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Synchronization in Tempered Fractional Complex Networks via Auxiliary System Approach.

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 a-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 by

[mathematical expression not reproducible], (1)

where x(t), y(t), z(t) [member of] [R.sup.n] are the states of drive, response, and auxiliary system, respectively. [sub.C][D.sup.[alpha],[lambda].sub.0,t] is the tempered fractional Caputo derivative operator which is defined later on. If the response system and auxiliary system reach complete synchronization, that is, [lim.sub.t[right arrow][infinity]] [parallel]y(t) - z(t)[parallel] = 0 for any initial conditions y([t.sub.0]) [not equal to] z([t.sub.0]), 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 n - 1 [less than or equal to] [alpha] < n, n [member of] N, and [lambda] [greater than or equal to] 0, the tempered fractional Caputo derivative is defined as

[mathematical expression not reproducible], (2)

where [GAMMA](*) denotes the Gamma function defined by

[GAMMA](z) = [[integral].sup.[infinity].sub.0] [e.sup.-t] [t.sup.z-1] dt. (3)

Remark 1. If the tempered parameter [lambda] = 0, the tempered fractional Caputo derivative (2) reduces to the corresponding fractional Caputo derivative.

Consider the tempered Caputo fractional nonautonomous system

[sub.C][D.sup.[alpha],[lambda].sub.0,t]x(t) = f(t, x), (4)

with initial condition x(0), where [alpha] [member of] (0,1), [lambda] [greater than or equal to] 0, f : [0, +[infinity]) is piecewise continuous in t and locally Lipschitz in x, and [OMEGA] [subset] [R.sup.n] is a domain that contains the origin x = 0.

Definition 2. The constant [x.sub.0] is an equilibrium point of the tempered Caputo fractional dynamic system (4) if and only if

[sub.C][D.sup.[alpha],[lambda].sub.0,t][x.sub.0] = f(t, [x.sub.0]). (5)

For convenience, assume the equilibrium point in all definitions and theorems is [x.sub.0] = 0. 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 [bar.x] [not equal to] 0, and consider the change of variable y = x - [bar.x]. The tempered Caputo fractional derivative of y is given by

[mathematical expression not reproducible], (6)

where [bar.g](t, 0) = 0, 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 if

[parallel]x(t)[parallel] [less than or equal to] [[m(x(0))[e.sup.-[lambda]t] [E.sub.[alpha]](-[kt.sup.[alpha])].sup.b], (7)

where t [greater than or equal to] 0, [lambda] [greater than or equal to] 0, [alpha] [member of] (0,1), b > 0, k [greater than or equal to] 0, m(0) = 0, m(x) [greater than or equal to] 0, and m(x) is locally Lipschitz on x [member of] B [subset] [R.sup.n] with Lipschitz constant [m.sub.0].

Remark 2. If [lambda] = 0, tempered Mittag-Leffler stability reduces to Mittag-Leffler stability.

Remark 3. Mittag-Leffler stability and tempered Mittag-Leffler stability imply asymptotic stability, that is, [lim.sup.t[right arrow][infinity]] [parallel]x(t)[parallel] = 0.

Definition 4 (see [30]). The Mittag-Leffler function is defined as

[E.sub.[alpha]](z) = [[infinity].summation over (k=1)] [z.sup.k]/[GAMMA](k[alpha] + 1), (8)

where [alpha] > 0. The Mittag-Leffler function with two parameters is defined as

[E.sub.[alpha],[beta]](z) = [[infinity].summation over (k=1)] [z.sup.k]/[GAMMA](k[alpha] + [beta]), (9)

where [alpha] > 0 and [beta] > 0.

For [beta] = 1, we have [E.sub.[alpha],1] z) = [E.sub.[alpha]](z) and [E.sub.1,1](z) = [e.sup.z]. The Laplace transform of the Mittag-Leffler function in two parameters is

L{[t.sup.[beta]-1][E.sub.[alpha],[beta]](-[lambda][t.sup.[alpha]])} = [s.sup.[alpha]-[beta]]/[s.sup.[alpha]] + [lambda],

R(s) > [[absolute value of [lambda]].sup.1/[alpha]], (10)

where t and 5 are variables in the time domain and Laplace domain, respectively. R(s) is the real part of s, [lambda] [member of] R, and L{*} is the Laplace transform.

