# Finite-Time Synchronization for Complex-Valued Recurrent Neural Networks with Time Delays

Authors: Ziye Zhang; Xiaoping Liu; Chong Lin; Bing Chen
Journal: Complexity
 (2018)

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## Abstract

This paper focuses on the finite-time synchronization analysis for complex-valued recurrent neural networks with time delays. First, two kinds of common activation functions appearing in the existing references are combined together and more general assumptions are given. To achieve our aim, a nonlinear delayed controller with two independent parameters different from the existing ones is provided, which leads to great difficulty. To overcome it, a newly developed inequality is used. Then, via Lyapunov function approach, some criteria are derived to guarantee the finite-time synchronization of the considered system, and the settling time for synchronization is also estimated. Finally, two numerical simulations are given to support the effectiveness and advantages of the obtained results.


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## Body

## 1. Introduction

In recent years, neural networks have been always very important to researchers in various fields, to name a few, control theory and signal processing in electrical engineering, parallel computation and pattern recognition in computer science, and modeling optimization problems in applied mathematics. Moreover, as everyone knows, most applications rely heavily on the dynamical behavior of recurrent neural networks. This sets off a research upsurge for the dynamics of neural networks. As a result, many researchers have put their efforts into the analysis and synthesis problems for the dynamics of neural networks for several decades; see [1–8] and the references therein.In most results considering the stability and the stabilization of neural networks, the convergent mode is asymptotically stable and the time for system trajectories to reach an equilibrium point is infinite. However, in practical problems, the convergent time is often required to be faster or even finite. Thus, so as to satisfy this requirement, the topic about finite-time stability appeared. Recently, finite-time stability and synthesis problems for various systems have been hot issues and attracted many scholars’ interests [9–29]. Among them, finite-time stability, stabilization, synchronization and robust passive control for time-delay systems, stochastic systems, chaotic systems, multiagent systems, and nonlinear systems were studied in [10–20]. For various neural networks models, finite-time stabilization, instabilizability, adaptive control, H∞ control, and state estimation were discussed in [22–26] and finite-time synchronization control problems were addressed in [27–29].On the other hand, a complex-valued recurrent neural networks model has produced a research upsurge for the past few years. The main feature is that it owns complex-valued states, connection weights, and activation functions. This decides that this kind of system can be applied to a wider range involving electromagnetic waves, radar imaging, quantum waves, and so on, but it also leads to more difficulties in analyzing its dynamical behaviors in contrast to real-valued systems. In a word, anyway, the occurrence of complex-valued neural networks draws a great deal of attention from all aspects. Accordingly, many fruitful achievements have been reported for this model [30–52]. For example, global stability, global exponential stability, and Hopf bifurcation problems for complex-valued neural networks systems with delays or without delays were investigated in [32–40]. Dissipativity, passivity, state estimation, exponential stability, and input-to-state stability for memristor-based complex-valued neural networks with or without delays were discussed and relative criteria were established in [43–48]. However, for finite-time stability analysis of delayed complex-valued neural networks models, until now, few results have been developed. The boundedness problem of these systems in finite-time interval is considered in [49, 50]. The finite-time synchronization problem for complex-valued neural networks (CVNNs) with infinite-time distributed delays is discussed in [51] and the finite-time stabilizability and instabilizability for complex-valued memristive neural networks with time delays are studied in [52]. Moreover, considering the rich background of complex-valued neural networks models and the present status of development for them, it is absolutely essential to explore the dynamics of these models deeply. Thus, these further inspire us to make deeper research.In this article, in view of the unavoidability of time delays [53–55], the finite-time synchronization problem for complex-valued recurrent neural networks with time delays is presented. The main contributions of our work are embodied in several points: (1) A nonlinear delayed controller with independent parameters μ1 and μ2 different from the most existing results is designed to achieve the finite-time synchronization of the considered system. (2) To deal with the difficulty involved by the contribution (1), a new inequality stated in Lemma 6 proposed and proved by us in [52] is used. (3) For real-imaginary separate-type activation functions, two kinds of common functions in the existing references are combined together and more general conditions are assumed in this paper. So the proposed results have broader applications.The rest of this article is organized as follows. Section2 provides some preliminaries. Section 3 presents main criteria about the finite-time synchronization for delayed complex-valued recurrent neural networks and estimates the settling time by a new controller. Section 4 gives two numerical simulations to show the validity of theoretical results. Section 5 concludes this paper.Notation. Throughout this paper, i=-1 denotes the imaginary unit. For a vector ξ=ξ1,ξ2,…,ξnT∈Rn, ξ is named a positive vector if ξl>0,l=1,2,…,n.


