# Predefined-Time Antisynchronization of Two Different Chaotic Neural Networks

Authors: Lixiong Lin
Journal: Complexity
 (2020)

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## Abstract

This paper is concerned with antisynchronization in predefined time for two different chaotic neural networks. Firstly, a predefined-time stability theorem based on Lyapunov function is proposed. With the help of the definition of predefined time, it is convenient to establish a direct relationship between the tuning gain of the system and the fixed stabilization time. Then, the antisynchronization is achieved between two different chaotic neural networks via active control Lyapunov function design. The designed controller presents the practical advantage that the least upper bound for the settling time can be explicitly defined during the control design. With the help of the designed controller, the antisynchronization errors converge within a predefined-time period. Numerical simulations are presented in order to show the reliability of the proposed method.


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## Body

## 1. Introduction

Synchronization and control is one of the most important research topics in the last decades due to its wide applications in engineering [1–4], including secure communications [3] and cryptograph [4], and a variety of complex chemical and physical systems. The primary purpose of synchronization is to achieve the master and the slave systems coupled by designing an appropriate controller. The history of chaos synchronization could be traced back to the research of the interaction between coupled chaotic systems [5–7]. Since then, many different types of synchronization for chaotic systems have been proposed in the references, such as generalized synchronization [8], complete synchronization [7], lag synchronization, combination synchronization [9], and projective synchronization [10]. In practice, there also exists an interesting phenomenon in symmetrical oscillators and antisynchronization [11–15], which means that the absolute value of the state vectors between two synchronized systems is equal, but their signs are opposite. According to the related references [16, 17], antisynchronization has important applications in the communication system and laser field. Therefore, how to solve the problems of synchronization and antisynchronization in different chaotic neural networks is also an open issue to consider [12, 18].In order to realize the synchronization of two chaotic systems, several synchronization methods have been developed such as adaptive control [19], sliding mode control [20], backstepping method control [21], active control [2], and impulsive control [22]. However, it is worth to remark that although these methods provide finite-time synchronization [23–25], the convergence time is lack of uniform boundedness. And the convergence time of finite-time synchronization depends on the initial values of two chaotic systems. In other words, if the initial values of two chaotic systems are unknown, the convergence time cannot be estimated in advance. A new stability named fixed-time stability was proposed in [26] to solve this problem. The convergence time of fixed-time stability could be globally uniformly bounded without the initial values of the system [27–30]. In order to estimate the convergence time more accurately, Xu et al. [28] proposed a simple controller without sign function and linear parts to achieve complex networks fixed-time synchronization. In [29], the inverse function of the Lyapunov function was used to realize the synchronization in finite time and fixed time. To achieve fixed-time synchronization for a class of complex networks with impulsive effects, a controller, which was continuous and did not include the sign function, was designed by time-varying Lyapunov functions and convex combination techniques in [30]. Nevertheless, the fixed-time stability also has its disadvantages. Due to the difficulties in theoretical analysis, the convergence time cannot be accurately estimated.For several engineering applications, such as control systems and secure communications, it is required that the system reaches the origin within a predefined time. In order to assure synchronization between two chaotic systems within a predefined time, an active controller based on predefined-time stability was proposed in [31]. Predefined-time stability is a special kind of fixed-time stability with the settling time that can be defined in advance as a parameter of the control design [32–34]. Up to present, the research on predefined-time stability is still in its infancy. Sanchez-Torres et al. [35] proposed first-order controllers based on predefined-time sliding mode for a class of dynamical systems. Muñoz-Vázquez et al. [36] proposed a controller based on second-order predefined-time stability to solve the problem of fully exact tracking of actuated mechanical systems. To the best of our knowledge, there is still no research on predefined-time antisynchronization of two different chaotic neural networks. Motivated by the above discussions, based on [31], a predefined-time stability theorem for antisynchronization of two different chaotic neural networks is proposed in this paper. The main contributions of this paper are summarized in the following two aspects:(1)
A predefined-time stability theorem by Lyapunov function in [37] is proposed, the advantage of which is that it is more general and the predefined-time stability theorem in [31] is a special case of it. And its predefined-time stability criterion for the predefined-time is different from those in [31–36].(2)
Different from [14], this paper investigates the predefined-time antisynchronization schemes for two different chaotic neural networks, which assure the antisynchronization execution before a time that is predefined as a parameter during the control design.This paper is organized as follows. Useful results in relation to the predefined-time stability and other preliminaries are presented in Section2. In Section 3, a predefined-time stability theorem based on Lyapunov function is proposed, which is applied to achieve two different chaotic neural networks antisynchronized. Section 4 gives some simulation results, and finally, some conclusions are drawn in Section 5.