To obtain the main results, several lemmas are given below.

Lemma 1 (see [31]). Assume that Q = [([q.sub.ij]).sub.NxN] is symmetric. Let [mathematical expression not reproducible], and [mathematical expression not reproducible] diag ([m.sup.*.sub.1],..., [m.sup.*.sub.1]), and [m.sup.*] = [min.sub.1[less than or equal to]i[less than or equal to]l] {[m.sup.*.sub.i]}. [Q.sub.l] is a block matrix of Q obtained by removing its first l(1 [less than or equal to] l [less than or equal to] N) row-column pairs, and E and S are matrices with appropriate dimensions. When [m.sup.*] > [[lambda].sub.max](E - [SQ.sup.-l.sub.l][S.sup.T]), Q - [M.sup.*] < 0 is equivalent to [Q.sub.l] < 0.

Lemma 2 (see [32]). The Laplace tramform of tempered fractional Caputo derivative (2) is given as

[mathematical expression not reproducible], (11)

where X(s) = L{x(t)} denotes the Laplace transform of x(t).

We can obtain the following theorems for the tempered fractional derivative.

Theorem 1. Assume x(t) [member of] R is a continuous and differentiate function; then, for any time instant t [greater than or equal to] 0,

[sub.C][D.sup.[alpha],[lambda].sub.0,t](t) [less than or equal to] 2x(t) [sub.C][D.sup.[alpha],[lambda].sub.0,t]x(t), (12)

where 0 < [alpha] < 1 and [lambda] [greater than or equal to] 0.

Proof. Let y(s) = [e.sup.[lambda]s][(x(t) - x(s)).sup.2], s [member of] [0,t]. According to Definition 1, for any t [greater than or equal to] 0, we have

[mathematical expression not reproducible]. (13)

Because the function y (s) is differentiate, by L'Hopital's rule, we obtain

[mathematical expression not reproducible]. (14)

Combining (13) and (14), we obtain

[sub.C][D.sup.[alpha],[lambda].sub.0,t][x.sup.2](t) [less than or equal to] 2x(t) [sub.C][D.sup.[alpha],[lambda].sub.0,t]x(t). (15)

The proof is completed.

Theorem 2. Let x(t) [member of] [R.sup.n] be a differentiate vector function. Then, for any time instant t [greater than or equal to] 0, the following inequality holds:

[sub.C][D.sup.[alpha],[lambda].sub.0,t]([x.sup.T](t)Px(t)) [less than or equal to] 2[x.sup.T](t)P[sub.C][D.sup.[alpha],[lambda].sub.0,t]x(t), (16)

where 0 < [alpha] < 1, [lambda] [greater than or equal to] 0, and P is a symmetric and positive definite matrix.

Proof. Since P is a symmetric and positive definite matrix, there exists an orthogonal matrix B [member of] [R.sup.nxn] and a diagonal matrix [LAMBDA] [member of] [R.sup.nxn], such that

P = B[LAMBDA][B.sup.T], (17)

where [LAMBDA] = diag{[[lambda].sub.11], [[lambda].sub.22],..., [[lambda].sub.nn]}, [[lambda].sub.ii] > 0 (i = 1,2,..., n). Then,

[x.sup.T](t)Px(t) = [x.sup.T](t)B[LAMBDA][B.sup.T]x(t)

= [([B.sup.T]x(t)).sup.T] [LAMBDA]([B.sup.T]x(t)). (18)

Define an auxiliary variable y(t) = [B.sup.T]x(t). One can then write

[x.sup.T](t)Px(t) = [y.sup.T](t)[LAMBDA]y(t) = [n.summation over (i=1)] [[lambda].sub.ii][y.sup.2.sub.i](t). (19)

Using Theorem 1, one has

[mathematical expression not reproducible], (20)

which completes the proof.

