for fractional complex networks.

Synchronization of tempered fractional complex networks in a generalized sense leads to richer behavior than identical node dynamics in coupled networks. It may disclose a more complicated connection between the synchronized trajectories in the state spaces of coupled networks. In this paper, we first study the tempered fractional complex network and its generalized synchronization. Additionally, we derive some properties of tempered fractional calculus to construct the synchronization criteria of tempered fractional complex networks. Tempered Mittag-Leffler stable is proposed, which can better describe stability feature of tempered fractional systems. Third, tempered fractional chaotic systems have more alterable dynamical behaviors than fractional ones. It may be more useful in secure communication and control processing. Finally, an auxiliary system approach is used to consider the generalized synchronization for fractional complex networks. This is another difference between the fractional complex network synchronization in this paper and those of previous papers.

The rest of this paper is organized as follows. Some necessary preliminaries are given in Section 2. Generalized synchronization is discussed in Section 3. Simulation results are given in Section 4. Finally, the paper is concluded in Section 5.

2. Properties of the Tempered Fractional Derivative

Some definitions and properties are introduced in this section.

Definition 1 (see [9]). For , the tempered fractional Caputo derivative is defined aswhere denotes the Gamma function defined by

Remark 1. If the tempered parameter , the tempered fractional Caputo derivative (2) reduces to the corresponding fractional Caputo derivative.
Consider the tempered Caputo fractional nonautonomous systemwith initial condition , where , is piecewise continuous in t and locally Lipschitz in x, and is a domain that contains the origin .

Definition 2. The constant is an equilibrium point of the tempered Caputo fractional dynamic system (4) if and only ifFor convenience, assume the equilibrium point in all definitions and theorems is . Because any equilibrium point can be shifted to the origin via a change of variables, there is no loss of generality in doing so. Suppose the equilibrium point for (5) is , and consider the change of variable . The tempered Caputo fractional derivative of y is given bywhere , and in the new variable y, the system has equilibrium at the origin.

Definition 3. The solution of (4) is said to be tempered Mittag-Leffler stable ifwhere , and is locally Lipschitz on with Lipschitz constant .

Remark 2. If , tempered Mittag-Leffler stability reduces to Mittag-Leffler stability.

Remark 3. Mittag-Leffler stability and tempered Mittag-Leffler stability imply asymptotic stability, that is, .

Definition 4 (see [30]). The Mittag-Leffler function is defined aswhere . The Mittag-Leffler function with two parameters is defined aswhere and .
For , we have and . The Laplace transform of the Mittag-Leffler function in two parameters iswhere t and s are variables in the time domain and Laplace domain, respectively. is the real part of s, , and is the Laplace transform.
To obtain the main results, several lemmas are given below.

Lemma 1 (see [31]). Assume that is symmetric. Let , , and (), , , and . is a block matrix of Q obtained by removing its first row-column pairs, and E and S are matrices with appropriate dimensions. When , is equivalent to .

Lemma 2 (see [32]). The Laplace transform of tempered fractional Caputo derivative (2) is given aswhere denotes the Laplace transform of .
We can obtain the following theorems for the tempered fractional derivative.

Theorem 1. Assume is a continuous and differentiable function; then, for any time instant ,where and .

Proof. Let , . According to Definition 1, for any , we haveBecause the function is differentiable, by L’Hôpital’s rule, we obtainCombining (13) and (14), we obtain The proof is completed.

Theorem 2. Let be a differentiable vector function. Then, for any time instant , the following inequality holds:where , and P is a symmetric and positive definite matrix.

Proof. Since P is a symmetric and positive definite matrix, there exists an orthogonal matrix and a diagonal matrix , such thatwhere , . Then,Define an auxiliary variable . One can then writeUsing Theorem 1, one haswhich completes the proof.

Theorem 3. Let be a differentiable vector function. If a continuous function satisfiesthenwhere , and θ is a positive constant.

Proof. From inequality (21), there exists a nonnegative function satisfyingTaking the Laplace transform (11) of (23), we obtainwhere . Therefore,Taking the inverse Laplace transform of (25), we havewhere means convolution. Because both and are nonnegative functions, we can obtainThis completes the proof.
Actually, an appropriate function that satisfies the inequality (21) is difficult to find. Therefore, we present a second Lyapunov method to weaken the conditions in Theorem 3.