## 2. Preliminaries

Some basis on fixed-time stability and predefined-time stability are addressed below. Consider the system(1)x˙=fx;r,where x∈Rn is the system state. r∈Rb with r˙=0 represents the parameters of system (1). f:R⟶Rn is nonlinear function. For this system, the initial condition is x0=x0.Definition 1 (see [26]).
The origin is said to be a globally fixed-time stable equilibrium point for system (1) if it is globally finite-time stable and the settling time function is bounded and independent of the initial conditions, i.e.,∃Tmax>0: ∀x0∈Rn, Tx0≤Tmax.Definition 2 (see [38]).
For a predefined constantTc>0, the origin of system (1) is said to be predefined-time stable if it is fixed-time stable and the settling time function T:Rn⟶R is such that(2)Tx0≤Tc,∀x0∈Rn.
If this is the case,Tc is called a predefined time.Remark 1.
Let the origin of system (1) be fixed-time stable. Notice that it is difficult to find an explicit relationship between the system parameters and the fixed-time Tmax. In some cases, it is difficult to reduce time Tmax even by tuning system parameters. The predefined-time Tc, which is defined during the control design, can solve the above problems, so system (1) can achieve stability before the predefined-time Tc.Lemma 1 (inequalities [39]).
Ifz1,z2,…,zn≥0, 0<θ≤1, γ>1, then it gets that(3)∑i=1nziθ≥∑i=1nziθ,∑i=1nziγ≥n1−γ∑i=1nziγ.Lemma 2 (see [26]).
SupposingV⋅:Rn⟶R+∪0 is a continuous strictly monotonically decreased function and satisfies the following:(1)
Vx=0⟹x∈M, where M∈Rn is a nonempty set and said to be globally fixed time attractive for system (1).(2)
For anyVx>0, there exist α,η>0, p>1, and 0<q<1 such that(4)V˙≤−αVp+ηVq.
Then, system (1) can realize globally fixed-time stability and(5)T=1α1−p+1η1−q.Lemma 3 (see [31]).
Consider system (1) and Tc∈r1,…,rb>0 a user-defined parameter. Supposing V⋅:Rn⟶R+∪0 is a continuous strictly monotonically decreased function and satisfies(6)V˙≤−πgTcV1−g/2+V1+g/2for g∈0,1. Then, system (1) can realize globally predefined-time stability with the time Tc.


## 3. Main Results

### 3.1. Predefined-Time Stability

Theorem 1.
SupposingV⋅:Rn⟶R+∪0 is a continuous strictly monotonically decreased function and satisfies the following:(1)
Vx=0⟹x∈M, where M∈Rn is a nonempty set and said to be globally fixed time attractive for system (1).(2)
Tc∈r1,…,rb>0 is a user-defined parameter.(3)
For anyVx>0, there exist α,η,k,p,q>0, pk>1, and 0<qk<1 such that(7)V˙≤−CvTcαVp+ηVqk,where(8)Cv=11−qk2γ−1kα1/γγk−1η1−γk/γ,ifγ>1,11−qk1α1/γγk−1η1−γk/γ,if0<γ≤1,γ=p−q1−qk.
Then, system (1) is globally predefined-time stable, with predefined-time Tc.Proof.
As we know, for anyVx>0, it gets that(9)V˙≤−CvTcαVp+ηVqk=−VqkCvTcαVp−q+ηk.