Theorem 3. Let x(t) [member of] [R.sup.n] be a differentiate vector function. If a continuous function V : [0, +[infinity]) satisfies

[sub.C][D.sup.[alpha],[lambda].sub.0,t]V(t, x(t)) [less than or equal to] - [theta]V(t, x(t)), (21)


V(t, x(t)) [less than or equal to] V(0, x(0))[e.sup.-[lambda]t] [E.sub.[alpha]](-[theta][t.sup.[alpha]]), (22)

where 0 < [alpha] < 1, [lambda] [greater than or equal to] 0, and [theta] is a positive constant.

Proof. From inequality (21), there exists a nonnegative function M(t) satisfying

[sub.C][D.sup.[alpha],[lambda].sub.0,t]V(t, x(t)) + [theta]V(t, x(t)) + M(t) = 0. (23)

Taking the Laplace transform (11) of (23), we obtain

[(s + [lambda]).sup.[alpha]]V(s) - [(s + [lambda]).sup.[alpha]-1] V(0, x(0)) + [theta]V(s) + M(s) = 0, (24)

where V(s) = L{V(t, x(t))} and M(s) = L{M(t)}. Therefore,

V(s) = [(s + [lambda]).sup.[alpha]-1] V(0, x(0)) - M(s)/[(s + [lambda]).sup.[alpha]] + [theta]. (25)

Taking the inverse Laplace transform of (25), we have

V(t, x(t)) = V(0, x(0))[e.sup.-[lambda]t] [E.sub.[alpha]](-[theta][t.sup.[alpha]])

-M(t) * [[t.sup.[alpha]-1] [e.sup.-[lambda]t] [E.sub.[alpha],[alpha]](-[theta][t.sup.[alpha]])], (26)

where * means convolution. Because both M(t) and [t.sup.[alpha]-1] [e.sup.-[lambda]t] [E.sub.[alpha],[alpha]](-[theta][t.sup.[alpha]]) are nonnegative functions, we can obtain

V(t, x(t)) [less than or equal to] V(0, x(0))[e.sup.-[lambda]t] [E.sub.[alpha]](-[theta][t.sup.[alpha]]). (27)

This completes the proof.

Actually, an appropriate function V(t, x(t)) 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 D [subset] [R.sup.n] and system (4) all contain equilibrium point x = 0, and assume V(t, x(t)) : [0, +[infinity]) satisfies

[[alpha].sub.1][[parallel]x(t)[parallel].sup.a]] < V(t, x t)) [less than or equal to] [[alpha].sub.2]x [(t).sup.ab], (28)

[sub.C][D.sup.[alpha],[lambda].sub.0,t]V([t.sup.+], x([t.sup.+])) [less than or equal to] - [[alpha].sub.3][[parallel]x(t)[parallel].sup.ab]

(holding almost everywhere), (29)

and that V(t, x(t)) is locally Lipschitz with respect to x, V(t, x(t)) is piecewise continuous, [mathematical expression not reproducible] exists, and [mathematical expression not reproducible], where t [greater than or equal to] 0, x [member of] D, [alpha] [member of] (0,1), [lambda] [greater than or equal to] 0, [[alpha].sub.1], [[alpha].sub.2], [[alpha].sub.3], a, and b are given positive constants. Then, x = 0 is tempered Mittag-Leffler stable. If the assumptions hold globally on [R.sup.n], then x = 0 is globally tempered Mittag-Leffler stable.