## 2. Preliminaries

Consider a complex-valued recurrent neural networks model as the drive system represented by(1)z˙pt=-dpzpt+∑q=1napqhqzqt+∑q=1nbpqmqzqt-τq+Jptwith the initial conditions(2)zpθ=ψpθ,θ∈-τ,0,where p=1,2,…,n, n denotes the number of neurons; zp(t) is the state variable; dp>0 is a constant; apq and bpq are complex-valued connection weights; Jp(t) represents external input vector; τq is the time delay; τ=max⁡τ1,τ2,…,τn; ψp(θ)∈C([-τ,0],C); hq(zq(t)) and mq(zq(t-τq)) are complex-valued activation functions.Lethq(zq(t))=hqR(xq(t),yq(t))+ihqI(xq(t),yq(t)) and mq(zq(t-τq))=mqR(xq(t-τq),yq(t-τq))+imqI(xq(t-τq),yq(t-τq)). For simplicity, set xq=xq(t), yq=yq(t), xqτq=xq(t-τq), yqτq=yq(t-τq), x~q=x~q(t), y~q=y~q(t), x~qτq=x~q(t-τq), and y~qτq=y~q(t-τq). The complex-valued activation functions hq(zq(t)) and mq(zq(t-τq)) are assumed to satisfy the following assumption.Assumption 1.
For anyxq, x~q, yq, y~q and xqτq, x~qτq, yqτq, y~qτq, there exist scalars κqRR, κqRI, κqIR, κqII≥0 and πqRR, πqRI, πqIR, πqII≥0 such that the following inequalities hold:(3)hqRx~q,y~q-hqRxq,yq≤κqRRx~q-xq+κqRIy~q-yqhqIx~q,y~q-hqIxq,yq≤κqIRx~q-xq+κqIIy~q-yqmqRx~qτq,y~qτq-mqRxqτq,yqτq≤πqRRx~qτq-xqτq+πqRIy~qτq-yqτqmqIx~qτq,y~qτq-mqIxqτq,yqτq≤πqIRx~qτq-xqτq+πqIIy~qτq-yqτq.Remark 2.
For the dynamic analysis of complex-valued recurrent neural networks models, the most results are based on the existence, continuity, and boundedness of the partial derivatives for real and imaginary parts [32, 33, 46, 49]. Then, by the mean value theorem, it can be proved that they satisfy the inequalities similar to those in Assumption 1. In fact, only the inequalities can be used in the process of obtaining the main results. Thus, the existence, continuity, and boundedness of the partial derivatives are unnecessary and they also lead to limitations in choosing complex-valued activation functions. Here, we remove these constraints and provide a more suitable assumption, Assumption 1, made on the activation functions, so that the current work can be applied to solve more engineering problems. This advantage can be seen from example 2.Remark 3.
It should be noticed that such an assumption is made on the activation functions in the literature [44, 45]; i.e., hq(z)=hqR(Re(z))+ihqI(Im(z)), mq(z)=mqR(Re(z))+imqI(Im(z)) for the complex number z and the following inequalities hold:(4)ξqR-≤hqRx1-hqRx2x1-x2≤ξqR+,ξqI-≤hqIy1-hqIy2y1-y2≤ξqI+ζqR-≤mqRx1-mqRx2x1-x2≤ζqR+,ζqI-≤mqIy1-mqIy2y1-y2≤ζqI+for all q=1,2,…,n, where z1=x1+y1 and z2=x2+y2. Obviously, this assumption is a special case of Assumption 1. Therefore, in this paper, we will not discuss this case in detail, but a numerical example for this case will be provided to show the effectiveness.In this article, the corresponding response system is defined by(5)z~˙pt=-dpz~pt+∑q=1napqhqz~qt+∑q=1nbpqmqz~qt-τq+Jpt+uptwith the initial conditions(6)z~pθ=ψ~pθ,θ∈-τ,0,p=1,2,…,n.Letzp(t)=xp(t)+iyp(t), Jp(t)=JpR(t)+iJpI(t), ψp(s)=ψpR(s)+iψpI(s), apq=apqR+iapqI, bpq=bpqR+ibpqI, z~p(t)=x~p(t)+iy~p(t), ψ~p(s)=ψ~pR(s)+iψ~pI(s), hq(z~q(t))=hqR(x~q(t),y~q(t))+ihqI(x~q(t),y~q(t)), mq(z~q(t-τq))=mqR(x~q(t-τq),y~q(t-τq))+imqI(x~q(t-τq),y~q(t-τq)), and