SinceVqk>0, it follows that(10)1VqkdVdt≤−CvTcαVp−q+ηk.
Letz=V1−qk, and equation (10) can be written as(11)dzdt≤−CvTc1−qkαVp−q+ηk=−CvTc1−qkαzγ+ηk.
The settling time function is(12)Tx0=∫0Tx0dt.
Furthermore, since the functionVx is continuous and strictly monotonically decreased, one has that(13)Tx0=∫0Tx0dt≤−∫zx01/1−qk0TcCv11−qk⋅1αzγ+ηkdz.
The following proof is divided in two cases.Case 1. Ifγ>1, by Lemma 1, it gets that(14)αzγ+η≥21−γα1/γz+η1/γγ.Thus,(15)Tx0≤∫0zx01/1−qkTcCv11−qk⋅121−γk1α1/γz+η1/γγkdz.Then,(16)Tx0≤TcCv11−qk2γ−1kα1/γ1−γk⋅α1/γz+η1/γ1−γk0zx01/1−qk=TcCv11−qk2γ−1kα1/γ1−γk⋅α1/γVx0+η1/γ1−γk−η1−γk/γ.By virtue of(17)γk=p−q1−qk⋅k=1+pk−11−qk>1.Thus,(18)Tx0≤TcCv11−qk2γ−1kα1/γγk−1⋅η1−γk/γ−α1/γVx0+η1/γ1−γk.IfVx0=0, one can obtain that(19)limVx0⟶0Tx0=0.IfVx0⟶∞, one can obtain that(20)Tx0≤TcCv11−qk2γ−1kα1/γγk−1⋅η1−γk/γ−α1/γVx0+η1/γ1−γk=TcCv11−qk2γ−1kα1/γγk−1⋅η1−γk/γ−1α1/γVx0+η1/γγk−1=TcCv11−qk2γ−1kα1/γγk−1η1−γk/γ=Tc.It is clear that(21)limVx0⟶∞Tx0=Tc.Case 2. If0<γ≤1, by Lemma 1, it gets that(22)αzγ+η≥α1/γz+η1/γγ,and thus,(23)Tx0≤∫0zx01/1−qkTcCv11−qk1α1/γz+η1/γγkdz.Then,(24)Tx0≤TcCv11−qk1α1/γ1−γk⋅α1/γz+η1/γ1−γk0zx01/1−qk=TcCv11−qk1α1/γγk−1⋅η1−γk/γ−α1/γVx0+η1/γ1−γk.
IfVx0=0, one can obtain that(25)limVx0⟶0Tx0=0.
IfVx0⟶∞, one can obtain that(26)Tx0≤TcCv11−qk1α1/γγk−1η1−γk/γ−α1/γVx0+η1/γ1−γk=TcCv11−qk1α1/γγk−1η1−γk/γ−1α1/γVx0+η1/γγk−1=TcCv11−qk1α1/γγk−1η1−γk/γ=Tc.
It is clear that(27)limVx0⟶∞Tx0=Tc.
This proof is completed.Corollary 1.
SupposingV⋅:Rn⟶R+∪0 is a continuous strictly monotonically decreased function and satisfies the following:(1)
Vx=0⟹x∈M, where M∈Rn is a nonempty set and said to be globally fixed time attractive for system (1).(2)
Tc∈r1,…,rb>0 is a user-defined parameter.(3)
For anyVx>0, there exist α,η>0, p>1, and 0<q<1 such that(28)V˙≤−CvTcαVp+ηVq,where(29)Cv=11−q2γ−1α1/γγ−1η1−γ/γ,γ=1+p−11−q.
Then, system (1) is globally predefined-time stable, with predefined-time Tc.Proof.
Proof of Corollary 1 in Appendix A
Compared with Lemma2, predefined-time stability can be achieved in Corollary 1.Remark 2.
In Corollary1, α, η, p, and q are required to satisfy α,η>0, p>1, and 0<q<1. However, in Lemma 3, α, η, p, and q should satisfy α=η=π/gCv, g=p−q, and p+q=2, which will constrain the application of Lemma 3 in some scenarios.Remark 3.
Different from the proof of Lemma3, the proof of Theorem 1 uses Lemma 1 to solve the inequality problem. In order to reflect the convergence time of system (1) more intuitively, the convergence of Lyapunov function is regarded as an implicit inverse function of time. Furthermore, using these techniques, it easily derives the results of Lemma 2, Lemma 3, and Corollary 1.