Proof. From equations (28) and (29), we obtain

[sub.C][D.sup.[alpha],[lambda].sub.0,t] V([t.sup.+], x)[t.sup.+])) [less than or equal to] - [[alpha].sub.3]/[[alpha].sub.2] V(t, x(t))

(holding almost everywhere). (30)

There exists a nonnegative function M(t) satisfying

[mathematical expression not reproducible]. (31)

Taking the Laplace transform of (31), we have

[(s + [lambda]).sup.[alpha]] [V.sup.+](s) - [(s + [lambda]).sup.[alpha]-1] V([0.sup.+]) + M(s) = - [[alpha].sub.3]/[[alpha].sub.2] V(s), (32)

where [mathematical expression not reproducible] and [V.sup.+](s) = L {V([t.sup.+], x([t.sup.+]))}. Due to continuity of function V(t, x(t)) and (32), we obtain V([t.sup.+], x([t.sup.+])) = V (t, x(t)), [V.sup.+](s) = V(s) and

V(s) = V(0)[(s + [lambda]).sup.[alpha]-1] - M(s)/[(s + [lambda]).sup.[alpha]] + ([[alpha].sub.3]/[[alpha].sub.2]). (33)

Applying the inverse Laplace transform, the unique solution of (33) is

V(t) = V(0) [e.sup.-[lambda]t] [E.sub.[alpha]](-[[alpha].sub.3]/[[alpha].sub.2] [t.sup.[alpha]])

- M(t) * [[e.sup.-[lambda]t] [t.sup.[alpha]-1] [E.sub.[alpha],[alpha]](- [[alpha].sub.3]/[[alpha].sub.2][t.sup.[alpha]])]. (34)

Because [t.sup.[alpha]-1], [e.sup.-[lambda]t], and [E.sub.[alpha],[alpha]](- ([[alpha].sub.3]/[[alpha].sub.2])[t.sup.[alpha]]) are nonnegative functions, we obtain

V(t) [less than or equal to] V(0) [e.sup.-[lambda]t] [E.sub.[alpha]](-[[alpha].sub.3]/[[alpha].sub.2] [t.sup.[alpha]]). (35)

Substituting (35) into (28) gives

[parallel]x(t)[parallel] [less than or equal to] [[V(0)/[[alpha].sub.1] [e.sup.-[lambda]t] [E.sub.[alpha]](- [[alpha].sub.3]/[[alpha].sub.2] [t.sup.[alpha]])].sup.1/a]. (36)

Let m = (V(0)/[[alpha].sub.1]) [greater than or equal to] 0. It follows from (26) that

x(t) [less than or equal to] [[[me.sup.-[lambda]t] [E.sub.[alpha]](-[[alpha].sub.3]/[[alpha].sub.2] [t.sup.[alpha]])].sup.1/a], (37)

where x(0) = 0 if and only if (V(0)/[[alpha].sub.1]) = 0.

Since V(t, x) about x satisfies the local Lipschitz condition, we can obtain (V(0, x(0)))/[[alpha].sub.1] near x(0) 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 [lambda] = 0, 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 by

[mathematical expression not reproducible], (38)

where 0 < [[alpha].sub.1] < 1, [[lambda].sub.1] [greater than or equal to] 0, and [x.sub.i](t) = [([x.sub.i1](t), [x.sub.i2](t),..., [] (t)).sup.T] [member of] [R.sup.n] is the state vector of the i-th node. H[x.sub.i](t) and f([x.sub.i] (t)) = [([f.sub.1]([x.sub.i](t)), [f.sub.2]([x.sub.i](t)),..., [f.sub.n]([x.sub.i](t))).sup.T] : [R.sup.n] [right arrow] [R.sup.n] are the linear and nonlinear parts of the isolated i-th node, respectively. [[epsilon].sub.1] is the coupling strength. P [member of] [R.sup.nxn] is the positive definite diagonal inner-coupling matrix, and A = [([a.sub.ij]).sub.NxN] is the symmetric coupling configuration matrix, in which [a.sub.ij] is defined as follows: if there is a connection from node j to node i(i [not equal to] j), [a.sub.ij] = 1; otherwise, [a.sub.ij] = 0. The diagonal elements of matrix A are defined as [a.sub.ii] = -[[summation].sup.N.sub.j=1,j[not equal to]i] [a.sub.ij], i = 1,2,... , N.