Theorem 4. Let domain and system (4) all contain equilibrium point , and assume satisfiesand that is locally Lipschitz with respect to x, is piecewise continuous, exists, and , where , and b are given positive constants. Then, is tempered Mittag-Leffler stable. If the assumptions hold globally on , then is globally tempered Mittag-Leffler stable.

Proof. From equations (28) and (29), we obtainThere exists a nonnegative function satisfyingTaking the Laplace transform of (31), we havewhere and . Due to continuity of function and (32), we obtain andApplying the inverse Laplace transform, the unique solution of (33) isBecause , and are nonnegative functions, we obtainSubstituting (35) into (28) givesLet . It follows from (26) thatwhere if and only if .
Since about x satisfies the local Lipschitz condition, we can obtain near which satisfies the local Lipschitz condition. This result also means system (4) is tempered Mittag-Leffler stable, which completes the proof.

Remark 4. In Theorems 1–4, if , one has some properties about fractional derivative and fractional system, which have been discussed [33–36]. So, the results are more general.

Remark 5. It should be noted that we pay special attention to the tempered Mittag-Leffler stability for the following reasons. On one hand, the tempered Mittag-Leffler stability is essential in the synchronization analysis of tempered fractional complex networks. On the other hand, the tempered Mittag-Leffler stability is an important character when evaluating the tempered fractional system.

3. Generalized Synchronization Criteria for Two-Layer Tempered Fractional Networks via Pinning Control

In this section, a pinning control method for achieving the generalized synchronization between two-layer tempered fractional complex networks is presented.

Consider the following tempered fractional complex network composed of N unidirectionally coupled nodes that is described bywhere , and is the state vector of the i-th node. and are the linear and nonlinear parts of the isolated i-th node, respectively. is the coupling strength. is the positive definite diagonal inner-coupling matrix, and is the symmetric coupling configuration matrix, in which is defined as follows: if there is a connection from node j to node , ; otherwise, . The diagonal elements of matrix A are defined as

Equation (38) is regarded as the drive network, and the response network with N coupling nodes is chosen as follows:where is the state vector. and are the linear and nonlinear parts of the isolated i-th node, respectively. is the coupling strength. Q and have the same meanings as those of P and A in network (38), respectively.

Without loss of generality, the first l nodes are selected as the pinned nodes in the complex network (39). Therefore, the pinned network with pinning controllers is described as follows:

Output signals from the drive network (39) are taken as input for the response network (40), so the latter will reach harmonious coexistence with the former.

To realize the generalized synchronization between the drive layer (39) and the response layer (40), an auxiliary network layer is selected as the following form:where is the state vector of the i-th node and are controllers of the same form as that of .

According to the auxiliary system approach, the drive layer (38) and response layer (40) are said to achieve generalized synchronization if the response network (40) and the auxiliary network (41) reach complete outer synchronization. If for any initial conditions , the generalized synchronization between the two-layer tempered fractional networks (38) and (40) is realized.

Hereafter, suppose that the nonlinear function satisfies the Lipschitz condition. Then, there exists a nonnegative constant L such thatfor any time-varying vectors with the norm .

Define the synchronization error between (40) and (41) as Then, the error dynamical network is described by

Theorem 5. The pinning controllers are chosen asLet , , be an identity matrix, be a modified matrix of matrix B by replacing the diagonal elements with , and . If Lipschitz condition (42) and the conditionare satisfied, then the drive network (38) and the response network (40) can reach generalized synchronization with controllers (44) and (45).

Specifically, for a suitable k, if there exists a natural number such that the largest eigenvalue of satisfies , then the tempered fractional response network (40) asymptotically synchronizes to the drive network (38) with the pinning scheme (44) and (45).

Proof. According to (44) and (45), the error network (43) can be rewritten asConsider the Lyapunov function candidateAccording to Theorem 2, its tempered fractional calculus along the trajectory of (47) iswhere is the maximum eigenvalue of matrix , is the minimum eigenvalue of matrix , and .
If the condition holds, we have andAccording to Theorem 3, we havethat is, as , which means that the generalized synchronization between the drive network (38) and the response network (40) is realized.
Specifically, let and be the reduced matrix of obtained by removing the first l row-column pairs. According to Lemma 1, is equivalent to for suitable k. Since is a real symmetric matrix, the largest eigenvalue of is real and . Thus, if there exists a positive constant such that , one has . Therefore, , and we have obtained the desired results.