### 3.2. Controller Design for Predefined-Time Antisynchronization

In this subsection, the predefined-time antisynchronization of two different delayed chaotic neural networks is realized by using active control strategy. Consider the following delayed chaotic neural network(30)x˙it=−aixit+∑j=1nbijgjxjt+∑j=1ncijgjxjt−τ,i=1,2,…,n, where xit represents the state of the ith neuron; ai>0 denotes the self-inhibition; gj⋅ is the activation function; bij∈R stands for the connection weight; cij∈R stands the delayed connection weight; and τt represents the discrete delay and satisfies 0≤τt≤τ.Let (30) be the master delayed chaotic neural network, and the slave delayed chaotic neural network is given by(31)y˙it=−diyit+∑j=1nhijfjyjt+∑j=1nmijfjyjt−τ+uit,i=1,2,…,n, where yit represents the state of the i-th neuron; di>0 denotes the self-inhibition; fj⋅ denotes the neuron activation function; hij∈R stands for the connection weight; mij∈R stands the delayed connection weight; and uit is a controller to be designed late.Assumption 1.
(see [40]). The neuron activation functions in system (30) and (31) satisfy(32)gj⋅≤Nj,fj⋅≤Mj,where Nj>0 and Mj>0, j=1,2,…,n.Theorem 2.
The slave delayed chaotic neural network (31) can achieve predefined-time antisynchronization with the master delayed chaotic neural network (30) via the control law as below:Ifk>1,(33)uit=−k1iei−k2isignei−2k−1Cv2Tcαisigneieipk+ηisigneieiqk−di−aixi.If0<k≤1,(34)uit=−k1iei−k2isignei−Cv2Tcαisigneieipk+ηisigneieiqk−di−aixi.Proof.
The antisynchronization error system between (30) and (31) can be defined as(35)eit=yit+xit.
According to the error system (35) of master-slave delayed chaotic neural networks (30) and (31), it gets that(36)e˙it=−diyi+∑j=1nhijfjyj+∑j=1nmijfjyjt−τ+uit+−aixi+∑j=1nbijgjxj+∑j=1ncijgjxjt−τ=−diei+∑j=1nhijfjyj+∑j=1nmijfjyjt−τ+uit+di−aixi+∑j=1nbijgjxj+∑j=1ncijgixjt−τ.
To simplify the formula, the variableG is defined as follows:(37)G=∑j=1nhijfjyj+∑j=1nmijfjyjt−τ+∑j=1nbijgjxj+∑j=1ncijgixjt−τ.
If Assumption1 holds, it can be derived that(38)G≤G≤∑j=1nhijM+∑j=1nmijM+∑j=1nbijN+∑j=1ncijN=∑j=1nhij+mijM+∑j=1nbij+cijN.Thus,(39)e˙it=−diei+di−aixi+G+ui.
Consider the candidate Lyapunov function(40)Vet=et=∑i=1neit,where et=e1t,e2t,…,entT. Then, the derivative of the candidate Lyapunov function is(41)V˙et=∑i=1nsigneie˙i=∑i=1nsignei−diei+di−aixi+G+ui.
Ifk>1, according to the controller (33), it gets that(42)V˙et=∑i=1nsigneie˙i=∑i=1nsignei−diei+di−aixi+G+−k1iei−k2isignei−2k−1Cv2Tcαisigneieipk+ηisigneieiqk−di−aixi≤−∑i=1nsigneidi+k1iei+∑i=1nsigneik2i+∑j=1nhij+mijM+∑j=1nbij+cijN−2k−1CvTc∑i=1nαieipk+∑i=1nηieiqk.
Sincek1i≥−di and k2i≤−∑j=1nhij+mijM−∑j=1nbij+cijN, it follows that(43)V˙et≤−2k−1CvTc∑i=1nαieipk+∑i=1nηieiqk.
Letα=minα1,α2,…,αn, η=minη1,η2,…,ηn, and by Lemma 1, then(44)V˙et≤−2k−1CvTcαVetpk+ηVetqk≤−CvTcαVetp+ηVetqk.
If0<k≤1, according to the controller (34), a similar proof can be obtained:(45)V˙et=∑i=1nsignei−diei+di−aixi+G+−k1iei−k2isignei−Cv2Tcαisigneieipk+ηisigneieiqk−di−aixi≤−CvTcαVetp+ηVetqk.
Thus, in accordance to Theorem1, the master delayed chaotic neural network (30) and the slave delayed chaotic neural network (31) can achieve predefined-time antisynchronization under the controller (33) (if k > 1) or (34) (if 0 < k≤ 1).
This proof is completed.Remark 4.
Theorem2 states that two different chaotic neural networks can achieve antisynchronization within a predefined-time Tc. Compared to other antisynchronization of two different chaotic neural networks, such as asymptotic antisynchronization, finite-time antisynchronization, or fixed-time antisynchronization, the new predefined-time antisynchronization Theorem 2 can provide a tuning stable time Tc, which could be tuned according to the actual needs of the antisynchronization of neural networks.
Li et al. [22] realized the exponential synchronization of dynamical networks by hybrid pinning impulsive controller, and the convergence time was uncertain. In [30], fixed-time synchronization of complex networks with impulsive effects was realized by nonchattering control, but the convergence time could not be accurately estimated. Theorem 2 can achieve the predefined-time antisynchronization of two different chaotic networks with the convergence time explicitly defined during the control design. Theorem 2 also has many applications, such as communication system or image field.(1)
By using predefined-time antisynchronization in communication system, one may transmit digital signals continuously by setting the synchronization and antisynchronization time in advance, which will strengthen the security and secrecy.(2)
In image secure communication, due to the complex behavior of neural networks, it is possible to add a predefined-time antisynchronization of neural networks in the process of image encryption and decryption to build a strong encryption and decryption model.


