With the development of information technology and the higher requirement of information transmission, it is vital to ensure the security of communication. The field of secure communication has also attracted a large number of scientific researchers. Meanwhile, some important scientific research achievements have been obtained in the aspect of secure communication based on the synchronization performance of a chaotic system (Tao and Chua, 1997; Feki, 2003; Femat et al., 2004; Zheng et al., 2009). The types of chaotic systems include simple three-dimension chaotic systems, complex dynamical networks, and general neural networks, etc. Similar to a general neural network, the memristive neural network is also a kind of chaotic system, but differs in that the parameters of MNN are state-dependent. Memristor was first proposed by Chua (1971). Unlike ordinary resistors, which have fixed resistance values, the memristor is nonlinear and its value is not unique; the memristor is also considered to be the electronic equivalent of the synapse. As the memristor is often used to mimic the synapse, the model of MNN is widely applied in associative memory, next-generation computers, and powerful brain-like “neural” computers. MNN is the more realistic model for the description of real neural systems. Moreover, MNN with a coupling term is more suitable for real complex neural networks, which is known as a coupled memristive neural network (CMNN). Therefore, there has been an upsurge in the study of dynamic behavior based on the model of CMNN. In addition, the main considered problems also include secure transmission performance in synchronization time and the encryption performance based on a chaotic complex system. The speed of synchronization control time is also a key factor affecting communication security and quality.

The types of synchronization time include asymptotic time synchronization, exponential synchronization, finite-time synchronization, fixed-time synchronization, and predefined-time synchronization. Asymptotic time synchronization and exponential synchronization are types of infinite time synchronization, but the convergence rate of exponential time is slightly faster than that of asymptotic time. However, finite-time stability has more practical significance in secure communication. Finite time was introduced in 1961 (Dorato, 1961), which has a much faster convergence time. In secure communication, compared with asymptotic synchronization and exponential synchronization, the finite-time synchronization technique enables us to recover the transmitted signals in a setting time, which improves the efficiency and the confidentiality greatly. The concept of fixed-time stability was proposed by Polyakov (2012), and the criteria for determining whether the system can achieve fixed-time stability were also attained. Fixed-time stability is a special kind of finite-time stability. It is different from general finite-time stability in that its stability time has a definite upper bound, and the settling upper bound time is not dependent on the initial value of the system, but can be calculated by the system parameters and controller parameters. Therefore, the research of fixed-time is still fascinating to many researchers, who have committed to exploring the smaller conservative of the fixed-time upper bound value. Unfortunately, the control issues of fixed-time algorithms are still challenges related to convergence time estimation. The main drawback is that convergence time is not explicit and controllable. In order to overcome the problems presented above, a new stability with a tunable parameter is proposed based on fixed-time stability, which is defined as predefined-time stability (Sanchez-Torres et al., 2014).

Tao and Chua (1997) attained the general stabilization of chaotic systems, and it was applied to secure communication. Then, many kinds of synchronization of more complex systems were also studied, which can also be applied to secure communication schemes. Many previous studies have made important considerations to the convergence of synchronization time. Therefore, during a period of time, many researchers have found many excellent results in the aspect of finite-time and fixed-time stabilities of coupled neural networks. Yang et al. (2015) and Yang and Lu (2016) focused on the finite-time synchronization of coupled neural networks and gave some novel conclusions. Some effective guiding conclusions are given in Lv et al. (2019) and Zhang et al. (2020). Since the existence of memristor was confirmed by the HP laboratory in 2008 (Strukov et al., 2008), many scholars introduced memristor in the study of neural networks, which can more truly simulate the memory characteristics of biological neural networks. Guo and Gao (2014), Wang et al. (2016), Chen et al. (2018), Zhang and Qi (2021), and Peng et al. (2020) considered the asymptotic time synchronization of CMNN and gave full consideration to various cases of random disturbances. Various results about exponential synchronization of CMNN (Wang and Shen, 2014; Bao et al., 2016; Feng et al., 2016; Guo et al., 2018; Chen et al., 2021) were given, and Guo et al. (2018) also considered multiple coupled terms. Especially, with the requirement of the development of converge speed, these results about finite/fixed time synchronization of CMNN are discussed (Li et al., 2019; Lu et al., 2019; Yang et al., 2019; Huang et al., 2020; Gong et al., 2021). Since the concept of predefined-time stability was proposed in 2014 (Sanchez-Torres et al., 2014), predefined-time stability in dynamical systems has been studied (Juan et al., 2018;), and some theorems of predefined-time stability have been obtained. Recently, Anguiano-Gijon et al. (2019), Lin (2021b), Assali (2021), Lin et al. (2021), and Muoz-Vazquez et al. (2021) went further and discussed the predefined-time synchronization of two chaotic systems, one of them obtained the projective synchronization criterion about predefined-time (Lin et al., 2021), and another two of them considered chaotic systems of fractional-order forms (Lin et al., 2021; Muoz-Vazquez et al., 2021). In 2020, Lin defined the novel criterion of predefined-time synchronization in different kinds of neural networks, including a chaos neural network and memristive neural network (Lin et al., 2020; Lin, 2021a).