up(t)=upR(t)+iupI(t). Then, by separating system (1) with (2) and system (5) with (6) into the real and imaginary parts, respectively, we have(7)x˙p=-dpxp+∑q=1napqRhqRxq,yq-∑q=1napqIhqIxq,yq+∑q=1nbpqRmqRxqτq,yqτq-∑q=1nbpqImqIxqτq,yqτq+JpRty˙p=-dpyp+∑q=1napqIhqRxq,yq+∑q=1napqRhqIxq,yq+∑q=1nbpqImqRxqτq,yqτq+∑q=1nbpqRmqIxqτq,yqτq+JpItxpθ=ψpRθ,ypθ=ψpIθ,θ∈-τ,0,p=1,2,…,nand(8)x~˙p=-dpx~p+∑q=1napqRhqRx~q,y~q-∑q=1napqIhqIx~q,y~q+∑q=1nbpqRmqRx~qτq,y~qτq-∑q=1nbpqImqIx~qτq,y~qτq+JpRt+upRty~˙p=-dpy~p+∑q=1napqIhqRx~q,y~q+∑q=1napqRhqIx~q,y~q+∑q=1nbpqImqRx~qτq,y~qτq+∑q=1nbpqRmqIx~qτq,y~qτq+JpIt+upItx~pθ=φ~pRθ,y~pθ=φ~pIθ,θ∈-τ,0,p=1,2,…,n.Define the error vector between the drive system (1) and the response system (5) as ep(t)=z~p(t)-zp(t)=epR(t)+iepI(t)=x~p(t)-xp(t)+i(y~p(t)-yp(t)); it follows from (7) and (8) that(9)e˙pRt=-dpepRt+∑q=1napqRhqRx~q,y~q-hqRxq,yq-∑q=1napqIhqIx~q,y~q-hqIxq,yq+∑q=1nbpqRmqRx~qτq,y~qτq-mqRxqτq,yqτq-∑q=1nbpqImqIx~qτq,y~qτq-mqIxqτq,yqτq+upRte˙pIt=-dpepIt+∑q=1napqIhqRx~q,y~q-hqRxq,yq+∑q=1napqRhqIx~q,y~q-hqIxq,yq+∑q=1nbpqImqRx~qτq,y~qτq-mqRxqτq,yqτq+∑q=1nbpqRmqIx~qτq,y~qτq-mqIxqτq,yqτq+upItepRθ=ϕpRθ,epIθ=ϕpIθ,θ∈-τ,0,p=1,2,…,n.Let ωpt=epR(t)epI(t); then equation (9) can be rewritten as(10)ω˙pt=-dp00dpωpt+∑q=1napqR-apqIapqIapqRhqRx~q,y~q-hqRxq,yqhqIx~q,y~q-hqIxq,yq+∑q=1nbpqR-bpqIbpqIbpqRmqRx~qτq,y~qτq-mqRxqτq,yqτqmqIx~qτq,y~qτq-mqIxqτq,yqτq+upRtupItωpθ=ϕpθ=ϕpRθϕpIθ,θ∈-τ,0for p=1,2,…,n.Before deriving the main results, the following definition and lemmas are provided.Definition 4 (see [27]).
Consider the drive-response systems (1) with (2) and (5) with (6). If for a suitable controller up(t)=upR(t)+iupI(t), there exists a function T=T(ϕp)>0 depending on the initial value ϕp, such that(11)limt→T⁡epRt=limt→T⁡epIt=0and epRt=epIt=0 for t>T, p=1,2,…,n, then the drive system (1) and the response system (5) achieve synchronization in finite time.Lemma 5 (see [24]).
An open setU∈C([-τ,0],R2n). The class G denotes all the strictly increasing continuous functions s:R+→R+ and s(0)=0. If there exist s,ϱ∈G and a continuous function V:[0,+∞)×U→R+ for system (10) such that(1)
V(t,0)=0, s(x)≤V(t,x),t∈[0,+∞),(2)
D+V(t,x)≤-ϱ(V(t,x)) with ∫0ϵ(dz/ϱ(z))<+∞,
for allϵ>0, x∈U, then, system (10) is finite-time stable and the settling time T≤∫0V(0,ϕ)(dz/ϱ(z)). In particular, if ϱ(V)=kVμ(k>0,0<μ<1), then the settling time(12)T≤∫0V0,ϕdzϱz=V1-μ0,ϕk1-μ.Lemma 6 (see [52]).
Ifa>0, b>0, 0<α1<1, 0<α2<1, then there must exist a scalar 0<α<1 depending on α1 and α2, such that(13)aα1+bα2>a+bα2α.Remark 7.
In order to deal with the finite-time stabilization problem for complex-valued neural networks system and estimate the upper bound of Dini-derivative of Lyapunov function, the inequality in Lemma6 was proposed and proved by us in Lemma 8 of [52]. Moreover, from the proof of the inequality stated in [52], we can see that either α=α1 or α=α2, which can be used to compute the settling time. This point can be shown in examples 1 and 2.Lemma 8 (see [24]).
For real numbersc1,c2,…,cn and 0<k1<k2, the following inequality holds:(14)∑q=1ncqk21/k2≤∑q=1ncqk11/k1.