Equation (38) is regarded as the drive network, and the response network with N coupling nodes is chosen as follows:

[mathematical expression not reproducible], (39)

where [y.sub.i](t) = [([y.sub.i1](t), [y.sub.i2](t),..., [](t)).sup.T] [member of] [R.sup.n] is the state vector. M[y.sub.i](t) and g([y.sub.i](t)) are the linear and nonlinear parts of the isolated i-th node, respectively. [[epsilon].sub.2] is the coupling strength. Q and B = [([b.sub.ij]).sub.NxN] 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 [u.sub.i]([x.sub.i], [y.sub.i]), where i = 1,2,..., l, is described as follows:

[mathematical expression not reproducible]. (40)

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:

[mathematical expression not reproducible], (41)

where [z.sub.i] is the state vector of the i-th node and [u.sub.i]([x.sub.i], [z.sub.i])(i = 1,2,..., l) are controllers of the same form as that of [u.sub.i]([x.sub.i], [y.sub.i]).

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 [lim.sub.t[right arrow][infinity]] [parallel][y.sub.i](t) - [z.sub.i](t)[parallel] = 0 for any initial conditions [y.sub.i](0) [not equal to] [z.sub.i](0)(i = 1,2,..., N), the generalized synchronization between the two-layer tempered fractional networks (38) and (40) is realized.

Hereafter, suppose that the nonlinear function g(x(t)) satisfies the Lipschitz condition. Then, there exists a nonnegative constant L such that

[parallel]g(x(t)) - g(y(t))[parallel] [less than or equal to] L[parallel]x(t) - y(t)[parallel], (42)

for any time-varying vectors x(t), y(t) [member of] [R.sup.n] with the norm [parallel]x[parallel] = [square root of ([x.sup.T]x)].

Define the synchronization error between (40) and (41) as [e.sub.i](t) = [z.sub.i](t)- [y.sub.i](t) (i = 1,2,..., N). Then, the error dynamical network is described by

[mathematical expression not reproducible]. (43)

Theorem 5. The pinning controllers are chosen as

[u.sub.i]([x.sub.i], [z.sub.i]) = -[k.sub.i]([z.sub.i] - [x.sub.i]), i = 1,2,..., l, (44)

[u.sub.i]([x.sub.i], [y.sub.i]) = -[k.sub.i]([y.sub.i] - [x.sub.i]), i = 1,2,..., l. (45)

Let [mathematical expression not reproducible] be an N x N identity matrix, [??] be a modified matrix of matrix B by replacing the diagonal elements [b.sub.ii] with [[rho].sub.min][b.sub.ii]/q, and k = [min.sub.1[less than or equal to]i[less than or equal to]l]{[k.sub.i]}. If Lipschitz condition (42) and the condition

(L + [[lambda].sub.M])[I.sub.N] + [[epsilon].sub.2]q[??] - K < 0, (46)

are 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 1 [less than or equal to] l < N such that the largest eigenvalue of [??] satisfies L + [[lambda].sub.M] + [[epsilon].sub.2]q[[lambda].sub.max]([[??].sub.l+1]) < 0, 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 as

[mathematical expression not reproducible]. (47)

Consider the Lyapunov function candidate

V(t) = 1/2 [N.summation over (i=1)] [e.sup.T.sub.i](t) [e.sub.i](t). (48)

According to Theorem 2, its tempered fractional calculus along the trajectory of (47) is

[mathematical expression not reproducible], (49)

where [[lambda].sub.M] is the maximum eigenvalue of matrix (M + [M.sup.T])/2, [[rho].sub.min] is the minimum eigenvalue of matrix (Q + [Q.sup.T])/2, and e = [([e.sub.1], [e.sub.2],..., [parallel][e.sub.N][parallel]).sup.T].