For nearly 2 years, many effective conclusions were given in the research of synchronous control in a coupled memristive neural network. Zhou et al. (2020) constructed a novel synchronization about weighted sum synchronization for CMNN. Chen et al. (2021) and Feng et al. (2021) obtained some results in exponential and fixed-time synchronization of CMNN. Bao et al. (2021) completed further research on prescribed-time synchronization in CMNN. But, few papers fully take into account coupled topology and multi-links performance to explore the predefined-time stability of the systems, and give some application on image encryption and decryption schemes. The detailed contents of multi-links performance of complex networks are referred to by Zhao et al. (2015) and Zhao et al. (2016).

Motivated by the above discussions, multi-links performance, coupled forms, and synchronization time of systems are taken fully into account in this paper. We investigate the predefined-time synchronization of MCMNN and design an efficient secure communication scheme based on predefined-time stability. The contributions of this paper are given as follows: Firstly, we overcome the complexity of factor interaction including multi-links performance and coupled forms, dealing with some parameter mismatches and complex topological structure problems, the network model is more general; secondly, the predefined-time synchronization issue of drive-response MCMNN is first studied based on the feedback controller and give a new predefined-time stability theorem. The predefined-time is a case of fixed-time, the bound time can be explicitly tuned; thirdly, the secure communication scheme is designed based on predefined-time synchronization of drive-response MCMNN. The preset synchronization time can be used as an important common key, the plaintext signal can be recovered after the settling time; finally, numerical simulations are given to verify the effectiveness of our theoretical results and the feasibility of the secure communication scheme. Therefore, our work aims to fill some gaps in the research of coupled memristive neural networks and predefined-time synchronization.

The paper is organized as follows. In section 2, the model of MCMNN and preliminaries are explained. In section 3, the predefined-time synchronization theorem and corollaries are respectively shown. In section 4, we designed the security communication scheme based on predefined-time synchronization of MCMNN. The numerical example of predefined-time stability and application example of secure communication scheme are given to show the effectiveness of our theoretical results in section 5. Finally, the conclusion and prospects are given in section 6.

In the paper, according to the property of multi-links coupled topology, we consider a model of MCMNN as follows:

where xi=(xi1,xi2,...,xin)T∈Rn,i=1,2,...N is the state vector of the ith node; C = diag(c₁, c₂, …, c_n) is a positive matrix and denotes the decay rates to the ith neuron. ḡ(xi(t))=(ḡ1(xi1(t)),ḡ2(xi(t)),...,ḡn(xin(t)))∈Rn and g(xi(t-τ0))=(g1(xi1(t-τ0)),g2(xi2(t-τ0)),...,gn(xin(t-τ0)))∈Rn are the discontinuous feedback functions, τ₀ is time delay, σ represents the coupling strength, and Γ = diag(γ₁, γ₂, …, γ_n) > 0 is the inner coupling matrix between each pair of nodes. W0=(wij0)N×N,Wl=(wijl)N×N,l=1,...,m represents the outer coupling configuration matrix of MNN, which is the different sub-network's Laplacian matrices, τ_l(l = 1…, m) > 0 denote different time-delays in the sub-networks. If nodes i and j are linked by an edge, then wijl=wjil>0(i≠j), otherwise, wijl=wjil=0, and the diagonal elements of matrix W_l are defined as wiil=-∑j=1,j≠iNwijl. If there are no isolated nodes in the network, then all of the matrix W_l(l = 0, 1, …, m) is an irreducible real symmetric matrix. I(t)=(I1(t),I2(t),...,IN(t))T∈Rn is the external input.

The parameters a_kq(x_ik(t)) and b_kq(x_ik(t−τ₀)) denote the non-delayed and delayed memristor-based synaptic connection weights, respectively. They can be described as follows:

where the switching jumps T_i > 0, â_kq, ǎ_kq, b^kq, b^kq, k, q = 1, 2, …, n, are all constants.

If Equation (1) denotes the drive system, the corresponding response system with a control input can be characterized by:

where i, j = 1, 2, …, N, yi=(yi1,yi2,...,yin)T∈Rn is the state vector of the ith node of the response network and u_ik(t) is the controller for node i. The remaining parameters of Equation (4) have the same meanings as those in Equation (1).

Definition 1. Filippov (1960) For a differential system: ẋ(t) = f(t, x), where f(t, x) is discontinuous in x(t), and x(t) is a solution of the differential system on [t₀, t₁] in Filippov's sense, if x(t) is absolutely continuous on any compact interval [t₀, t₁], for almost all t ∈ [t₀, t₁] such that

where

where co¯[·] is the convex closure hull of a set, B(x, δ) = {y:||y − x|| ≤ δ} is the ball of center x and radius δ, the intersection is takes over all sets N of measure zero and over all δ > 0, and μ(N) is the Lebesgue measure of set N.