## 3. Main Result

In this section, new sufficient conditions will be derived to ascertain the finite-time synchronization for system (1) and (5). The new nonlinear controller up(t) in response system (5) is designed as the following form:(15)upt=upRt+iupItupRt=-α1pepRt-β1pepRtμ1sgn⁡epRt-γ1p∑q=1neqRt-τqsgn⁡epRtupIt=-α2pepIt-β2pepItμ2sgn⁡epIt-γ2p∑q=1neqIt-τqsgn⁡epItwhere α1p, α2p, β1p>0, β2p>0, γ1p, γ2p, 0<μ1<1, and 0<μ2<1 are constants for p=1,2,…,n.Remark 9.
In this paper, we design the controller (15) with independent parameters μ1 and μ2, which are different from the most existing results based on μ1=μ2, such as [17, 18, 22]. This is one of the main features for our work, which also leads to difficulty in studying finite-time dynamic behavior of the considered system. To solve this difficulty, we use the inequality in Lemma 6 proposed and proved by us in [52].First, for convenience, we denote(16)κRR=diagκ1RR,κ2RR,…,κnRR,κRI=diagκ1RI,κ2RI,…,κnRI,AR=apqRn×n,κIR=diagκ1IR,κ2IR,…,κnIR,κII=diagκ1II,κ2II,…,κnII,AI=apqIn×n,πRR=diagπ1RR,π2RR,…,πnRR,πRI=diagπ1RI,π2RI,…,πnRI,BR=bpqRn×n,πIR=diagπ1IR,π2IR,…,πnIR,πII=diagπ1II,π2II,…,πnII,BI=bpqIn×n,Γ1=diagγ11,γ12,…,γ1n,Γ¯1=trΓ1En,Γ2=diagγ21,γ22,…,γ2n,Γ¯2=trΓ2En,En=diag1,1,…,1,D1=diagd1+α11,d2+α12,…,dn+α1n,D2=diagd1+α21,d2+α22,…,dn+α2n,In=1,1,…,1T,I=col⁡In,In,D=D100D2,Γ¯=Γ¯100Γ¯2,A=ARAIAIAR,B=BRBIBIBR,κ=κRRκRIκIRκII,π=πRRπRIπIRπII,G1=D-AκT,G2=Γ¯-BπT.Based on the controller (15), the main result of our paper can be obtained as follows.Theorem 10.
Under Assumption1, if the vectors G1I>0 and G2I>0, then via the controller (15), the drive system (1) and the response system (5) are synchronized in finite time. Moreover, the settling time satisfies T2≤(2/β(1-μ))V1-μ(0), where β=min⁡β1,β2, β1=min⁡β1p, β2=min⁡β2p, p=1,2,…,n, and 0<μ<1 depending on μ1 and μ2.Proof.
IfG1I>0 holds, then for all p=1,2,…,n, we have(17)dp+α1p-∑q=1naqpRκpRR+aqpIκpIR+aqpIκpRR+aqpRκpIR>0dp+α2p-∑q=1naqpIκpRI+aqpRκpII+aqpRκpRI+aqpIκpII>0.Similarly, if G2I>0 holds, then for all p=1,2,…,n, we have(18)∑q=1nγ1q-∑q=1nbqpRπpRR+bqpIπpIR+bqpIπpRR+bqpRπpIR>0∑q=1nγ2q-∑q=1nbqpRπpRI+bqpIπpII+bqpIπpRI+bqpRπpII>0.
For the error system (9) or (10), choose the following Lyapunov function:(19)Vt=∑p=1nepRt+∑p=1nepIt.