If the condition H = (L + [[lambda].sub.M])[I.sub.N] + [[epsilon].sub.2]q[??] - K < 0 holds, we have [eta] = -[[lambda].sub.max](H) > 0 and

[mathematical expression not reproducible]. (50)

According to Theorem 3, we have

[mathematical expression not reproducible], (51)

that is, [parallel][e.sub.i](t)[parallel] [right arrow] 0 (i = 1,2,..., N) as t [right arrow] [infinity], which means that the generalized synchronization between the drive network (38) and the response network (40) is realized.

Specifically, let k = [min.sub.1[less than or equal to]i[less than or equal to]l]{[k.sub.i]} and [[??].sub.l+1] be the reduced matrix of [??] obtained by removing the first l row-column pairs. According to Lemma 1, H < 0 is equivalent to (L + [[lambda].sub.M])[I.sub.N-l] + cq[[??].sub.l+1] < 0 for suitable k. Since [H.sub.l+1] = (L + [[lambda].sub.M])[I.sub.N-l] + [[epsilon].sub.2]q[[??].sub.l+1] is a real symmetric matrix, the largest eigenvalue [[lambda].sub.max]([H.sub.l+1]) of [H.sub.l+1] is real and [[lambda].sub.max]([H.sub.l+1]) = (L+ [[lambda].sub.M]) + [[epsilon].sub.2]q[[lambda].sub.max] ([B.sub.l+1]). Thus, if there exists a positive constant 1 [less than or equal to] l [less than or equal to] N such that L + [[lambda].sub.M] + [[epsilon].sub.2] q[[lambda].sub.max]([B.sub.l+1]) < 0, one has (L + [[lambda].sub.M])[I.sub.N-l] + [[epsilon].sub.2]q[[??].sub.l+1] < 0. Therefore, H < 0, and we have obtained the desired results.

Remark 6. In Theorem 5, if [lambda] = 0, 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 equation

[sub.C][D.sup.[alpha],[lambda].sub.0,t] = f(t, x(t)), 0 < t < T, (52)

with initial conditions

[d.sup.k]/[dt.sup.k][([e.sup.[lambda]t] x(t))|.sub.t=0] = [c.sub.k], k = 0,1,..., [alpha] - 1, (53)

where n [less than or equal to] [alpha] < n, n [member of] [N.sup.+], [lambda] > 0, and [c.sub.k] are arbitrary real numbers.

The tempered fractional equations (52) and (53) are equivalent to the following Volterra integral equation [32]:

x(t) = [x.sub.0](t) + 1/[GAMMA]([alpha]) [[integral].sup.t.sub.0] [e.sup.-[lambda](t-[tau]][(t - [tau].sup.[alpha]-1] f([tau], x([tau]))d[tau], (54)

where [x.sub.0](t) = [[summation].sup.[alpha]-1.sub.k=0] [c.sub.k](([e.sup.-[lambda]t][t.sup.k])/k!). For uniform nodes [t.sub.n+1] = (n + 1)h(n = 0,1,..., N), with h = T/N being the step size of the numerical computation. Then, the corrector formula for (54) to compute [x.sub.n+1] [approximately equal to] x([t.sub.n+1]) is

[mathematical expression not reproducible], (55)

where the predicted value [x.sup.P.sub.n+1] is determined by

[mathematical expression not reproducible]. (57)

The estimation error of this approximation is

[mathematical expression not reproducible]. (58)

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 as

[mathematical expression not reproducible], (59)

where i = 1,2,..., 10, [[alpha].sub.1] = 0.99, [a.sub.1] = 5, [b.sub.1] = 10, and [c.sub.1] = 3.8. One can obtain the attractor of tempered fractional chaotic systems with different [lambda], as shown in Figure 2.

In the layer of the response network (39), tempered fractional Chen system [33] is chosen as the node dynamics:

[mathematical expression not reproducible], (60)

where i = 1,2,..., 10, [[alpha].sub.2] = 0.995, [a.sub.2] = 35, [b.sub.2] = 3, and [c.sub.2] = 28. 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.