Remark 6. In Theorem 5, if , the results hold too, that is, the generalized synchronization in two-layer fractional complex networks via pinning control can be realized.

4. Simulation Results

In this section, a predictor–corrector algorithm and an example are provided to illustrate the effectiveness of the proposed method.

4.1. Predictor–Corrector Algorithm

The numerical calculation for tempered fractional systems is not as simple as that of an ordinary differential equation. Here, we use the generalized Adams–Bashforth–Moulton method [37, 38]. A brief introduction of this algorithm is given.

Consider the differential equationwith initial conditionswhere , and are arbitrary real numbers.

The tempered fractional equations (52) and (53) are equivalent to the following Volterra integral equation [32]:where . For uniform nodes , with being the step size of the numerical computation. Then, the corrector formula for (54) to compute iswhere the predicted value is determined by

The estimation error of this approximation is

The numerical solution of a tempered fractional system can be determined by applying the above method.

4.2. Numerical Simulations

In this section, an example is provided to illustrate the effectiveness of the pinning strategies for the generalized synchronization in two-layer tempered fractional complex networks.

Now, we obtain a new tempered fractional chaotic system, that is, a generalization of the corresponding integer order system [39]. This chaotic system is adopted as the node dynamics in the layer of drive network (38), which is described aswhere , , and . One can obtain the attractor of tempered fractional chaotic systems with different λ, as shown in Figure 2.

(a)

(b)

(c)

(d)

In the layer of the response network (39), tempered fractional Chen system [33] is chosen as the node dynamics:where , , and . One can obtain the attractor of the fractional chaotic Chen system, as shown in Figure 3. From Figures 2 and 3, we can see that the tempered fractional system has more abundant dynamical behavior than the corresponding fractional one. Additionally, in the layer of the response network (41), the dynamics of each node are in the same form as those in the response layer.

(a)

(b)

(c)

(d)

For convenience, let inner-coupling matrices . The coupling configuration matrices A and B are given asFractional Chen systems are bounded. Actually, , and that is, . The out-degrees of nodes 2, 3, and 4 are larger than their in-degrees, so we select them as the pinned nodes [31]. After rearranging the network nodes, the new order is 4, 3, 2, 1, 6, 10, 5, 7, 8, and 9.

Let , and ; we can obtain . From Theorem 5, the drive network (38) and the response network (40) can achieve generalized synchronization. The simulation results are shown in Figure 4, which illustrates the time waveforms of errors .

(a)

(b)

(c)

The temper parameter λ has a direct effect on the chaotic behavior of nonlinear dynamical systems. For convenience, we define the average error norm as

We let the parameters depict as above, except . From Figure 5, the synchronization behaviors of fractional complex networks are realized with and different tempered parameters . Furthermore, from Figure 6, fractional complex networks can reach synchronization for and different fractional orders . The figures show that fractional complex networks (38) and (40) are synchronized, which demonstrate the effectiveness and robustness of the proposed method.

5. Conclusion

In this paper, the tempered fractional derivative is introduced to chaotic systems and complex networks. Synchronization of tempered fractional complex networks may be more useful in secure communication and control processing due to the addition of the tempered parameter. The pinning control scheme and auxiliary system approach are used to verify the existence of generalized synchronization in tempered fractional complex networks. Some properties of the tempered fractional derivative and tempered fractional system are discussed. Generalized synchronization criteria are established based on the proposed theory. The numerical results demonstrate the effectiveness of the proposed method. Furthermore, generalized synchronization can be achieved with different . Future work will study synchronization-based parameter identification [29] of tempered fractional delayed complex networks.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (grant nos. 11872234 and 61966032), the Natural Science Foundation of Gansu Province, China (grant no. 17JR5RA284), and Gansu Provincial First-Class Discipline Program of Northwest Minzu University (no. 31920190047).

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Copyright © 2019 Weiyuan Ma et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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