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.Then, by taking the common factors, one has(23)D+Vt≤∑p=1nepRt-dp+α1p+∑q=1naqpRκpRR+aqpIκpIR+aqpIκpRR+aqpRκpIR+∑p=1nepIt-dp+α2p+∑q=1naqpIκpRI+aqpRκpII+aqpRκpRI+aqpIκpII+∑p=1n∑q=1nbqpRπpRR+bqpIπpIR-γ1q+bqpIπpRR+bqpRπpIRepRt-τp+∑p=1n∑q=1nbqpRπpRI+bqpIπpII+bqpIπpRI+bqpRπpII-γ2qepIt-τp-∑p=1nβ1pepRtμ1-∑p=1nβ2pepItμ2.Then, by (17) and (18), one can obtain(24)D+Vt≤-∑p=1nβ1pepRtμ1-∑p=1nβ2pepItμ2≤-β1∑p=1nepRtμ1-β2∑p=1nepItμ2≤-β∑p=1nepRtμ1+∑p=1nepItμ2where β=min⁡β1,β2, β1=min⁡β1p, β2=min⁡β2p, p=1,2,…,n. In addition, by Lemma 8, for 0<μ1<1, 0<μ2<1, the following inequalities hold:(25)-∑p=1nepRtμ1≤-∑p=1nepRtμ1,-∑p=1nepItμ2≤-∑p=1nepItμ2.Then, substituting (25) into (24), one has(26)D+Vt≤-β∑p=1nepRtμ1+∑p=1nepItμ2.By Lemma 6, there exists a 0<μ<1 depending on μ1 and μ2, such that(27)D+Vt≤-β2μ∑p=1nepRt+∑p=1nepItμ≤-β2Vμt.Moreover, ∫0ε2/βzμdz=2ε1-μ/β(1-μ)<+∞, for all ε>0. Thus, from Lemma 5, we know that the error system (9) or (10) is finite-time stable. That is to say, the drive system (1) and the response system (5) can be synchronized in finite time via the controller (15). This completes the proof.Remark 11.
Recently, the finite-time stabilization and synchronization problems of real-valued neural networks have been investigated and the time for system trajectories tending to an equilibrium point is finite [22–29]. However, for finite-time stability analysis of complex-valued neural networks models, until now, few results have been developed. For instance, the finite-time synchronization problem for complex-valued neural networks (CVNNs) with infinite-time distributed delays is discussed in [51] and the finite-time stabilizability and instabilizability for complex-valued memristive neural networks with time delays are studied in [52]. Here, Theorem 10 further provides sufficient conditions for the finite-time synchronization of the considered system and their effectiveness can be illustrated in numerical examples. From the point of theory development, our result is significant in dynamical behavior analysis of complex-valued models and it can be regarded as the extension of the previous work.