## 3.1. Predefined-Time Stability

Theorem 1.
SupposingV⋅:Rn⟶R+∪0 is a continuous strictly monotonically decreased function and satisfies the following:(1)
Vx=0⟹x∈M, where M∈Rn is a nonempty set and said to be globally fixed time attractive for system (1).(2)
Tc∈r1,…,rb>0 is a user-defined parameter.(3)
For anyVx>0, there exist α,η,k,p,q>0, pk>1, and 0<qk<1 such that(7)V˙≤−CvTcαVp+ηVqk,where(8)Cv=11−qk2γ−1kα1/γγk−1η1−γk/γ,ifγ>1,11−qk1α1/γγk−1η1−γk/γ,if0<γ≤1,γ=p−q1−qk.
Then, system (1) is globally predefined-time stable, with predefined-time Tc.Proof.
As we know, for anyVx>0, it gets that(9)V˙≤−CvTcαVp+ηVqk=−VqkCvTcαVp−q+ηk.
SinceVqk>0, it follows that(10)1VqkdVdt≤−CvTcαVp−q+ηk.
Letz=V1−qk, and equation (10) can be written as(11)dzdt≤−CvTc1−qkαVp−q+ηk=−CvTc1−qkαzγ+ηk.
The settling time function is(12)Tx0=∫0Tx0dt.
Furthermore, since the functionVx is continuous and strictly monotonically decreased, one has that(13)Tx0=∫0Tx0dt≤−∫zx01/1−qk0TcCv11−qk⋅1αzγ+ηkdz.
The following proof is divided in two cases.Case 1. Ifγ>1, by Lemma 1, it gets that(14)αzγ+η≥21−γα1/γz+η1/γγ.Thus,(15)Tx0≤∫0zx01/1−qkTcCv11−qk⋅121−γk1α1/γz+η1/γγkdz.Then,(16)Tx0≤TcCv11−qk2γ−1kα1/γ1−γk⋅α1/γz+η1/γ1−γk0zx01/1−qk=TcCv11−qk2γ−1kα1/γ1−γk⋅α1/γVx0+η1/γ1−γk−η1−γk/γ.By virtue of(17)γk=p−q1−qk⋅k=1+pk−11−qk>1.Thus,(18)Tx0≤TcCv11−qk2γ−1kα1/γγk−1⋅η1−γk/γ−α1/γVx0+η1/γ1−γk.IfVx0=0, one can obtain that(19)limVx0⟶0Tx0=0.IfVx0⟶∞, one can obtain that(20)Tx0≤TcCv11−qk2γ−1kα1/γγk−1⋅η1−γk/γ−α1/γVx0+η1/γ1−γk=TcCv11−qk2γ−1kα1/γγk−1⋅η1−γk/γ−1α1/γVx0+η1/γγk−1=TcCv11−qk2γ−1kα1/γγk−1η1−γk/γ=Tc.It is clear that(21)limVx0⟶∞Tx0=Tc.Case 2. If0<γ≤1, by Lemma 1, it gets that(22)αzγ+η≥α1/γz+η1/γγ,and thus,(23)Tx0≤∫0zx01/1−qkTcCv11−qk1α1/γz+η1/γγkdz.Then,(24)Tx0≤TcCv11−qk1α1/γ1−γk⋅α1/γz+η1/γ1−γk0zx01/1−qk=TcCv11−qk1α1/γγk−1⋅η1−γk/γ−α1/γVx0+η1/γ1−γk.
IfVx0=0, one can obtain that(25)limVx0⟶0Tx0=0.
IfVx0⟶∞, one can obtain that(26)Tx0≤TcCv11−qk1α1/γγk−1η1−γk/γ−α1/γVx0+η1/γ1−γk=TcCv11−qk1α1/γγk−1η1−γk/γ−1α1/γVx0+η1/γγk−1=TcCv11−qk1α1/γγk−1η1−γk/γ=Tc.
It is clear that(27)limVx0⟶∞Tx0=Tc.
This proof is completed.Corollary 1.
SupposingV⋅:Rn⟶R+∪0 is a continuous strictly monotonically decreased function and satisfies the following:(1)
Vx=0⟹x∈M, where M∈Rn is a nonempty set and said to be globally fixed time attractive for system (1).(2)
Tc∈r1,…,rb>0 is a user-defined parameter.(3)
For anyVx>0, there exist α,η>0, p>1, and 0<q<1 such that(28)V˙≤−CvTcαVp+ηVq,where(29)Cv=11−q2γ−1α1/γγ−1η1−γ/γ,γ=1+p−11−q.
Then, system (1) is globally predefined-time stable, with predefined-time Tc.Proof.
Proof of Corollary 1 in Appendix A
Compared with Lemma2, predefined-time stability can be achieved in Corollary 1.Remark 2.
In Corollary1, α, η, p, and q are required to satisfy α,η>0, p>1, and 0<q<1. However, in Lemma 3, α, η, p, and q should satisfy α=η=π/gCv, g=p−q, and p+q=2, which will constrain the application of Lemma 3 in some scenarios.Remark 3.
Different from the proof of Lemma3, the proof of Theorem 1 uses Lemma 1 to solve the inequality problem. In order to reflect the convergence time of system (1) more intuitively, the convergence of Lyapunov function is regarded as an implicit inverse function of time. Furthermore, using these techniques, it easily derives the results of Lemma 2, Lemma 3, and Corollary 1.


## 3.2. Controller Design for Predefined-Time Antisynchronization

In this subsection, the predefined-time antisynchronization of two different delayed chaotic neural networks is realized by using active control strategy. Consider the following delayed chaotic neural network(30)x˙it=−aixit+∑j=1nbijgjxjt+∑j=1ncijgjxjt−τ,i=1,2,…,n, where xit represents the state of the ith neuron; ai>0 denotes the self-inhibition; gj⋅ is the activation function; bij∈R stands for the connection weight; cij∈R stands the delayed connection weight; and τt represents the discrete delay and satisfies 0≤τt≤τ.Let (30) be the master delayed chaotic neural network, and the slave delayed chaotic neural network is given by(31)y˙it=−diyit+∑j=1nhijfjyjt+∑j=1nmijfjyjt−τ+uit,i=1,2,…,n, where yit represents the state of the i-th neuron; di>0 denotes the self-inhibition; fj⋅ denotes the neuron activation function; hij∈R stands for the connection weight; mij∈R stands the delayed connection weight; and uit is a controller to be designed late.Assumption 1.
(see [40]). The neuron activation functions in system (30) and (31) satisfy(32)gj⋅≤Nj,fj⋅≤Mj,where Nj>0 and Mj>0, j=1,2,…,n.Theorem 2.
The slave delayed chaotic neural network (31) can achieve predefined-time antisynchronization with the master delayed chaotic neural network (30) via the control law as below:Ifk>1,(33)uit=−k1iei−k2isignei−2k−1Cv2Tcαisigneieipk+ηisigneieiqk−di−aixi.If0<k≤1,(34)uit=−k1iei−k2isignei−Cv2Tcαisigneieipk+ηisigneieiqk−di−aixi.Proof.
The antisynchronization error system between (30) and (31) can be defined as(35)eit=yit+xit.
According to the error system (35) of master-slave delayed chaotic neural networks (30) and (31), it gets that(36)e˙it=−diyi+∑j=1nhijfjyj+∑j=1nmijfjyjt−τ+uit+−aixi+∑j=1nbijgjxj+∑j=1ncijgjxjt−τ=−diei+∑j=1nhijfjyj+∑j=1nmijfjyjt−τ+uit+di−aixi+∑j=1nbijgjxj+∑j=1ncijgixjt−τ.
To simplify the formula, the variableG is defined as follows:(37)G=∑j=1nhijfjyj+∑j=1nmijfjyjt−τ+∑j=1nbijgjxj+∑j=1ncijgixjt−τ.
If Assumption1 holds, it can be derived that(38)G≤G≤∑j=1nhijM+∑j=1nmijM+∑j=1nbijN+∑j=1ncijN=∑j=1nhij+mijM+∑j=1nbij+cijN.Thus,(39)e˙it=−diei+di−aixi+G+ui.
Consider the candidate Lyapunov function(40)Vet=et=∑i=1neit,where et=e1t,e2t,…,entT. Then, the derivative of the candidate Lyapunov function is(41)V˙et=∑i=1nsigneie˙i=∑i=1nsignei−diei+di−aixi+G+ui.
Ifk>1, according to the controller (33), it gets that(42)V˙et=∑i=1nsigneie˙i=∑i=1nsignei−diei+di−aixi+G+−k1iei−k2isignei−2k−1Cv2Tcαisigneieipk+ηisigneieiqk−di−aixi≤−∑i=1nsigneidi+k1iei+∑i=1nsigneik2i+∑j=1nhij+mijM+∑j=1nbij+cijN−2k−1CvTc∑i=1nαieipk+∑i=1nηieiqk.
Sincek1i≥−di and k2i≤−∑j=1nhij+mijM−∑j=1nbij+cijN, it follows that(43)V˙et≤−2k−1CvTc∑i=1nαieipk+∑i=1nηieiqk.
Letα=minα1,α2,…,αn, η=minη1,η2,…,ηn, and by Lemma 1, then(44)V˙et≤−2k−1CvTcαVetpk+ηVetqk≤−CvTcαVetp+ηVetqk.
If0<k≤1, according to the controller (34), a similar proof can be obtained:(45)V˙et=∑i=1nsignei−diei+di−aixi+G+−k1iei−k2isignei−Cv2Tcαisigneieipk+ηisigneieiqk−di−aixi≤−CvTcαVetp+ηVetqk.
Thus, in accordance to Theorem1, the master delayed chaotic neural network (30) and the slave delayed chaotic neural network (31) can achieve predefined-time antisynchronization under the controller (33) (if k > 1) or (34) (if 0 < k≤ 1).
This proof is completed.Remark 4.
Theorem2 states that two different chaotic neural networks can achieve antisynchronization within a predefined-time Tc. Compared to other antisynchronization of two different chaotic neural networks, such as asymptotic antisynchronization, finite-time antisynchronization, or fixed-time antisynchronization, the new predefined-time antisynchronization Theorem 2 can provide a tuning stable time Tc, which could be tuned according to the actual needs of the antisynchronization of neural networks.
Li et al. [22] realized the exponential synchronization of dynamical networks by hybrid pinning impulsive controller, and the convergence time was uncertain. In [30], fixed-time synchronization of complex networks with impulsive effects was realized by nonchattering control, but the convergence time could not be accurately estimated. Theorem 2 can achieve the predefined-time antisynchronization of two different chaotic networks with the convergence time explicitly defined during the control design. Theorem 2 also has many applications, such as communication system or image field.(1)
By using predefined-time antisynchronization in communication system, one may transmit digital signals continuously by setting the synchronization and antisynchronization time in advance, which will strengthen the security and secrecy.(2)
In image secure communication, due to the complex behavior of neural networks, it is possible to add a predefined-time antisynchronization of neural networks in the process of image encryption and decryption to build a strong encryption and decryption model.


## 4. Numerical Simulations

In this section, some simulation results are used to verify the correctness of our conclusions.


### 4.1. Predefined-Time Antisynchronization of Two Different Chaotic Neural Networks

Consider the following delayed chaotic neural network:(46)xi.t=−aixit+∑j=1nbijgjxjt+∑j=1ncijgjxjt−τt.Let (46) be the master delayed chaotic neural network, the slave delayed chaotic neural network is designed as(47)yi.t=−diyit+∑j=1nhijfjyjt+∑j=1nmijfjyjt−τt+uit,where i,j=1,2,…,n, denotes the number of neurons. In this example, we set n=2, and some parameters are set as follows: a1=a2=1.0, b11=2, b12=−0.1, b21=−5, b22=3, c11=−1.5, c12=−0.1, c21=−0.2, c22=−2.5, d1=d2=1.0, h11=0.02, h12=−0.01, h21=0.02, h22=0.1, m11=−0.01, m12=0.01, m21=0.01, m22=0.05, τt=1, p=2/3, q=2, and(48)gxt=tanhx1ttanhx2t,fyt=tanh0.2∗y1ttanh0.4∗y2t.The initial values of the master delayed chaotic neural network (46) are xt=2,−6T,t∈−1,0. The initial values of the slave delayed chaotic neural network (47) are yt=6,−2T,t∈−1,0. The antisynchronization of master-slave delayed chaotic neural networks begins at t0=0, and the goal is to achieve antisynchronization within the predefined-time Tc. Simulation results in Figures 1 and 2 verify the correctness and advantages of the proposed methods. Note that, for the case of(49)uit=−kiei−wisignei,simulation results are shown in Figure 3 with the initial values yt=6,−2T,t∈−1,0, and xt=2,−6T and in Figure 4 with the initial values yt=−6,−4T,t∈−1,0, and xt=4,2T. The antisynchronization is finite-time stable in theory, but the drawback of which is impossible to estimate the settling time in advance. For the case of(50)uit=−kiei−wisignei−αisigneieip−ηisigneieiq,simulation results are shown in Figure 5. Theoretically, the antisynchronization is fixed-time stable and the settling time can be calculated as follows:(51)Tset=11−q2γ−1α1/γγ−1η1−γ/γ=1.9,with the parameters q=0.8,p=1.5,α=6,andη=6. As can be seen from Figure 5, the master-slave delayed chaotic neural networks can realize fixed-time antisynchronization under controller (50) and the convergence time T≤Tset.Figure 1
Predefined-time stability based on the controller (52) with time Tc=0.5.Figure 2
Predefined-time stability based on the controller (52) with time Tc=1.Figure 3
Finite-time stability based on the controller (49) with initial values yt=6,−2T,t∈−1,0, and xt=2,−6T.Figure 4
Finite-time stability based on the controller (49) with initial values yt=−6,−4T,t∈−1,0, and xt=4,2T.Figure 5
Fixed-time stability based on the controller (50).The proposed controller (33) based on predefined-time stability(52)ui=−k1iei−k2isignei−2k−1Cv2Tcaisigneieipk+bisigneieiqk−di−aixican solve the above problems and guarantee that the master-slave delayed chaotic neural networks are antisynchronized before a predefined-time Tc, also verified Theorem 2. Simulation results are shown in Figures 1 and 2.


### 4.2. Predefined-Time Synchronization of Two Lorenz Chaotic Systems

To verify the reliability of Theorem1, the synchronization of two Lorenz chaotic systems is considered in this subsection. The master system is described by the following equations:(53)x˙1=σx2−x1,x˙2=x1ρ−x3−x2,x˙3=x1x2−βx3,where x=x1x2x3T∈R3 the system state vector and σ=10, ρ=28, and β=8/3 are constant parameters. The slave system can be written as follows:(54)x^˙1=σx^2−x^1+u1,x^˙2=x1ρ−x^3−x^2+u2,x^˙3=x1x^2−βx^3+u3,where x^=x^1x^2x^3T∈R3 the slave system state vector. Then, the active controller is proposed as(55)u=2k−1Cv2TcαkVpk−1+ηkVqk−1e,where e=e1e2e3T and V=1/2eTe the candidate Lyapunov function for the synchronization of the two Lorenz chaotic systems. It can be seen from controller (55) that the design process of the predefined-time controller depends on the actual systems, and different systems will have different forms of the predefined-time controllers. In Figure 6, it is appreciable that the proposed controller (55) enforces the synchronization of two Lorenz chaotic systems before the predefined-time Tc for several different initial conditions. And the predefined-time Tc can be explicitly defined during the control design, and the simulation results are shown in Figure 6 with Tc=0.06 and Figure 7 with Tc=0.02. The predefined-time synchronization can be achieved by the different settling time Tc. In theory, the choice of Tc should only satisfy the conditions of Theorem 1, that is, Tc=limVx0⟶∞Tx0. In practice, the choice of Tc should consider the actual system parameters and requirements simultaneously. It is shown in addition that Theorem 1 is also applied to other chaotic systems.Figure 6
Synchronization error for several different initial conditions with the predefined-timeTc=0.06.Figure 7
Synchronization error for several different initial conditions with the predefined-timeTc=0.02.


## 4.1. Predefined-Time Antisynchronization of Two Different Chaotic Neural Networks

Consider the following delayed chaotic neural network:(46)xi.t=−aixit+∑j=1nbijgjxjt+∑j=1ncijgjxjt−τt.Let (46) be the master delayed chaotic neural network, the slave delayed chaotic neural network is designed as(47)yi.t=−diyit+∑j=1nhijfjyjt+∑j=1nmijfjyjt−τt+uit,where i,j=1,2,…,n, denotes the number of neurons. In this example, we set n=2, and some parameters are set as follows: a1=a2=1.0, b11=2, b12=−0.1, b21=−5, b22=3, c11=−1.5, c12=−0.1, c21=−0.2, c22=−2.5, d1=d2=1.0, h11=0.02, h12=−0.01, h21=0.02, h22=0.1, m11=−0.01, m12=0.01, m21=0.01, m22=0.05, τt=1, p=2/3, q=2, and(48)gxt=tanhx1ttanhx2t,fyt=tanh0.2∗y1ttanh0.4∗y2t.The initial values of the master delayed chaotic neural network (46) are xt=2,−6T,t∈−1,0. The initial values of the slave delayed chaotic neural network (47) are yt=6,−2T,t∈−1,0. The antisynchronization of master-slave delayed chaotic neural networks begins at t0=0, and the goal is to achieve antisynchronization within the predefined-time Tc. Simulation results in Figures 1 and 2 verify the correctness and advantages of the proposed methods. Note that, for the case of(49)uit=−kiei−wisignei,simulation results are shown in Figure 3 with the initial values yt=6,−2T,t∈−1,0, and xt=2,−6T and in Figure 4 with the initial values yt=−6,−4T,t∈−1,0, and xt=4,2T. The antisynchronization is finite-time stable in theory, but the drawback of which is impossible to estimate the settling time in advance. For the case of(50)uit=−kiei−wisignei−αisigneieip−ηisigneieiq,simulation results are shown in Figure 5. Theoretically, the antisynchronization is fixed-time stable and the settling time can be calculated as follows:(51)Tset=11−q2γ−1α1/γγ−1η1−γ/γ=1.9,with the parameters q=0.8,p=1.5,α=6,andη=6. As can be seen from Figure 5, the master-slave delayed chaotic neural networks can realize fixed-time antisynchronization under controller (50) and the convergence time T≤Tset.Figure 1
Predefined-time stability based on the controller (52) with time Tc=0.5.Figure 2
Predefined-time stability based on the controller (52) with time Tc=1.Figure 3
Finite-time stability based on the controller (49) with initial values yt=6,−2T,t∈−1,0, and xt=2,−6T.Figure 4
Finite-time stability based on the controller (49) with initial values yt=−6,−4T,t∈−1,0, and xt=4,2T.Figure 5
Fixed-time stability based on the controller (50).The proposed controller (33) based on predefined-time stability(52)ui=−k1iei−k2isignei−2k−1Cv2Tcaisigneieipk+bisigneieiqk−di−aixican solve the above problems and guarantee that the master-slave delayed chaotic neural networks are antisynchronized before a predefined-time Tc, also verified Theorem 2. Simulation results are shown in Figures 1 and 2.


## 4.2. Predefined-Time Synchronization of Two Lorenz Chaotic Systems

To verify the reliability of Theorem1, the synchronization of two Lorenz chaotic systems is considered in this subsection. The master system is described by the following equations:(53)x˙1=σx2−x1,x˙2=x1ρ−x3−x2,x˙3=x1x2−βx3,where x=x1x2x3T∈R3 the system state vector and σ=10, ρ=28, and β=8/3 are constant parameters. The slave system can be written as follows:(54)x^˙1=σx^2−x^1+u1,x^˙2=x1ρ−x^3−x^2+u2,x^˙3=x1x^2−βx^3+u3,where x^=x^1x^2x^3T∈R3 the slave system state vector. Then, the active controller is proposed as(55)u=2k−1Cv2TcαkVpk−1+ηkVqk−1e,where e=e1e2e3T and V=1/2eTe the candidate Lyapunov function for the synchronization of the two Lorenz chaotic systems. It can be seen from controller (55) that the design process of the predefined-time controller depends on the actual systems, and different systems will have different forms of the predefined-time controllers. In Figure 6, it is appreciable that the proposed controller (55) enforces the synchronization of two Lorenz chaotic systems before the predefined-time Tc for several different initial conditions. And the predefined-time Tc can be explicitly defined during the control design, and the simulation results are shown in Figure 6 with Tc=0.06 and Figure 7 with Tc=0.02. The predefined-time synchronization can be achieved by the different settling time Tc. In theory, the choice of Tc should only satisfy the conditions of Theorem 1, that is, Tc=limVx0⟶∞Tx0. In practice, the choice of Tc should consider the actual system parameters and requirements simultaneously. It is shown in addition that Theorem 1 is also applied to other chaotic systems.Figure 6
Synchronization error for several different initial conditions with the predefined-timeTc=0.06.Figure 7
Synchronization error for several different initial conditions with the predefined-timeTc=0.02.


## 5. Conclusion

In this paper, a predefined-time stability theorem based on strict theoretical derivations has been presented. Then, a controller has been designed to achieve the antisynchronization of two different chaotic neural networks. The antisynchronization errors converge within a finite-time period and present the practical advantage that the least upper bound for this settling time can be explicitly defined during the control design. In order to show better performance of predefined-time controller, some comparisons have been made with the controllers by other stability theorems. Future work in this direction will apply predefined-time stability theorem to secure communications and be concerned with the following:(1)
Designing output-based predefined-time controllers [41, 42] for the systems with only the measured output utilized.(2)
Designing active controller for other types of systems with predefined-time stability, such as fractional order neural networks, fuzzy sets, and systems.(3)
Designing predefined-time active controller for discrete-time systems, such as discrete-time Boolean control networks [43, 44].



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