For convenience, let inner-coupling matrices P = Q = I. The coupling configuration matrices A and B are given as

[mathematical expression not reproducible]. (61)

Fractional Chen systems are bounded. Actually, [parallel][y.sub.i1][parallel] [less than or equal to] 60, [parallel][y.sub.i2][parallel] [less than or equal to] 40, [parallel][y.sub.i3][parallel] [less than or equal to] 40, [parallel][z.sub.i1][parallel] [less than or equal to] 60, [parallel][z.sub.i2][parallel] [less than or equal to] 40, [parallel][z.sub.i3][parallel] [less than or equal to] 40, i = 1,2,..., 10, and [parallel]f([x.sub.i]) - f([y.sub.i])[parallel] [less than or equal to] [square root of [(-[y.sub.i1] [y.sub.i3] + [z.sub.i1][z.sub.i3]).sup.2] + [([y.sub.i1][y.sub.i2] - [z.sub.i1][z.sub.i2]).sup.2])] [less than or equal to] 84.85[parallel][e.sub.i][parallel], that is, L = 84.85. 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, 1o, 5, 7, 8, and 9.

Let [k.sub.i] = 120, [[epsilon].sub.1] = 0.45, and [[epsilon].sub.2] = 5; we can obtain [[lambda].sub.max]((L + [[lambda].sub.M])[I.sub.N] + [[epsilon].sub.2]q[??] - K) = -26.2501 < 0. 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 [e.sub.i1], [e.sub.i2], and [e.sub.i3] (i = 1,2,..., 10).

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

[[parallel]e[parallel].sub.a] = 1/N [N.summation over (i=1)] [parallel][e.sub.i][parallel] = 1/N [N.summation over (i=1)] [parallel][y.sub.i] - [z.sub.i][parallel]. (62)

We let the parameters depict as above, except [[lambda].sub.2]. From Figure 5, the synchronization behaviors of fractional complex networks are realized with [[alpha].sub.2] = 0.995 and different tempered parameters [[lambda].sub.2]. Furthermore, from Figure 6, fractional complex networks can reach synchronization for [[lambda].sub.2] = 2 and different fractional orders [[alpha].sub.2]. 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 [[epsilon].sub.2]. 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.


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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Weiyuan Ma [ID], (1) Changpin Li [ID], (2) and Jingwei Deng [ID] (1)

(1) School of Mathematics and Computer Science, Northwest Minzu University, Lanzhou 730000, China

(2) Department of Mathematics, Shanghai University, Shanghai 200444, China

Correspondence should be addressed to Changpin Li;

Received 29 October 2018; Revised 25 January 2019; Accepted 26 September 2019; Published 25 November 2019

Academic Editor: Thierry Floquet

Caption: Figure 1: A schematic diagram [28] of generalized synchronization via the auxiliary system approach.

Caption: Figure 2: The attractor of the new tempered fractional system with different [[lambda].sub.1]. (a) [[lambda].sub.1] = 0. (b) [[lambda].sub.1] = 1. (c) [[lambda].sub.1] = 2. (d) [[lambda].sub.1] = 3.

Caption: Figure 3: The attractor of the tempered fractional Chen system with different [[lambda].sub.2]. (a) [[lambda].sub.2] = 0. (b) [[lambda].sub.2] = 2. (c) [[lambda].sub.2] = 4. (d) [[lambda].sub.2] = 6.

Caption: Figure 4: Time evolution of the error states [e.sub.i1], [e.sub.i2], and [e.sub.i3] with [[lambda].sub.1] = 1 and [[lambda].sub.2] = 2. (a) [e.sub.i1](i = 1,2,..., 10). (b) [e.sub.i2] (i = 1,2,..., 10). (c) [e.sub.i3] (i = 1,2,...,10).

Caption: Figure 5: Time evolution of the average error norms [e.sub.a] with different tempered parameters [[lambda].sub.2].

Caption: Figure 6: Time evolution of the average error norms [[parallel]e[parallel].sub.a] with different fractional orders [[alpha].sub.2].
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Title Annotation:Research Article
Author:Ma, Weiyuan; Li, Changpin; Deng, Jingwei
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Date:Dec 31, 2019
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