## 4. Numerical Examples

This section will provide two examples to show the effectiveness of the given conditions of Theorem10.Example 12.
Consider complex-valued neural networks as the drive system given by(28)z˙pt=-dpzpt+∑q=12apqhqzqt+∑q=12bpqmqzqt-τq+Jpt,p=1,2where d1=1.5, d2=1, a11=1+2i, a12=-2-i, a21=-2.5-i, a22=-1-0.5i, b11=2-i, b12=-1+i, b21=-1+2i, b22=2+i, τ1=0.1, τ2=0.2, J1(t)=6sin⁡(t+1)-4cos⁡(t-1)i, and J2(t)=5cos⁡t+1-6sin⁡(t-1)i. Moreover, the activation functions are taken as(29)hqzq=1-e-xq1+e-xq+i11+e-yq,mqzq=1-e-yq1+e-yq+i11+e-xq,q=1,2.Then, from Assumption 1, it can be computed that κqRR=0.5, κqII=0.25, κqRI=κqIR=0, πqRI=0.5, πqIR=0.25, and πqRR=πqII=0 for q=1,2.The response system is provided by(30)z~˙pt=-dpz~pt+∑q=12apqhqz~qt+∑q=12bpqmqz~qt-τq+Jpt+upt.Under the initial conditionsz1(θ)=-2+2.4i, z2(θ)=-1-1.4i, z~1(θ)=-2.3-1.8i, and z~2(θ)=1.2-i for θ∈[-0.2,0], Figures 1 and 2 depict the trajectories of systems (28) and (30) without controller, which show that the drive system (28) does not synchronize with the response system (30).Figure 1
Time responses of the variablesz1, z2, z~1, and z~2 in 3D and 2D spaces without controller for example 1.Figure 2
Time responses of real and imaginary parts of the variablesz1, z2, z~1, and z~2 without controller for example 1.By choosing the parameters of the controller (15) as α11=2.3, α12=1.6, α21=1.1, α22=0.8, β11=β12=β21=β22=1, γ11=1, γ12=0.6, γ21=1.5, γ22=1.6, μ1=0.5, and μ2=0.6, it is easy to check that the conditions of Theorem 10 are satisfied. Then, under the corresponding controller, the drive system (28) can be synchronized with the response system (30) in finite time. In addition, according to Theorem 10 and Remark 7, it can be computed that T1≤max⁡10.88,10.95=10.95 seconds. The synchronization curves between the drive system (28) and the response system (30) are displayed in Figure 3. The time responses of the synchronization errors between them are shown in Figure 4. From Figures 3 and 4, it can be seen that system (28) can synchronize with system (30) in finite time via the corresponding controller under the given initial values. These also further show that the obtained result is effective and our work is meaningful.Figure 3
Synchronization curves of real and imaginary parts ofz1, z2, z~1, and z~2 under controller for example 1.Figure 4
Time responses of synchronization errorse1 and e2 under controller for example 1.Next, one example with special activation functions stated as in Remarks2 and 3 is given to illustrate the validity of our result.Example 13.
Consider a drive system as follows:(31)z˙pt=-dpzpt+∑q=12apqhqzqt+∑q=12bpqmqzqt-τq+Jpt,p=1,2where d1=1, d2=0.5, a11=-0.5+i, a12=2-i, a21=1.5-1.8i, a22=-1+2i, b11=-0.8-0.5i, b12=-1+1.5i, b21=2+1.5i, b22=-1+2i, τ1=0.1, τ2=0.2, J1(t)=3-i, and J2(t)=-2+5i. The activation functions are chosen as(32)hqzq=mqzq=xq+1-xq-12+iyq+1-yq-12,q=1,2.Obviously, (∂/∂xq)((xq+1-xq-1)/2)xq=±1 and (∂/∂yq)((yq+1-yq-1)/2)yq=±1 do not exist. Hence, the obtained results in many references, such as [32, 33, 46, 49], cannot be applied in these functions. This leads to limitations for their results in applications whereas our work has just filled the gap. Moreover, from Assumption 1, it is easy to obtain that κqRR=κqII=πqRR=πqII=1, κqRI=κqIR=πqRI=πqIR=0 for q=1,2.The response system is given by(33)z~˙pt=-dpz~pt+∑q=12apqhqz~qt+∑q=12bpqmqz~qt-τq+Jpt+upt.Under the initial conditionsz1(θ)=0.8-0.9i, z2(θ)=1.1-i, z~1(θ)=-1+i, and z~2(θ)=-0.6+1.1i for θ∈[-0.2,0], Figures 5 and 6 depict the trajectories of systems (31) and (33) without controller, which shows that the drive system (31) does not synchronize with the response system (33).Figure 5
Time responses of the variablesz1, z2, z~1, and z~2 in 3D and 2D spaces without controller for example 2.Figure 6
Time responses of real and imaginary parts of the variablesz1, z2, z~1, and z~2 without controller for example 2.By choosing the parameters of the controller (15) as α11=4.2, α12=6.5, α21=4.4, α22=5.9, β11=β12=β21=β22=1, γ11=2.5, γ12=3.1, γ21=2, γ22=3.6, μ1=0.5, μ2=0.6, it is easy to check that the conditions of Theorem 10 are satisfied. Thus, under the corresponding controller, the drive system (31) can be synchronized with the response system (33) in finite time. Moreover, according to Theorem 10 and Remark 7, we get T2≤max⁡10.95,11.19=11.19 seconds by simple calculation. The synchronization curves between the drive system (31) and the response system (33) are shown in Figure 7. The time responses of the synchronization errors between them are depicted in Figure 8. From Figures 7 and 8, it can be seen that system (31) can synchronize with system (33) in finite time via the corresponding controller under the given initial values. This example also further illustrates the effectiveness and superiority of our proposed result.Figure 7
Synchronization curves of real and imaginary parts ofz1, z2, z~1, and z~2 under controller for example 2.Figure 8
Time responses of synchronization errorse1 and e2 under controller for example 2.


## 5. Conclusion

In this paper, the finite-time synchronization problem of complex-valued recurrent neural networks with time delays has been studied. Based on the more general assumptions for activation functions, a nonlinear controller with independent parameters, and a new inequality proposed and proved by us in [52], some sufficient conditions have been established and the settling time for synchronization has been estimated. The obtained results have been shown to be effective and superior by two examples. Moreover, it is well known that the systems with stochastic terms are more extensive in practical applications [56–60], such as the noises, and we will discuss the addressed models with noises deeply in the future.



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