Definition 2. Polyakov (2012) considering the nonlinear system v˙=f(v,r) is said to suggest global fixed-time stability, if it has global finite-time stability and the settling time function T_max is bounded and independent of the initial conditions, i.e., there exists T_max > 0 such that

Definition 3. Sanchez-Torres et al. (2014) The drive-response systems (1) and (4) are said to achieve the predefined-time synchronization if there exists T_v in the case of fixed-time synchronization and if the settling time function T:Rn→R+ is such that

Assumption 1. For the neuron activation functions ḡ_q(·), g_q(·), there exist Lipschitz constants ḡ_q and g_q > 0 satisfying the following Lipschitz conditions:

Assumption 2. The neuron activation functions are bounded functions, there exists a real number Mq,M̄q for any g_q(x), ḡ_q(y) such that gq(x)≤Mq,ḡq(y)≤M̄q.

Lemma 1. Hu et al. (2017) If there exists a regular, positive definite, and radially unbounded function V(t):Rn→R+ and constants a > 0, b > 0, η > 1 meet

the ∀t ≥ T_max of the origin system V(t) is fixed-time stability, and the upper-bounded settling time T_max is estimated by

Lemma 2. Khalil and Grizzle (2002) Let a₁, a₂, …, a_N, η > 1, then the following inequality holds

Remark 1. The algebraic inequality of Lemma 2 is used in many studies to determine stability, such as Yang and Daniel (2016) and Yang et al. (2017), which focused on the research of exponential synchronization and finite-time synchronization in memristive neural networks without coupled links. Next, we extend the coupled memristive neural network model to explore predefined-time synchronization, in which the synchronization time can be adjusted in the controller.

Lemma 3. If there exists a regular, positive definite, and radially unbounded function V(t):Rn→R+ and constants a > 0, b > 0, η > 1 are satisfied

where T_v is a user-defined parameter and

Then, for ∀t > T_v, we have V(t) = 0. The origin system can achieve predefined-time synchronization.Proof: By Lemma 2, we have

Then

If V(x₀) = 0, then T(x₀) = 0. If V(x₀) → ∞, then 1(a1ηV(x0)+b1η)η-1→0, thus we have

The proof is completed.

Denote

Therefore, based on Definition 1 and the theory of differential inclusion, Equation (1) and Equation (4) can be written as

and

Remark 2. According to the state-dependence conditions of Equations (5) and (6), the variables cō[-ãkq,ãkq],cō[-b~kq,b~kq] may not reach their maximum and minimum values at the same time. Therefore, we give the following four different measurable functions to represent interval information.

According to the measurable selection theorem (Xiao and Zeng, 2017), there exist measurable functions ξkq1(t),ξkq2(t),ξkq3(t),ξkq4(t)∈cō[-1,1] such that

and

Let e_ik(t) = y_ik(t)−x_ik(t), the corresponding error system is given as follows:

Remark 3. Based on Definition 3, the issues of predefined-time synchronization between drive system (1) and response system (4) are transformed into the issues of predefined-time stability of error system (9).

In order to guarantee the predefined-time synchronization of drive-response systems, the controller is designed as follows:

where i = 1, 2, …, N, k = 1, 2, .., n. α_i, β_i, r_i, δ_i, η ≥ 0. T_v is the tunable predefined time, D_v is a positive constant given by other parameters. And sign(x) is the sign function which is defined as follows:

Remark 4. Designed controller (10) is discontinuous. To ensure the existence of the solutions of error system (11), the Dini derivative is used to ensure continuity at the breakpoint.

According to Assumptions 1 and 2, combined with designed controller (10), we obtain

Based on designed controller (10) applied on the response system, a theorem is presented to achieve the predefined-time synchronization for MCMNN.

Denote γmax=max1≤k≤n(γk), α = diag(α₁, α₂, …, α_N), β = diag(β₁, β₂, …, β_N),

r = diag(r₁, r₂, …, r_N), Wl=(wijl)N×N,l=1,2,...m.

Theorem 1. Under Assumptions 1 and 2, for a predefined-time T_v > 0 and controller (10), error system (9) can achieve predefined-time stability if

where

Proof: We construct a Lyapunov function as follows:

When e(t) = 0, V(e(t)) = 0 and the derivative of V(e(t)) is 0. Then, the derivative of V(e(t)) is along the trajectories of e(t) with e(t) ≠ 0. We have

According to the analysis above, we can obtain that

Let λ=min1≤i≤N{δi}, min1≤i≤N(βi-ϕ3)≥DvTvb, and by Lemma 2, we obtain

According to the above proof and a = λn^1−η, we have:

where

The proof of Theorem 1 is completed.

Remark 5. The predefined-time stability is a spacial case of fixed-time stability. Therefore, after removing the tuning parameters T_v and D_v, the error system is said to achieve fixed-time stability, and the upper bound of settling time can be indicated as T_max = D_v based on Definition 2 and Lemma 2. Therefore, error system (9) can achieve fixed-time stability based on the above controller (10).

The results of Theorem 1 can also easily extend to the general single coupled memristive neural network which does not include multi-links items. The drive-response systems are given as

The controller is designed as follows: