Loading [MathJax]/jax/element/mml/optable/Latin1Supplement.js
Research article

(α1, 2, β1, 2)-complex intuitionistic fuzzy subgroups and its algebraic structure

  • A complex intuitionistic fuzzy set is a generalization framework to characterize several applications in decision making, pattern recognition, engineering, and other fields. This set is considered more fitting and coverable to Intuitionistic Fuzzy Sets (IDS) and complex fuzzy sets. In this paper, the abstraction of (α1,2,β1,2) complex intuitionistic fuzzy sets and (α1,2,β1,2)-complex intuitionistic fuzzy subgroups were introduced regarding to the concept of complex intuitionistic fuzzy sets. Besides, we show that (α1,2,β1,2)-complex intuitionistic fuzzy subgroup is a general form of every complex intuitionistic fuzzy subgroup. Also, each of (α1,2,β1,2)-complex intuitionistic fuzzy normal subgroups and cosets are defined and studied their relationship in the sense of the commutator of groups and the conjugate classes of group, respectively. Furthermore, some theorems connected the (α1,2,β1,2)-complex intuitionistic fuzzy subgroup of the classical quotient group and the set of all (α1,2,β1,2)-complex intuitionistic fuzzy cosets were studied and proved. Additionally, we expand the index and Lagrange's theorem to be suitable under (α1,2,β1,2)-complex intuitionistic fuzzy subgroups.

    Citation: Doaa Al-Sharoa. (α1, 2, β1, 2)-complex intuitionistic fuzzy subgroups and its algebraic structure[J]. AIMS Mathematics, 2023, 8(4): 8082-8116. doi: 10.3934/math.2023409

    Related Papers:

    [1] Tao Xie, Wenqing Zheng . Robustness analysis of Cohen-Grossberg neural network with piecewise constant argument and stochastic disturbances. AIMS Mathematics, 2024, 9(2): 3097-3125. doi: 10.3934/math.2024151
    [2] Yijia Zhang, Tao Xie, Yunlong Ma . Robustness analysis of exponential stability of Cohen-Grossberg neural network with neutral terms. AIMS Mathematics, 2025, 10(3): 4938-4954. doi: 10.3934/math.2025226
    [3] Xingxing Song, Pengfei Zhi, Wanlu Zhu, Hui Wang, Haiyang Qiu . Exponential synchronization control of delayed memristive neural network based on canonical Bessel-Legendre inequality. AIMS Mathematics, 2022, 7(3): 4711-4734. doi: 10.3934/math.2022262
    [4] Shuo Ma, Jiangman Li, Qiang Li, Ruonan Liu . Adaptive exponential synchronization of impulsive coupled neutral stochastic neural networks with Lévy noise and probabilistic delays under non-Lipschitz conditions. AIMS Mathematics, 2024, 9(9): 24912-24933. doi: 10.3934/math.20241214
    [5] Yinjie Qian, Lian Duan, Hui Wei . New results on finite-/fixed-time synchronization of delayed memristive neural networks with diffusion effects. AIMS Mathematics, 2022, 7(9): 16962-16974. doi: 10.3934/math.2022931
    [6] Jiaqing Zhu, Guodong Zhang, Leimin Wang . Quasi-projective and finite-time synchronization of fractional-order memristive complex-valued delay neural networks via hybrid control. AIMS Mathematics, 2024, 9(3): 7627-7644. doi: 10.3934/math.2024370
    [7] Qike Zhang, Wenxiang Fang, Tao Xie . Robustness analysis of exponential synchronization in complex dynamic networks with random perturbations. AIMS Mathematics, 2023, 8(9): 20487-20509. doi: 10.3934/math.20231044
    [8] Jingyang Ran, Tiecheng Zhang . Fixed-time synchronization control of fuzzy inertial neural networks with mismatched parameters and structures. AIMS Mathematics, 2024, 9(11): 31721-31739. doi: 10.3934/math.20241525
    [9] Dong Pan, Huizhen Qu . Finite-time boundary synchronization of space-time discretized stochastic fuzzy genetic regulatory networks with time delays. AIMS Mathematics, 2025, 10(2): 2163-2190. doi: 10.3934/math.2025101
    [10] Ruoyu Wei, Jinde Cao, Wenhua Qian, Changfeng Xue, Xiaoshuai Ding . Finite-time and fixed-time stabilization of inertial memristive Cohen-Grossberg neural networks via non-reduced order method. AIMS Mathematics, 2021, 6(7): 6915-6932. doi: 10.3934/math.2021405
  • A complex intuitionistic fuzzy set is a generalization framework to characterize several applications in decision making, pattern recognition, engineering, and other fields. This set is considered more fitting and coverable to Intuitionistic Fuzzy Sets (IDS) and complex fuzzy sets. In this paper, the abstraction of (α1,2,β1,2) complex intuitionistic fuzzy sets and (α1,2,β1,2)-complex intuitionistic fuzzy subgroups were introduced regarding to the concept of complex intuitionistic fuzzy sets. Besides, we show that (α1,2,β1,2)-complex intuitionistic fuzzy subgroup is a general form of every complex intuitionistic fuzzy subgroup. Also, each of (α1,2,β1,2)-complex intuitionistic fuzzy normal subgroups and cosets are defined and studied their relationship in the sense of the commutator of groups and the conjugate classes of group, respectively. Furthermore, some theorems connected the (α1,2,β1,2)-complex intuitionistic fuzzy subgroup of the classical quotient group and the set of all (α1,2,β1,2)-complex intuitionistic fuzzy cosets were studied and proved. Additionally, we expand the index and Lagrange's theorem to be suitable under (α1,2,β1,2)-complex intuitionistic fuzzy subgroups.



    The memristors were initially introduced by Chua in 1971 to describe the relationship between charge and magnetic flux [1]. It was predicted to be the fourth fundamental circuit element, distinct from resistors, capacitors and inductors. In 2008, the research team at HP laboratories successfully created a practical memristor device with valuable applications [2]. Similar to conventional resistors, the memristors can handle safe currents through the device. Furthermore, its value changes based on the amount of charge passing through it, therefore the memristors have memory functionality [3,4,5,6,7]. As a result, an increasing number of researchers have been using memristors instead of traditional resistors to serve as connection weights between neurons and for self-feedback connection weights, forming a state-dependent nonlinear switching system known as a memristive neural networks (MNNs). Compared to conventional artificial neural networks (NNs), MNNs possess stronger computational capabilities and information capacity, thereby enhancing the applications of NNs in associative memory, signal processing and image processing [8,9,10,11,12,13].

    The dynamic behavior of MNNs is fundamental to their applications; therefore, it necessary to analyze their dynamic characteristics [10,14]. Among the various dynamic behaviors of MNNs, synchronization is an important and fundamental feature. The synchronization of MNNs has garnered extensive attention from researchers due to its numerous potential applications in artificial intelligence, information science, secure communications and various other fields [15,16,17]. In [5], Du et al. derived finite-time (FNT) synchronization criteria for fractional-order MNNs with delays using the fractional-order Gronwall inequality. The FNT/fixed time (FXT) stability of MNNs was tudied in [18], by designing a synovial membrane controller, the MNNs reaches the sliding-mode surface in FNT/FXT. In [10,11,12,13,14,15,16,17,18], synchronization criteria were obtained for MNNs with either delays or random interference. The above discussions mostly focus on the synchronization of MNNs with time delays, and few have considered the exponential synchronization (ESy) of MNNs with deviating arguments (DAs).

    The theory of DAs differential equations was proposed by Shah and Wiener in 1983 [19]. In [20], by transforming these equations into equivalent integral equations, new stability conditions were obtained. These equations involve DAs, combining the properties of discrete and continuous equations [21,22,23]. During the system's operation, the relevant arguments characteristics can be altered, allowing the system to become a combination of lag and advance equations [24,25]. As a result, systems with DAs have broader applications compared to systems with time delays. Reference [26] investigates recurrent neural networks with DAs and establishes criteria for the global exponential stability. In order to further explore the impact of DAs on the exponential stability (ESt) of the systems, the robustness analysis of a fuzzy cellular neural networks with DAs and stochastic disturbance is discussed in [23]. The signals transmitted between MNNs are inevitably subject to stochastic perturbations (SPs) caused by environmental uncertainties [27,28,29,30,31]. For systems with SPs, the feature can significantly impact the dynamic behavior of the system, leading to either synchronization or desynchronization under certain levels of SPs [32,33,34]. For example, for the following simple linear systems dx(t)=ax(t)dt and dy(t)=ay(t)dt, the error system is de(t)=ae(t)dt. The system is stable only when a<0. However, the stability of the system is affected by SPs. Therefore, consider the following system de(t)=ae(t)dt+be(t)dB(t). The system is almost surely ESt if and only if the condition b2>2a is satisfied [35]. Then, the error system is ESt, it implies that x(t) and y(t) are exponential synchronization (ESy). Based on the above discussions, we reach the following conclusion: SPs can disrupt the ESy of a system that was synchronized or facilitate the ESy of a system that was initially unsynchronized. If a MNNs with SPs are ESy, can we obtain upper bounds such that the MNNs remains ESy when the SPs are smaller than the bounds?

    Based on the discussion above, MNNs can lose synchronization when subjected to disturbances from external perturbations and DAs, provided that the intensity of perturbations and the width of arguments exceed certain limits. In [27,30,31,32,33,34,36,37], there are important results regarding the synchronization of MNNs under external disturbances. In [21,22,23], scholars research on MNNs with DAs. It is important to note that the aforementioned literature primarily focuses on the synchronization of MNNs than its robustness. Therefore, an interesting question arises: Under the control strategy, how much argument length and perturbation intensity can MNNs with ESy endure without losing synchronization?

    The major contributions of this paper include the following aspects:

    Compared to the references [5,6,7,11,16], we focus on the synchronization of MNNs with DAs. The systems with DAs have broader applications compared to traditionally time-delayed systems.

    The references [8,9,10,11,12,13,14,15,16,17] extensively investigated MNNs with time delays, providing various stability and synchronization criteria. In references [28,29,33,34], the robustness of ESt in systems with both time delays and SPs was further explored. In references [23,27], Fang et al. studied the robustness of ESt in fuzzy cellular neural networks with DAs. In contrast to the aforementioned literatures, we focus on the robustness of ESy in MNNs, utilizing the set-valued mapping method, differential inclusion theory and Gronwall inequalities, we derive the upper bounds for the DAs and SPs.

    Compared to the references [22,23,24,25,32,33,34]. The MNNs with state switching that we consider and results in a more complex system structure.

    The paper is organized as follows. In Section Ⅱ, we introduce the model, assumptions and some preliminary lemmas. In Section Ⅲ, we present the theorems and lemmas derived in this paper. In Section Ⅵ, we provide several examples to validate the feasibility of our results. Finally, in Section Ⅴ, we have summarized the work carried out in this paper.

    In the paper, Rn is Euclidean space, N represents integers, ||χ(t)|| is the norm of vector χ(t), where χ(t)Rn and ||χ(t)||=np=1|χ(t)|, the norm ||A|| of the matrix A is given by ||A||, where A=max1qnnp=1|apq|. For two real-valued sequences ρk,ηk, where kN, it holds that ρk<ρk+1, ρkηkρk+1 for all kN with ηk as k.

    Let (Ω,F,{Ft}tt0,P) is a complete probability space with a filtration {Ft}tt0 (the filtration contains all P-null sets and is right continuous). LPF0([τ,0];Rn) is the family of all F0-measurable C([τ,0];Rn), the state variable ξ={ξ(s);τξ(s)0} satisfies supτs0E(||ξ(s)||P). E() is the mathematical expectation in the probability space.

    Consider the MNNs as the derive system with the SPs,

    dwp(t)=[dpwp(t)+nq=1apq(wq(t))fq(wq(t))+Ip(t)]dt+σwp(t)dω(t), (2.1)

    where p=1,2,,n, wp(t) are the state variables, and fq(wq(t)) are the activation functions; dp>0 is a self-feedback connection weights and Ip(t) is the external inputs, apq(wq(t)) is the memristive connection weights, σ is the interference intensity. ωi(t) represents Brownian motion on the compete space.

    For convenience, we use wp,wq, apq(wq),up, ω to replace wp(t),wq(t),apq(wq(t)),up(t), ω(t), respectively. The initial conditions of (2.1) is

    wp(t0)=φp.

    The corresponding response system,

    dvp(t)=[dpvp(t)+nq=1apq(vq(t))fq(vq(t))+Ip(t)+up(t)]dt+σvp(t)dω(t). (2.2)

    For convenience, we use vp,vq, apq(vq),up to replace vp(t),vq(t),apq(vq(t)),up(t), respectively. Then, the memristive parameter of (2.1) and (2.2) are expressed as

    aij(wq)={aˊ

    where i, j\in \mathbb{N} , weights \acute{{a}_{pq}} , \grave{a}_{pq} and switching jumps T_{q} > 0 . The initial value of (2) are

    \begin{aligned} v_{p}(t_{0}) = \phi_{p}. \end{aligned}

    The linear feedback controller u_{p} is designed as follows

    \begin{array}{c} u_{p}(t) = -\xi_{p}(v_{p}(t)-w_{p}(t)). \end{array}

    The error system between the drive system (2.1) and the response system (2.2) is defined as

    \begin{equation} \begin{aligned} &e_{p} = v_{p}-w_{p},\; \psi_{p}^{1} = \phi_{p}-\varphi_{p},\; p\in \mathbb{N}.\\ \end{aligned} \end{equation} (2.3)

    Remark 1. We can observe that MNNs can be categorized as discontinuous switched systems, which necessitates considering the solution to MNNs (1) using the Filippov's sense. In the following, we will introduce certain definitions pertaining to set-valued maps and the Filippov solution.

    Definition 1. (Set-valued map [38]) Consider a set E\in\mathbb{R}^{n} . A set-valued map is defined as follows: For each point x in the set E , there exists a nonempty set F(x)\in \mathbb{R}^{n} such that x is mapped to F(x) .

    Definition 2. (Differential inclusion [38]) For a discontinuous differential system \dot{r}(t) = \mathbb{F}(t, r_{t}) , t\geq 0 , the function r_{t} is the solution of the differential equation in the Filippov sense, t\in [0, t_{1}] , t_{1}\geq 0 , if is absolutely continuous and satisfies the following differential inclusion:

    \begin{aligned} \dot{r}\in \mathbb{G}(t,r_{t}), \end{aligned}

    where t\in [0, +\infty] , r\in \mathbb{R}^{n} , the initial condition r(0) = r_{0}\in \mathbb{C}([-\tau, 0], \mathbb{R}^{n}) . The \mathbb{G}(t, r_{t}) is a set-valued mapping, satisfies

    \begin{aligned} \mathbb{G}(t, r_{t}) = \bigcap\limits_{\varGamma > 0} \bigcap\limits_{\delta (\mathbb{N}) = 0} \overline {\operatorname{co}}[f(\mathbb{B}(r_{t},\varGamma ) \backslash \mathbb{N})], \end{aligned}

    where \overline {\operatorname{co}} is the convex closure hull of a set, \mathbb{B}(r_{t}, \varGamma) = \{x:\|x-r_{t}\| \leq \varGamma\} , \varGamma > 0 and \delta(\mathbb{N}) is Lebesgue measure of set \mathbb{N} .

    The set value mapps of memristive parameters is as follows:

    K\left[a_{pq}\left(w_{q}\right)\right] = \left\{\begin{array}{ll} \acute{a}_{pq}, & \left|w_{q}\right| < T_{q}, \\ \overline{\operatorname{co}}\left\{\acute{a}_{pq}, \grave{a}_{pq}\right\}, & \left|w_{q}\right| = T_{q}, \\ \grave{a}_{pq}, & \left|w_{q}\right| > T_{q}, \end{array}\right.\; K\left[a_{pq}\left(v_{q}\right)\right] = \left\{\begin{array}{ll} \acute{a}_{pq}, & \left|v_{q}\right| < T_{q}, \\ \overline{\operatorname{co}}\left\{\acute{a}_{pq}, \grave{a}_{pq}\right\}, & \left|v_{q}\right| = T_{q}, \\ \grave{a}_{pq}, & \left|w_{q}\right| > T_{q}, \end{array}\right.

    where p, q\in \mathbb{N} . K[a_{pq} (w_{q})] and K[a_{pq}(v_{q})] are all closed, convex and compact about w_{q} , v_{q} .

    According to Definitions 1 and 2, the Filippov solution of the systems (1) and (2) can be written as:

    \begin{equation} \begin{aligned} dw_{p}(t)\in&[-d_{p}w_{p} +\sum\limits_{q = 1}^{n}K[a_{pq}(w_{q})] f_{q}(w_{q})+I_{p}]ds +\sigma w_{p}d\omega. \end{aligned} \end{equation} (2.4)
    \begin{equation} \begin{aligned} dv_{p}(t)\in&[-d_{p}v_{p} +\sum\limits_{q = 1}^{n}K[a_{pq}(v_{q})]f_{q}(v_{q}) +I_{p}+u_{p}]dt+\sigma w_{p}(t) d\omega.\\ \end{aligned} \end{equation} (2.5)

    Similarly, there exist

    \begin{array}{ll} \bar{a}_{pq}\left(w_{q}\right) \in K\left[a_{pq} \left(w_{q}\right)\right],\; \check{a}_{pq}\left(v_{q}\right) \in K\left[a_{pq} \left(v_{q}\right)\right], \end{array}

    such that

    \begin{equation} \begin{aligned} dw_{p} = &[-d_{p}w_{p} +\sum\limits_{q = 1}^{n}\bar{a}_{pq}(w_{q})f_{q}(w_{q})+I_{p}]ds +\sigma w_{p}d\omega,\\ d{v}_{p} = &[-d_{p}v_{p} +\sum\limits_{q = 1}^{n}\check{a}_{pq}(v_{q})f_{q}(v_{q}) +I_{p}+u_{p}]dt+\sigma w_{p}d\omega.\\ \end{aligned} \end{equation} (2.6)

    From (6), let e_{p} = v_{p}-w_{p} ,

    \begin{equation} \begin{aligned} de_{p} = -[(d_{p}+\xi_{p})e_{p} +\sum\limits_{q = 1}^{n}\hat{a}_{pq}(e_{q}) f_{q}(e_{p})]dt +\sigma e_{p}d\omega, \end{aligned} \end{equation} (2.7)

    where \hat{a}_{pq}(e_{q})f_{q}(e_{q}) = \bar{a}_{pq}(w_{q})f_{q}(w_{q}) -\check{a}_{pq}(v_{q})f_{q}({v_{q}}) .

    The following error system without random disturbance:

    \begin{equation} \begin{aligned} \dot{z}_{p} = &-(d_{p}+\xi_{p})z_{p} +\sum\limits_{q = 1}^{n}\hat{a}_{pq}(z_{q})f_{q}(z_{q}).\\ \end{aligned} \end{equation} (2.8)

    The initial conditions is

    \begin{aligned} z_{p}(t_{0}) = \psi_{p}^{2},\; p = 1,\cdots,n. \end{aligned}

    The (2.7) and (2.8) can be rewritten as

    \begin{equation} \begin{array}{l} de(t) = [-(D+C)e(t)+\hat{A} F(e(t))]dt +\sigma e(t) d\omega(t)\\ \dot{z}(t) = -(D+C)z(t)+\hat{A}F(z(t)), \end{array} \end{equation} (2.9)

    where e(t) = (e_{1}(t), \cdots, e_{n}(t))\; ^{T} , z(t) = (z_{1}(t), \cdots, z_{n}(t))^{T} , C = \operatorname{diag}\left\{\xi_{1}, \xi_{2}, \; \cdots\; , \xi_{n}\right\} , D = \operatorname{diag}\left\{d_{1}, d_{2}, \; \cdots\; , d_{n}\right\} , \hat{A} = \left(\hat{a}_{pq}\right)_{n \times n} , F(e(t)) = (f_{1}(e_{1}(t)), \cdots, f_{n}(e_{n}(t)))^{T} .

    The mian result of the paper are base on the following definitions, assumptions and lemmas.

    Definition 3. (Exponential synchronization (ESy)) If the error system \chi(t) is exponential stability (ESt). Then, the MNNs (2.1) and (2.2) are described as ESy, there exsit two nonnegative constant \alpha and \beta ,

    \begin{array}{c} ||\chi(t)||\le \alpha||\psi^{1}||\exp(-\beta(t-t_{0})), \end{array}

    where \psi^{1} = (\psi^{1}_{1}, \cdots, \psi^{1}_{n})^{T} is any initial condition, t\le 0 .

    Definition 4. (Mean square exponential synchronization (MSESy)) The state \chi(t) of system (2.7) is said to be MSESt. Then the MNNs (2.1) and (2.2) are described as MSESy, if for any t_{0}\in\mathbb{R}^{+} , \psi^{1}\in\mathbb{R}^{n} , there exist \theta > 0 and \vartheta > 0 such that

    \begin{aligned} E||\chi(t)||^{2}\le \theta||\psi^{1}||^{2}\exp\{-2\vartheta(t-t_{0})\}. \end{aligned}

    Assumption 1. The activation functions f_{q}(\cdot) and g_{q}(\cdot) satisfy

    \begin{aligned} ||f_{q}(w_{p}) -f_{q}(v_{p})||\le f^{*}_{q}||w_{p}-v_{p}||,\; ||g_{q}(w_{p}) -f_{q}(v_{p})||\le g^{*}_{q}||w_{p}-v_{p}||, \end{aligned}

    where f^{*}_{q}, g^{*}_{q} > 0 are Lipschitz constants, w_{p}, v_{p}\in \mathbb{R}^{n} .

    Under Assumption 1, we have the following conclude: The MSESt of system (2.7) implies the almost sure ESt of system (2.7), see [39].

    Assumption 2. f_{q}\left(\pm T_{q}\right) = g_{q}\left(\pm T_{q}\right) = 0 .

    Lemma 1. [14] From Assumptions 1 and 2, then

    \begin{array}{l} \left|K\left[a_{pq}\left(v_{q}\right)\right] f_{q}\left(v_{q}\right)-K\left[a_{pq}\left(w_{q}\right)\right] f_{q}\left(w_{q}\right)\right| \leq a_{pq}^{*} f_{q}^{*} \left|v_{q}-w_{q}\right| \end{array}

    for i, j \in \mathbb{N} .

    That is, for any \bar{a}_{pq}\left(w_{q}\right) \in K\left[a_{pq} \left(w_{q}\right)\right], \check{a}_{pq}\left(v_{q}\right) \in K\left[a_{pq}\left(v_{q}\right)\right] .

    \begin{array}{c} \left|\check{a}_{j}\left(v_{q}\right) f_{q} \left(v_{q}\right)-\bar{a}_{pq}\left(w_{q}\right) f_{q}\left(w_{q}\right)\right| \leq a_{pq}^{*} f_{q}^{*}\left|v_{q}-w_{q}\right|, \end{array}

    where a_{pq}^{*} = \max\left\{\left|\acute{a}_{pq}\right|, \left|\grave{a}_{pq}\right|\right\} .

    Lemma 2. [40] (Granwall inequalities) Suppose \vartheta(t) , \theta(t) and u(t) are continuous real-valued functions, and \theta(t) is integrable over the interval I = [t_{0}, t] , if \vartheta (t)\geq 0 and u (t) satisfies

    \begin{aligned} u(t) \leq \theta(t)+\int_{t_{0}}^{t}\vartheta (s) u(s) d s, \end{aligned}

    then,

    \begin{aligned} u(t) \leq \theta(t)+\int_{t_{0}}^{t} \theta(s) \vartheta (s) \exp \left(\int_{s}^{t} \vartheta (r) d r\right) d s, \end{aligned}

    where t > t_{0} .

    In addition, if \theta(t) is non-decreasing, then

    \begin{aligned} u(t) \leq \theta(t) \exp \left(\int_{t_{0}}^{t} \vartheta (s) d s\right) . \end{aligned}

    Assumption 3. There exists a nonnegative constant \rho that satisfies, \rho_{k+1}-\rho_{k}\le \rho , for all k\in\mathbb{N} .

    Assumption 4. h_{1}\rho+h_{2}\rho (1+h_{1}\rho)\exp\{h_{2}\rho\} < 1 ,

    where h_{1} = ||B^{*}||||G^{*}||\rho , h_{2} = ||C||+||D||+||A^{*}||||F^{*}|| .

    Assumption 5. \rho(3l_{1}+9l_{2}+l_{1}^{2} +3l_{1}l_{2}) < 1 ,

    where l_{1} = 9\rho^{2}||B^{*}||^{2}||G^{*}||^{2} , l_{2} = 3\rho(2||C||^{2} +2||D||^{2}+||A^{*}||^{2}||F^{*}||^{2}) +\theta^{2} .

    Under the influence of the linear controller, the error system (2.7) can achieve ESt, that is, systems (2.1) and (2.2) can achieve ESy. The next question is how much the SPs and DAs intensities can make the system can maintain ESy.

    Theorem 1. Let Assumption 1 hold, MNNs (2.8) is ESt. Then MNNs (2.7) is ESt, that is, the master system MNNs (2.1) and the slave system MNNs (2.2) are ESy. if |\sigma|\le\bar\sigma , \bar\sigma is the unique nonnegative solution of the transcendental equation below.

    \begin{equation} \begin{aligned} &(24\mathbb{T}||A^{*}||^{2} ||F^{*}||^{2} +2\sigma^{2})\theta/\vartheta \exp\{8\mathbb{T}(4\mathbb{T}(||D||^{2}\\ &+||C||^{2}+3||A^{*}||^{2}||F^{*}||^{2}) +\sigma^{2})\} +2\theta \exp\{-4\vartheta \mathbb{T}\} = 1, \end{aligned} \end{equation} (3.1)

    where \mathbb{T} > \ln(2\theta)/(4\vartheta) > 0 , A^{*} = (a_{pq}^{*})_{n\times n} , F^{*} = \max_{j\in\mathbb{N}}\{f_{q}^{*}\} , \mathbb{T} is the time interval.

    Proof. z(t) and e(t) have the same initial value, \psi^{1} = \psi^{2} ,

    \begin{aligned} z(t)-e(t) = &\int_{t_{0}}^{t} [-D(z(s)-e(s))+\hat{A}F(z(s))-\hat{A}F(e(s)) -C(z(s)-e(s))]dt-\int_{t_{0}}^{t}\sigma e(s)d\omega (t).\\ \end{aligned}

    The ESt of the MNNs (2.8), when t > t_{0} ,

    \begin{aligned} \int_{t_{0}}^{t}E||z(s)||^{2} = & \int_{t_{0}}^{t}\theta ||\psi^{2}||\exp\{-2\vartheta(t-t_{0})\} \le \theta ||\psi^{2}||^{2}/2\vartheta.\\ \end{aligned}

    When t\le t_{o}+2\mathbb{T} , by utilizing the Cauchy-Schwarz inequalities and Lemma 1, we can conclude the following:

    \begin{aligned} E||z(t)-e(t)||^{2}\le& 2E||\int_{t_{0}}^{t} \left[-D\left(z(s)-e(s)\right)+\hat{A}F(z(s)) -\hat{A}F(e(s)) -C(z(s)-e(s))\right]ds||^{2}\\ &-2E||\int_{t_{0}}^{t}\sigma e(t)d\omega (s)||^{2}\\ \le&2E\int_{t_{0}}^{t}1ds\int_{t_{0}}^{t} \left[||-D(z(s)-e(s)) +\hat{A}F(z(s))-\hat{A}F(e(s)) -C(z(s) -e(s))||\right]^{2}dt\\ &+2\sigma^{2}\int_{t_{0}}^{t} E||e(s) -z(s)+z(s)||^{2}ds\\ \le&4\mathbb{T}E\int_{t_{0}}^{t} 4(||D||^{2}||z(s)-e(s)||^{2} +||A^{*}||^{2} ||F^{*}||^{2}||z(s)||^{2} +||A^{*}||^{2} ||F^{*}||^{2}||e(s)||^{2}\\ &+||C||^{2} ||z(s)-e(s)||^{2})ds+2\sigma^{2}\int_{t_{0}}^{t} E||e(s) -z(s)+z(s)||^{2}ds.\\ \end{aligned}

    Then,

    \begin{equation} \begin{aligned} E||z(t)-e(t)||^{2}\le&\left[16\mathbb{T}(||D||^{2} +||C||^{2}+2||A^{*}||^{2}||F^{*}||^{2}) +4\sigma^{2}\right] \int_{t_{0}}^{t}E||z(s)-e(s)||^{2}ds\\ &+\left(16\mathbb{T}(3||A^{*}||^{2}||F^{*}||^{2}) +4\sigma^{2}\right)\int_{t_{0}}^{t} E||z(s)||^{2}ds\\ \le&\left[16\mathbb{T}(||D||^{2} +||C||^{2}+2||A^{*}||^{2}||F^{*}||^{2}) +4\sigma^{2}\right] \int_{t_{0}}^{t}E||z(s)-e(s)||^{2}ds\\ &+\left(8\mathbb{T}(3||A^{*}||^{2}||F^{*}||^{2}) +2\sigma^{2}\right)\theta||\psi^{2}|| ^{2}/\vartheta, \end{aligned} \end{equation} (3.2)

    When t_{0}+\mathbb{T}\le t\le t_{0}+2\mathbb{T} , by applying Lemma 2,

    \begin{equation} \begin{aligned} E||z(t)-e(t)||^{2} \le& \left(8\mathbb{T}(3||A^{*}||^{2}||F^{*}||^{2}) +2\sigma^{2}\right)\theta||\psi^{2}|| ^{2}/\vartheta \exp\Big\{(16\mathbb{T}(||D||^{2}\\ &+||C||^{2}+2||A^{*}||^{2}||F^{*}||^{2}) +4\sigma^{2})(t-t_{0})\Big\}\\ \le&\left(24\mathbb{T}||A^{*}||^{2} ||F^{*}||^{2} +2\sigma^{2}\right)\theta/\vartheta \exp\Big\{8\mathbb{T}(4(||D||^{2} +||C||^{2}+2||A^{*}||^{2}||F^{*}||^{2}) +\sigma^{2}\Big\}\\ &\times\biggl(\sup\limits_{t_{0}\le t\le t_{0}+2\mathbb{T}}E||z(t)||^{2}\biggr). \end{aligned} \end{equation} (3.3)

    When t_{0}+\mathbb{T}\le t\le t_{0}+2\mathbb{T} ,

    \begin{equation} \begin{aligned} E||e(t)||^{2}\le& 2E||z(t)-e(t)||^{2} +2E||z(t)||^{2}\\ \le&\biggl((24\mathbb{T}||A^{*}||^{2} ||F^{*}||^{2} +2\sigma^{2})\theta/\vartheta \exp\{8\mathbb{T}(4\mathbb{T}(||D||^{2} +||C||^{2}+2||A^{*}||^{2}||F^{*}||^{2}) +\sigma^{2})\}\biggr)\\ &\times\biggl(\sup\limits_{t_{0}\le t\le t_{0}+\mathbb{T}}E||z(t)||^{2}\biggr) +2\theta ||\psi^{2}||^{2}\exp\{-2\vartheta (t-t_{0})\}\\ \le&\biggl(\Big(24\mathbb{T}||A^{*}||^{2} ||F^{*}||^{2} +2\sigma^{2}\Big)\theta/\vartheta \exp\{8\mathbb{T}(4\mathbb{T}(||D||^{2} +||C||^{2}+2||A^{*}||^{2}||F^{*}||^{2}) +\sigma^{2})\}\\ &+2\theta \exp\{-4\vartheta \mathbb{T}\}\biggr) \times\biggl(\sup\limits_{t_{0}\le t\le t_{0}+\mathbb{T}}||e(t)||^{2}\biggr). \end{aligned} \end{equation} (3.4)

    From (11), when |\sigma| < \bar\sigma ,

    \begin{aligned} &(24\mathbb{T}||A^{*}||^{2} ||F^{*}||^{2} +2\sigma^{2})\theta/\vartheta \exp\{8\mathbb{T}(4\mathbb{T}(||D||^{2} +||C||^{2}+3||A^{*}||^{2}k^{2}) +\sigma^{2})\} +2\theta \exp\{-4\vartheta \mathbb{T}\} < 1. \end{aligned}

    Let

    \begin{aligned} \gamma = -\ln\biggl\{(24\mathbb{T}||A^{*}||^{2} ||F^{*}||^{2} +2\sigma^{2})\theta/\vartheta \exp\{8\mathbb{T}(4\mathbb{T}(||D||^{2} +||C||^{2}+3||A^{*}||^{2}k^{2}) +\sigma^{2})\} +2\theta \exp\{-4\vartheta \mathbb{T}\} \biggr\}/\mathbb{T}. \end{aligned}

    So, \gamma > 0 ,

    \begin{equation} \begin{aligned} \sup _{t_{0}+\mathbb{T} \leq t \leq t_{0} +2 \mathbb{T}}||e(t)|| \leq \exp (-\gamma \mathbb{T}) \left(\sup _{t_{0} \leq t \leq t_{0}+\mathbb{T}}||e(t)||\right). \end{aligned} \end{equation} (3.5)

    Then, for any nonnegative integer \mathbb{M} = 1, 2, \cdots , when t\geq t_{0}+(\mathbb{M}-1)\mathbb{T} ,

    \begin{equation} \begin{aligned} e\left(t ; t_{0}, e_{0}\right) = e\left(t ; t_{0}+(\mathbb{M}-1)\mathbb{T}, e(t_{0}+(\mathbb{M}-1)\mathbb{T})\right). \end{aligned} \end{equation} (3.6)

    From (3.5) and (3.6)

    \begin{aligned} \sup\limits_{t_{0}+\mathbb{M}\mathbb{T} \leq t \leq t_{0} +(m+1)\mathbb{T}}||e\left(t ; t_{0}, e_{0}\right)|| & = \Biggl(\sup _{t_{0}+(\mathbb{M}-1)\mathbb{T}+\mathbb{T} \leq t \leq t_{0}+(\mathbb{M}-1)\mathbb{T}+2 T} ||e\left(t ; t_{0}+(\mathbb{M}-1)\mathbb{T}, e\left(t_{0}+(\mathbb{M}-1)\mathbb{T} ; t_{0}, e_{0} \right)\right)|| \Biggr) \\ &\leq \exp (-\gamma \mathbb{T})\left(\sup _{t_{0}+(\mathbb{M}-1)\mathbb{T} \leq t \leq t_{0} +m\mathbb{T}}||e\left(t ; t_{0}, e_{0}\right)||\right) \\ &\leq \exp (-\gamma \mathbb{M}\mathbb{T})\left(\sup _{t_{0} \leq t \leq t_{0}+\mathbb{T}}||e\left(t ; t_{0}, e_{0}\right)||\right) \\ & = \aleph \exp (-\gamma \mathbb{M}\mathbb{T}), \\ \end{aligned}

    where

    \begin{aligned} \aleph = \sup\limits_{t_{0} \leq t \leq t_{0}+\mathbb{T}}||e\left(t ; t_{0}, e_{0}\right)||. \end{aligned}

    So for \forall t > t_{0}+\mathbb{T} , there have a nonnegative integer \mathbb{M} such that t_{0}+\mathbb{M}\mathbb{T} \leq t \leq t_{0}+(\mathbb{M}+1)\mathbb{T} ,

    \begin{array}{l} ||e\left(t ; t_{0}, e_{0}\right)|| \leq \aleph \exp \left(-\gamma t+\gamma t_{0}+\gamma T\right) = (\aleph \exp (\gamma T)) \exp \left(-\gamma\left(t-t_{0}\right)\right) . \end{array}

    The condition is also genuine when t_{0} \leq t \leq t_{0}+\mathbb{T} . So system (2.7) is ESt.

    The following analysis considers the influence of the DAs on the ESy of the master-slave system.

    Consider the derive system with the DAs,

    \begin{equation} \begin{aligned} \dot{w}_{p}(t) = &-d_{p}w_{p}(t) +\sum\limits_{q = 1}^{n}a_{pq}(w_{q}(t))f_{q}(w_{q}(t)) +\sum\limits_{q = 1}^{n}b_{pq}(w_{q}(\gamma(t)))g_{q}(w_{q}(\gamma(t))) +I_{p}(t)\\ w_{p}(t_{0}) = &\varphi _{0}, \end{aligned} \end{equation} (3.7)

    where g_{q}(w_{q}(t)) is the activation functions with DAs, a_{pq}(w_{q}(t)) and b_{pq}(w_{q}(\gamma(t))) are the memristive connection weights without and with DAs respectively.

    The corresponding response system:

    \begin{equation} \begin{aligned} &\dot{v}_{p}(t) = -d_{p}v_{p}(t) +\sum\limits_{q = 1}^{n}a_{pq}(v_{q}(t)) f_{q}(v_{q}(t)) +\sum\limits_{q = 1}^{n}b_{pq}(v_{q}(\gamma(t))) g_{q}(v_{q}(\gamma(t))) +I_{p}(t)+u_{p}(t)\\ &v_{p}(t_{0}) = \phi_{0}. \end{aligned} \end{equation} (3.8)

    The Filippov solution of the systems (3.7) and (3.8) are

    \begin{equation} \begin{aligned} \dot{w}_{p}\in&-d_{p}w_{p} +\sum\limits_{q = 1}^{n}K[a_{pq}(w_{q})] f(w_{q}) +\sum\limits_{q = 1}^{n}K[b_{pq}(w_{q}(\gamma))] g(w_{q}(\gamma))+w_{p}, \end{aligned} \end{equation} (3.9)
    \begin{equation} \begin{aligned} \dot{v}_{p}\in&-d_{p}v_{p} +\sum\limits_{q = 1}^{n}K[a_{pq}(v_{q})]f(v_{q}) +\sum\limits_{q = 1}^{n}K[b_{pq}(v_{q}(\gamma))] g(v_{q}(\gamma))+w_{p}+u_{p},\\ \end{aligned} \end{equation} (3.10)

    in which w(\gamma) = w(\gamma(t)) , v(\gamma) = v(\gamma(t)) .

    The set-valued maps be defined as follows

    K\left[b_{pq}\left(\tilde{w}_{q}\right)\right] = \left\{\begin{array}{ll} \acute{b}_{pq}, & \left|\tilde{w}_{q}\right| < T_{q}, \\ \overline{\operatorname{co}}\left\{\acute{b}_{pq}, \grave{b}_{pq}\right\}, & \left|\tilde{w}_{q}\right| = T_{q}, \\ \grave{b}_{pq}, & \left|\tilde{w}_{q}\right| > T_{q}, \end{array}\right. K\left[b_{pq}\left(\tilde{v}_{q}\right)\right] = \left\{\begin{array}{ll} \acute{b}_{pq}, & \left|\tilde{v}_{q}\right| < T_{q}, \\ \overline{\operatorname{co}}\left\{\acute{b}_{pq}, \grave{b}_{pq}\right\}, & \left|\tilde{v}_{q}\right| = T_{q}, \\ \grave{b}_{pq}, & \left|\tilde{v}_{q}\right| > T_{q}, \end{array}\right.

    where \tilde{w}_{q}, \tilde{v}_{q} to replace w_{q}(\gamma), v_{q}(\gamma) . K[a_{pq} (\tilde{w}_{q})] and K[a_{pq}(\tilde{v}_{q})] are all closed, convex and compact about \tilde{w}_{q} , \tilde{v}_{q} .

    There exist

    \begin{array}{ll} \bar{a}_{pq}\left(w_{q}\right) \in K\left[a_{pq} \left(w_{q}\right)\right], & \bar{b}_{pq} \left(\tilde{w}_{q}\right) \in K\left[b_{pq}\left(\tilde{w}_{q}\right)\right], \\ \\\check{a}_{pq}\left(v_{q}\right) \in K\left[a_{pq} \left(v_{q}\right)\right], &\check{b}_{pq}\left(\tilde{v}_{q}\right) \in K\left[b_{pq} \left(\tilde{v}_{q}\right)\right]. \end{array}

    Let e_{p} = v_{p}-w_{p} ,

    \begin{equation} \begin{aligned} \dot{e}_{p} = &-(d_{p}+\xi_{p})e_{q} +\sum\limits_{q = 1}^{n}\hat{a}_{pq}(e_{q}) f_{q}(e_{q}) +\sum\limits_{q = 1}^{n}\hat{b}_{pq} (e_{q}(\gamma)) g_{q}(e_{p}(\gamma)), \end{aligned} \end{equation} (3.11)

    where \hat{b}_{pq}(e_{q})g_{q}(e_{p}) = \bar{b}_{pq}(w_{q})g_{q}(w_{p}) -\check{b}_{pq}(v_{q})g_{q}({v_{p}}) .

    The following error system without DAs,

    \begin{equation} \begin{aligned} \dot{z}_{p} = -(d_{p}+\xi_{p})z_{p} +\sum\limits_{q = 1}^{n}\hat{a}_{pq}(z_{q})f_{q}(z_{q}) +\sum\limits_{q = 1}^{n}\hat{b}_{pq}(z_{q})g_{q}(z_{q}).\\ \end{aligned} \end{equation} (3.12)

    The (3.11) and (3.12) can be rewritten as

    \begin{equation} \begin{aligned} &\dot{e}(t) = -De(t)+\hat{A}Fe(t)+\hat{B}G(e(\gamma(t)))-Ce(t),\\ \end{aligned} \end{equation} (3.13)
    \begin{equation} \begin{aligned} &\dot{z}(t) = -Dz(t)+\hat{A}Fz(t)+\hat{B}G(e(t))-Cz(t),\\ \end{aligned} \end{equation} (3.14)

    where \hat{B} = \left(\hat{b}_{pq}\right)_{n \times n} , G(e(t)) = (g_{1}(e_{1}(t)), \cdots, g_{n}(e_{n}(t)))^{T} .

    Lemma 3. Consider the MNNs (3.11) with DAs and the Assumptions 3 and 4 hold, the following inequality is established,

    \begin{equation} \begin{aligned} e(\gamma(t))\le \mu e(t). \end{aligned} \end{equation} (3.15)

    Proof. For \gamma(t) = \eta_{k} and \eta_{k} \in [\rho_{k}, \rho_{k+1}] ,

    \begin{aligned} e(t) = &e(\eta_{k})+\int_{\eta_{k}}^{t} (-De(s)+\hat{A}F(e(s)) +\hat{B}(G(e(s)))-Ce(s))ds. \end{aligned}

    Utilizing Lemmas 1 and 2,

    \begin{aligned} ||e(t)||\le& ||e(\eta_{k})||+||\int_{\eta_{k}}^{t} (-De(s)+\hat{A}F(e(s)) +\hat{B}Ge(\eta_{k})-Ce(s))ds||\\ \le&(1+||B^{*}||||G^{*}||\rho)||e(\eta_{k})||+\int_{\eta_{k}}^{t} (||C||+||D|| +||A^{*}||||F^{*}||)||e(s)||ds\\ \le& [(1+||B^{*}||G^{*}||\rho)||e(\eta_{k})||] \exp\{(||C||+||D|| +||A^{*}||||F^{*}||)\rho\},\\ \end{aligned}

    where B^{*} = (b_{pq}^{*})_{n\times n} , G^{*} = \max_{q\in\mathbb{N}}\{g_{q}^{*}\} , then

    \begin{aligned} ||e(\eta_{k})||\le&||e(t)||+\rho||B^{*}||||G^{*}|| ||e(\eta_{k})|| +\int_{\eta_{k}}^{t}(||C||+||D||+ ||A^{*}||||F^{*}||)||e(s)||ds\\ \le&||e(t)||+(h_{1}\rho+h_{2}\rho(1+h_{1}\rho)) ||e(\eta_{k})||\exp\{h_{2}\rho\}, \end{aligned}

    where h_{1} = ||B^{*}||||G^{*}|| , h_{2} = ||C||+||D||+||A^{*}||||F^{*}|| .

    \begin{aligned} \biggl(1-(h_{1}\rho+h_{2}\rho (1+h_{1}\rho)\exp\{h_{2}\rho\})\biggr)||e(\eta_{k})|| \le||e(t)||. \end{aligned}

    Therefore, for Assumption 4,

    \begin{equation} \begin{aligned} ||e(\eta_{k})||\le& \biggl(1-(h_{1}\rho+h_{2}\rho (1+h_{1}\rho)\exp\{h_{2}\rho\})\biggr)^{-1}||e(t)||\\ = &\mu ||e(t)||, \end{aligned} \end{equation} (3.16)

    where \mu = \biggl(1-(h_{1}\rho+h_{2}\rho (1+h_{1}\rho)\exp\{h_{2}\rho\})\biggr)^{-1} , for t \in [\rho_{k}, \rho_{k+1}] . With regards to arbitrary values of t and k , (3.16) holds for t\in \mathbb{R}^{+} .

    Remark 2. When considering MNNs (3.11) on the interval [\rho_{k}, \rho_{k+1}] , where k\in\mathbb{N} , if \rho_{k}\le t < \eta_{k} , MNNs (3.11) behaves as an advanced system. Conversely, if \eta_{k} < t \le \rho_{k+1} , MMN (3.11) behaves as a retarded system.

    Theorem 2. If Assumptions 1–4 hold, MNNs (3.11) is ESt. Then MMNs (3.12) is ESt, that is, the derive system MNNs (3.7) and the response system MNNs (3.8) are ESy. If |\rho|\le \min\{\bar{\rho}, \tilde{\rho}\} , where \bar{\rho} is a unique nonnegative solution of the transcend equation:

    \begin{equation} \begin{aligned} k_{2}\alpha/\beta\exp\{2k_{1}\mathbb{T}\}+ \alpha\exp\{-\beta \mathbb{T}\} = 1, \end{aligned} \end{equation} (3.17)

    where \mathbb{T} > \ln(\alpha)/\beta , k_{1} = ||C||+||D||+||A^{*}||||F^{*}||+ \mu||B^{*}||||G^{*}|| , k_{2} = (1+\mu)||B^{*}||||G^{*}|| .

    The \tilde{\rho} is a unique positive solution of the transcend equation:

    \begin{equation} \begin{aligned} (h_{1}\rho+h_{2}\rho (1+h_{1}\rho)\exp\{h_{2}\rho\}) = 1. \end{aligned} \end{equation} (3.18)

    Proof. Utilizing Lemmas 1 and 3, initial value \psi^{1} = \psi^{2} , we can conclude the following

    \begin{equation} \begin{aligned} ||z(t)-e(t)||\le&||\int_{t_{0}}^{t} \left[-(C+D)(z(s)-e(s))+\hat{A}F(z(s)) -\hat{A}F(e(s))+\hat{B}G(z(s))- \hat{B}G((\gamma(s)))\right]ds\\ \le&\int_{t_{0}}^{t} \left((||C||+||D|| +||A^{*}|||F^{*}||) ||z(s)-e(s)||ds +||B^{*}||||G^{*}|| ||z(s)|| +||B^{*}||||G^{*}||||e(\gamma(s))||\right)ds\\ \le&\int_{t_{0}}^{t} ((||C||+||D||+||A^{*}|||F^{*}||)||z(s) -e(s)||\\ &+||B^{*}||||G^{*}|| ||z(s)|| +\mu||B^{*}||||G^{*}||||e(s)-z(s)+z(s)||)ds\\ \le&\int_{t_{0}}^{t} (||C||+||D||+||A^{*}|||F^{*}|| +\mu||B^{*}||||G^{*}||) ||z(s)-e(s)||ds\\ &+\int_{t_{0}}^{t}((1+\mu)||B^{*}||||G^{*}||) ||z(s)||ds\\ \le&\int_{t_{0}}^{t}k_{1}(z(s)-e(s))ds +k_{2}||\psi^{2}||\alpha/\beta, \end{aligned} \end{equation} (3.19)

    where k_{1} = ||C||+||D||+||A^{*}||||F^{*}||+ \mu||B^{*}||||G^{*}|| , k_{2} = (1+\mu)||B^{*}||||G^{*}|| .

    By Lemma 2, when t_{0}+\mathbb{T}-\rho\le t\le t_{0}+2\mathbb{T} ,

    \begin{aligned} ||e(t)-z(t)||\le k_{2}\alpha/\beta||\psi^{2}||\exp\{2k_{1} \mathbb{T}\}. \end{aligned}

    So, when t_{0}+\mathbb{T} -\rho\le t\le t_{0}+2\mathbb{T} , from (3.19) and the global exponential stability of (3.12),

    \begin{equation} \begin{aligned} ||e(t)|| = &||e(t)-z(t)+z(t)||\\ \le&k_{2}\alpha/\beta||\psi^{2}||\exp\{2k_{1}\mathbb{T}\} +\alpha||\psi^{2}||\exp\{-\beta \mathbb{T}\}\\ \le&\Big(k_{2}\alpha/\beta\exp\{2k_{1}\mathbb{T}\} +\alpha\exp\{-\beta \mathbb{T}\}\Big) \biggl(\sup _{t_{0}-\rho \leq t \leq t_{0} +\mathbb{T}}||e(t)||\Biggr). \end{aligned} \end{equation} (3.20)

    From (3.20), when |\rho|\le\bar{\rho} ,

    \begin{aligned} k_{2}\alpha/\beta\exp\{2k_{1}T\}+ \alpha\exp\{-\beta T\} < 1. \end{aligned}

    Let \kappa_{1} = -(\ln (k_{2}\alpha/\beta\exp\{2k_{1}T\}+ \alpha\exp\{-\beta T\}))/\mathbb{T}, \kappa_{1} > 0 , when t_{0}-\rho+\mathbb{T}\le t\le t_{0}+ 2\mathbb{T} ,

    \begin{equation} \begin{aligned} \sup\limits_{t_{0}-\rho+\mathbb{T}\le t\le t_{0}+2\mathbb{T}}||e(t)|| \le \exp (-\rho \mathbb{T}) \sup\limits_{t_{0}-\rho\le t\le t_{0}+\mathbb{T}}||e(t)||. \end{aligned} \end{equation} (3.21)

    Consider the existence and uniqueness of the solution e(t) of (8), when t > t_{0}-\xi+(\mathbb{M}-1)\mathbb{T} ,

    \begin{equation} \begin{aligned} e(t,t_{0},x_{0}) = &e(t,t_{0}-\xi+(\mathbb{M}-1)\mathbb{T}, e(t_{0}-\xi+(\mathbb{M}-1)\mathbb{T},t_{0},x_{0})). \end{aligned} \end{equation} (3.22)

    From (3.21) and (3.22),

    \begin{aligned} \sup\limits_{t_{0}-\rho+m\mathbb{T}\le t\le t_{0}+(m+1)\mathbb{T} }||e(t,t_{0},e_{0})|| & = \sup\limits_{t_{0}-\rho+(\mathbb{M}-1)\mathbb{T}+\mathbb{T} \le t\le t_{0}+(\mathbb{M}-1)\mathbb{T}+2\mathbb{T}} ||e(t,t_{0}-\rho+(\mathbb{M}-1)\mathbb{T}, \\e(t_{0}-\rho +(\mathbb{M}-1)\mathbb{T} ;t_{0},e_{0}))|| &\le \exp(-\rho \mathbb{T})\sup\limits_{t_{0}-\rho+(\mathbb{M}-1)\mathbb{T} \le t\le t_{0}+m\mathbb{T}} ||e(t;t_{0},e_{0})||\\ &\le \exp (-\rho m\mathbb{T})\sup\limits_{t_{0}-\rho\le t\le t_{0}+ \mathbb{T}}||e(t;t_{0},e_{0})||\\ & = \kappa_{2} \exp {-\rho m\mathbb{T}}, \end{aligned}

    where \kappa_{2} = \sup_{t_{0}-\rho\le t\le t_{0}+\mathbb{T}} ||e(t; t_{0}, e_{0})|| .

    To go a step further, there is the only scalar m\in \mathbb{N} such that t_{0}-\rho +(\mathbb{M}-1)\mathbb{T} \le t\le t_{0}+\mathbb{M}\mathbb{T} ,

    \begin{equation} \begin{aligned} ||e(t;t_{0},x_{0})||&\le \kappa_{2}\exp(-\rho \mathbb{M}\mathbb{T} ) \le \kappa_{2}\exp(\rho\mathbb{T})\exp(-\rho(t-t_{0})). \end{aligned} \end{equation} (3.23)

    Clerly, (3.23) holds for t_{0}-\rho\le t\le t_{0}+\mathbb{T} .

    The following consider MNNs with SPs and DAs,

    \begin{equation} \begin{aligned} dw_{p}(t) = &[-d_{p}w_{p}(t) +\sum\limits_{q = 1}^{n}a_{pq}(w_{q}(t))f_{q}(w_{q}(t)) +\sum\limits_{q = 1}^{n}b_{pq}(w_{q}(\gamma(t))) g_{q}(w_{q}(\gamma(t)))+w_{p}(t)]dt +\sigma w_{p}(t)d\omega(t).\\ \end{aligned} \end{equation} (3.24)

    The corresponding response system,

    \begin{equation} \begin{aligned} d{v}_{p}(t) = &[-d_{p}v_{p}(t) +\sum\limits_{q = 1}^{n}a_{pq}(v_{q}(t))f(v_{q}(t))+\sum\limits_{q = 1}^{n}b_{pq}(v_{q}(\gamma(t))) g_{q}(v_{q}(\gamma(t)))+w_{p}(t)+u_{p}(t)]ds\\ & +\sigma v_{p}(t)d\omega(t).\\ \end{aligned} \end{equation} (3.25)

    Let e_{p} = v_{p}-w_{p} ,

    \begin{equation} \begin{aligned} de_{p} = &[-d_{p}e_{p}+\sum\limits_{q = 1}^{n} \hat{a}_{pq}(e_{q})f_{q}(e_{p}) +\sum\limits_{q = 1}^{n}\hat{b}_{pq}(e_{q}(\gamma)) g_{q}(e_{q}(\gamma))-\xi_{p}e_{p}]ds +\sigma e_{p}d\omega.\\ \end{aligned} \end{equation} (3.26)

    The original system is

    \begin{equation} \begin{aligned} \dot{z}_{p} = &-d_{p}z_{p}+\sum\limits_{q = 1}^{n} \hat{a}_{pq}(z_{q})f_{q}(z_{q}) +\hat{b}_{pq}(z_{q}) g_{q}(z_{q})-\xi_{p}e_{p}.\\ \end{aligned} \end{equation} (3.27)

    Further,

    \begin{aligned} &de(t) = [-De(t)+\hat{A}Fe(t)+\hat{B}G((\gamma(t))) -Ce(t)]dt+\sigma e(t)d\omega(t)\\ &\dot{z}(t) = -De(t)+\hat{A}Fe(t)+\hat{B}G((t))-Ce(t). \end{aligned}

    Lemma 4. Let Assumptions 3 and 5 hold, then the following inequality

    \begin{equation} \begin{aligned} E||e(\gamma(t))||^{2}\le \lambda ||e(t)||^{2} \end{aligned} \end{equation} (3.28)

    holds for all t\in \mathbb{R}^{+} , where \lambda = 3(1-\varpi)^{-1} , \varpi = \rho(3l_{1}+9l_{2}+l_{1}^{2} +3l_{1}l_{2}) .

    Proof. For \gamma(t) = \eta_{k} , t\in [\rho_{k}, \rho_{k+1}] , \forall t\in \mathbb{R}^{+} , \exists k\in \mathbb{N} , we have

    \begin{aligned} E||e(t)||^{2}\le& E||e(\eta_{k})+ \int_{\eta_{k}}^{t} \left[-(C+D)e(s) +\hat{A}F(e(s)) +\hat{B}G((\eta_{k}))\right]ds +\int_{\eta_{k}}^{t}\sigma e(s)d\omega(s)||^{2}\\ \le&3\left[E||e(\eta_{k})||^{2}+E||\int_{\eta_{k}}^{t} \left[-(C+D)e(s)+\hat{A}F(e(s)) +\hat{B}G((\eta_{k}))\right]ds||^{2}+E||\int_{\eta_{k}}^{t} \sigma e(s)d\omega(s)||^{2}\right]\\ \le &3\Big[E||e(\eta_{k})||^{2}+3\rho E\int_{\eta_{k}} ^{t}2(||C||^{2}+||D||^{2})||e(s)||^{2} +||A^{*}||^{2}||F^{*}||^{2}||e(s)||^{2}\\ &+||B^{*}||^{2}||G^{*}||^{2}||e(\eta_{k})||^{2} +\sigma^{2}\int_{\eta_{k}}^{t}E||e(s)||^{2}ds\Big]\\ \le&3(1+3\rho^{2}||B^{*}||^{2}||G^{*}||^{2})E||e(\eta_{k})||^{2} +3(3\rho(2||C||^{2}+2||D||^{2}+ ||A^{*}||^{2} ||F^{*}||^{2}) +\sigma^{2})\int_{\eta_{k}}^{t}E||e(s)|| ^{2}ds.\\ \end{aligned}

    Applying Lemma 2,

    \begin{aligned} E||e(t)||^{2}\le&(3+9\rho^{2}||B^{*}||^{2}||G^{*}||^{2}) E||e(\eta_{k})||^{2} \exp\{3\rho(3\rho(2||C||^{2} +2||D||^{2} +||A^{*}||^{2}||F^{*}||^{2}) +\sigma^{2})\}\\ = &(3+l_{1})E||e(\eta_{k})||^{2}\exp\{3\rho l_{2}\}, \end{aligned}

    where l_{1} = 9\rho^{2}||B^{*}||^{2}||G^{*}||^{2} , l_{2} = 3\rho(2||C||^{2} +2||D||^{2}+||A^{*}||^{2}||F^{*}||^{2}) +\sigma^{2} .

    Similarly, for t \in [\rho_{k}, \rho_{k+1}] ,

    \begin{aligned} E||e(\eta_{k})||^{2}\le&3\biggl[E||e(s)||^{2} +E||\int_{eta_{k}}^{t}(-(C+D)e(s) +\hat{A}Fe(s)+\hat{B}Ge(s))ds||^{2} +E||\int_{\eta_{k}}^{t}\sigma e(s)d\omega(s)||^{2} \biggr]\\ \le&3\biggl[E||e(s)||^{2} +3\rho E\int_{eta_{k}}^{t}(2(||C||^{2} +||D||^{2})||e(s)||^{2} +||A^{*}||^{2}||F^{2}||||e(s)||^{2}\\ &+||B^{*}||^{2}||G^{*}||^{2}||e(\eta_{k})||^{2})ds +\sigma^{2}\int_{\eta_{k}}^{t} E||e(s)||^{2}ds\biggr]\\ = &3E||e(s)||^{2}+9\rho^{2}||B^{*}||^{2} ||G^{*}||^{2}||e(\eta_{k})||^{2} +3(3\rho((2||C^{*}||^{2}+2||D^{*}||^{2})\\ &+||A^{*}||^{2}||F^{*}||^{2}) +\sigma^{2})\int_{\eta_{k}}^{t} E||e(s)||^{2}ds\\ = &3E||e(s)||^{2}+\rho(3l_{1}+9l_{2}+l_{1}^{2} +3l_{1}l_{2})\exp\{3\rho l_{2}\} ||e(\eta_{k})||^{2}. \end{aligned}

    By the Assumption 5,

    \begin{equation} \begin{aligned} E||e(\eta_{k})||^{2}\le&3(1-\varpi)^{-1} E||e(s)||^{2}\\ = &\lambda E||e(s)||^{2},\\ \end{aligned} \end{equation} (3.29)

    where \varpi = \rho(3l_{1}+9l_{2}+l_{1}^{2} +3l_{1}l_{2}) , \lambda = 3(1-\varpi)^{-1} . Therefore, (3.29) holds for t \in [\rho_{k}, \rho_{k+1}] . By the randomicities of t and k , (3.29) holds for all t\in\mathbb{R} .

    In the following, we investigate the effects of DAs and SPs on the robustness of ESy of MNNs (3.26).

    Theorem 3. If {Assumptions 3–5} and {Definition 4} hold, MNNs (3.27) is ESt. Then MMNs (3.26) is ESt, that is, the derive system MNNs (3.24) and the response system MNNs (3.25) are ESy. If |\sigma|\le \bar{\sigma} , |\rho| \le\min\{\bar{\rho}, \tilde{\rho}\} where \bar{\rho} is a unique nonnegative solution of the transcend equation:

    \begin{equation} \begin{array}{c} \upsilon_{2}\theta ||\psi^{2}||^{2} /\beta\exp\{2\upsilon_{1}\mathbb{T}\} +2\theta ||\psi^{2}||^{2}\exp\{-2\mathbb{T}\beta\} = 1, \end{array} \end{equation} (3.30)

    where \upsilon_{1} = (24\mathbb{T}(||C||^{2}+||D||^{2} +2||A^{*}||^{2}||F^{*}||^{2} +2\lambda^{2}||B^{*}||^{2}||G^{*}||^{2})+4\sigma^{2}) , \upsilon_{2} = (12\mathbb{T}(2+2\lambda^{2}) ||B^{*}||^{2}||G^{*}||^{2} +2\sigma^{2}) . The \tilde{\rho} is a unique nonnegative solution of the transcend equation:

    \begin{equation} \begin{aligned} l_{1} +9\rho l_{1}l_{2}\exp\{3\rho l_{2}\} = 1. \end{aligned} \end{equation} (3.31)

    The \bar{\sigma} is a unique nonnegative solution of the transcend equation:

    \begin{equation} \begin{aligned} &(36\mathbb{T}(||A^{*}||^{2} ||F^{*}||^{2}+||B^{*}||^{2}||G^{*}||^{2}) +2\sigma^{2})\theta/\vartheta \exp\{12\mathbb{T}(4\mathbb{T}(||D||^{2}\\ &+||C||^{2}+3||A^{*}||^{2}||F^{*}||^{2} +3||B^{*}||^{2}||G^{*}||^{2}) +\sigma^{2})\} +2\theta \exp\{-4\vartheta \mathbb{T}\} = 1. \end{aligned} \end{equation} (3.32)

    Proof. When t_{0}-\rho\le t\le t_{0}+2\mathbb{T} , By applying Lemma 4, initial value \psi^{1} = \psi^{2} ,

    \begin{aligned} E||z(t)-e(t)||^{2}\le& 2E||\int_{t_{0}}^{t} [-(C+D)(z(s)-e(s)) +\hat{A}F(z(s))-\hat{A}F(e(s)) +\hat{B}G(z(s))-\hat{B}G((\gamma(s)))]ds||^{2}\\ &+2E||\int_{t_{0}}^{t} \sigma e(s)d\omega(s)||^{2}\\ \le&2E\int_{t_{0}}^{t}1^{2}ds\int_{t_{0}}^{t} 6[(||C||^{2}+||D||^{2}) +||A^{*}||^{2}||F^{*}||^{2} ||z(s)||^{2} +||A^{*}||^{2}||F^{*}||^{2}||e(s)||^{2}\\ & +||B^{*}||^{2}||G^{*}||^{2}||z(s)||^{2}+||B^{*}||^{2}||G^{*}||^{2}|| e(\gamma(s))||^{2}]ds +2\sigma^{2}||\int_{t_{0}}^{t}\ E|| e(s)||^{2}ds\\ \le&4\mathbb{T}E\int_{t_{0}}^{t} 6[(||C||^{2}+||D||^{2} +2||A^{*}||^{2}||F^{*}||^{2}) ||z(s)-e(s)||^{2} +(2||A^{*}||^{2}||F^{*}||^{2}\\ &+||B^{*}||^{2}||G^{*}||^{2})||z(s)||^{2} +2\lambda^{2}||B^{*}||^{2}||G^{*}||^{2}|| e(s)-z(s)+z(s)||^{2}]ds\\ &+2\sigma^{2}\int_{t_{0}}^{t}\ E|| e(s)-z(s)+z(s)||^{2}ds\\ \le&(24\mathbb{T}(||C||^{2}+||D||^{2} +2||A^{*}||^{2}||F^{*}||^{2} +2\lambda^{2}||B^{*}||^{2}||G^{*}||^{2})+4\sigma^{2}) \int_{t_{0}}^{t}E||z(s)-e(s)||^{2}\\ &+(24\mathbb{T}(2+2\lambda^{2})||B^{*}||^{2}||G^{*}||^{2} +4\sigma^{2})\int_{t_{0}}^{t}E||e(s)||^{2}\\ \le&(24\mathbb{T}(||C||^{2}+||D||^{2} +2||A^{*}||^{2}||F^{*}||^{2} +2\lambda^{2}||B^{*}||^{2}||G^{*}||^{2})+4\sigma^{2}) \int_{t_{0}}^{t}E||z(s) -e(s)||^{2}\\ &+(12\mathbb{T}(2+2\lambda^{2})||B^{*}||^{2}||G^{*}||^{2} +2\sigma^{2})\theta ||\psi^{2}||^{2}/\vartheta \\ = &\upsilon _{1}\int_{t_{0}}^{t}E||z(s)-e(s)||^{2}ds +\upsilon_{2}\theta ||\psi^{2}||^{2}/\vartheta, \end{aligned}

    where \upsilon_{1} = (24\mathbb{T}(||C||^{2}+||D||^{2} +2||A^{*}||^{2}||F^{*}||^{2} +2\lambda^{2}||B^{*}||^{2}||G^{*}||^{2})+4\sigma^{2}) , \upsilon_{2} = (12\mathbb{T}(2+2\lambda^{2}) ||B^{*}||^{2}||G^{*}||^{2} +2\sigma^{2}) .

    When t_{0}-\rho\le t\le t_{0}+2\mathbb{T} , By applying Lemma 2,

    \begin{aligned} E||z(t)-e(t)||^{2} \le &\upsilon_{2}\theta ||\psi^{2}||^{2} /\vartheta \exp\{2\upsilon_{1}\mathbb{T}\} \times \sup\limits_{t_{0}-\rho\le t\le t_{0}+\mathbb{T}} E||z(t)||^{2}. \end{aligned}

    Then,

    \begin{aligned} E||e(t)||^{2}\le&2E||z(t)-e(t)||^{2}+2E||z(t)||^{2}\\ \le&\upsilon_{2}\theta ||\psi^{2}||^{2} /\vartheta \exp\{2\upsilon_{1}\mathbb{T}\} \sup\limits_{t_{0}-\rho\le t\le t_{0}+\mathbb{T}} E||z(t)||^{2} +2\theta ||\psi^{2}||^{2}\exp\{-2\vartheta (t-t_{0})\}\\ \le&(\upsilon_{2}\theta ||\psi^{2}||^{2} /\vartheta \exp\{2\upsilon_{1}\mathbb{T}\} +2\theta\exp\{-2\mathbb{T}\vartheta \}) \sup\limits_{t_{0}-\rho\le t\le t_{0}+\mathbb{T}} E||e(t)||^{2}.\\ \end{aligned}

    When |\rho|\le\min\{\bar{\rho}, \tilde{\rho}\} , |\sigma|\le\bar{\sigma} ,

    \begin{aligned} \upsilon_{2}\theta ||\psi^{2}||^{2} /\vartheta \exp\{2\upsilon_{1}\mathbb{T}\} +2\theta ||\psi^{2}||^{2}\exp\{-2\mathbb{T}\beta\}\le 1. \end{aligned}

    We demonstrate the aforementioned theoretical results through three numerical simulations.

    Example 1. Consider two dimensional MNNs with SPs.

    \begin{equation} \begin{aligned} dw_{p}(t) = &[-d_{p}w_{p}(t) +\sum\limits_{q = 1}^{2}a_{pq}(w_{q}(t))f_{q}(w_{q}(t)) +w_{p}(t)]dt+\sigma w_{p}(t)d\omega(t),\; i = 1,2,\\ \end{aligned} \end{equation} (4.1)

    where

    a_{11}\left(w_{1}\right) = \left\{ \begin{array}{l} 0.1, \quad\left|w_{1}\right| \leq 1, \\ -0.1, \quad\left|w_{1}\right| > 1, \end{array}\right. a_{12}\left(w_{2}\right) = \left\{ \begin{array}{l} 0.2, \quad\left|w_{2}\right| \leq 1, \\ -0.2, \quad\left|w_{2}\right| > 1, \end{array}\right.
    a_{21}\left(w_{1}\right) = \left\{ \begin{array}{l} 0.1, \quad\left|w_{1}\right| \leq 1, \\ -0.1, \quad\left|w_{1}\right| > 1, \end{array}\right. a_{22}\left(w_{2}\right) = \left\{ \begin{array}{l} 0.2, \quad\left|w_{2}\right| \leq 1, \\ -0.2, \quad\left|w_{2}\right| > 1, \end{array}\right.

    w_{1} = (w_{1}, w_{2}) , d_{1} = d_{2} = 1, q = 1, 2 , f_{q}(w_{q}) = tanh(w_{q}) , w_{1} = w_{2} = 0 , \phi_{1} = (0.3, 0.35)^{T}, \phi_{2} = (0.2, 0.25)^{T} .

    The response system is

    \begin{equation} \begin{aligned} dv_{p}(t) = &[-d_{p}v_{p}(t) +\sum\limits_{q = 1}^{2}a_{pq}(v_{q}(t))f(v_{q}(t)) +w_{p}(t)+u_{p}(t)]dt +\sigma w_{p}(t)d\omega(t),\; i = 1,2,\\ \end{aligned} \end{equation} (4.2)

    where \varphi_{1} = (-0.1, -0.15)^{T}, \varphi_{2} = (-0.2, -0.25)^{T} .

    Let \theta = 1.1 , \vartheta = 0.2 , \mathbb{T}\le\ln(\theta)/\vartheta = 0.01 , ||A^{*}|| = 0.1, ||F^{*}|| = 1, ||C|| = 0.7, ||D|| = 1 .

    Solving the following transcedental equation,

    \begin{aligned} 2.2(24&\times0.01\times0.01+2\sigma)\exp\{ 0.08(0.04(1+0.49 +0.03)+\sigma)\} +2.2\exp\{-4.4\times 0.2\} = 1. \end{aligned}

    We can obtain \bar{\sigma} = 0.1364 , let \sigma = 0.04 , \sigma = 0.06 , \sigma = 0.1 , The state trajectories of MMNs (4.1) and MNNs (4.2) are shown in Figures 13, respectively. It can be seen from the figures that when time tends to infinity, the states of a and b tend to 0 This can show that when the perturbations intensity \sigma is less than \bar{\sigma} , the drive-response systems can achieve Esy. When \sigma = 0.25 , The states are illustrated in Figure 4.

    Figure 1.  State of x(t) and y(t) with \sigma = 0.04 .
    Figure 2.  State of x(t) and y(t) with \sigma = 0.06 .
    Figure 3.  State of x(t) and y(t) with \sigma = 0.1 .
    Figure 4.  State of x(t) and y(t) with \sigma = 0.25 .

    Example 2. Consider two-dimensional MNNs with DAs,

    \begin{equation} \begin{aligned} \dot{w}_{p}(t) = &-d_{p}w_{p}(t) +\sum\limits_{q = 1}^{2}a_{pq}(w_{q}(t)) f_{q}(w_{q}(t)) +\sum\limits_{q = 1}^{2}b_{pq}(w_{q}(\gamma(t))) g_{q}(w_{q}(\gamma(t))) +w_{p}(t),\; i = 1,2,\\ \end{aligned} \end{equation} (4.3)

    where

    a_{11}\left(w_{1}\right) = \left\{ \begin{aligned} 0.125, &\quad\left|w_{1}\right| \leq 1, \\ -0.125, &\quad\left|w_{1}\right| > 1, \end{aligned}\right.\; a_{12}\left(w_{2}\right) = \left\{ \begin{aligned} 0.15, &\quad\left|w_{2}\right| \leq 1, \\ -0.15, &\quad\left|w_{2}\right| > 1, \end{aligned}\right.
    a_{21}\left(w_{1}\right) = \left\{ \begin{aligned} 0.125, &\quad\left|w_{1}\right| \leq 1, \\ -0.125, &\quad\left|w_{1}\right| > 1, \end{aligned}\right.\; a_{22}\left(w_{2}\right) = \left\{ \begin{aligned} 0.15, &\quad\left|w_{2}\right| \leq 1, \\ -0.15, &\quad\left|w_{2}\right| > 1, \end{aligned}\right.
    b_{11}\left(\tilde{w}_{1}\right) = \left\{ \begin{aligned} 0.2, &\quad\left|\tilde{w}_{1}\right| \leq 1, \\ -0.2, &\quad\left|\tilde{w}_{1}\right| > 1, \end{aligned}\right.\; b_{12}\left(\tilde{w}_{2}\right) = \left\{ \begin{aligned} 0.1, &\quad\left|\tilde{w}_{2}\right| \leq 1, \\ -0.1, &\quad\left|\tilde{w}_{2}\right| > 1, \end{aligned}\right.
    b_{21}\left(\tilde{w}_{1}\right) = \left\{ \begin{aligned} 0.2, \quad\left|\tilde{w}_{1}\right| \leq 1, \\ -0.2, \quad\left|\tilde{w}_{1}\right| > 1, \end{aligned}\right.\; b_{22}\left(\tilde{w}_{2}\right) = \left\{ \begin{aligned} 0.1, &\quad\left|\tilde{w}_{2}\right| \leq 1, \\ -0.1, &\quad\left|\tilde{w}_{2}\right| > 1, \end{aligned}\right.

    where \tilde{w}_{q} = w_{q}(\gamma) , w_{1} = (w_{1}, w_{2}) , d_{1} = d_{2} = 1, q = 1, 2 , f_{q}(w_{q}) = tanh(w_{q}) , g_{q}(w_{q}) = |w_{q}+1|-|w_{q}-1| , w_{1} = w_{2} = 0 , \phi_{1} = (3, 4)^{T}, \phi_{2} = (1, 2)^{T} .

    The response system is

    \begin{equation} \begin{aligned} \dot{v}_{p}(t) = &-d_{p}v_{p}(t) +\sum\limits_{q = 1}^{2}a_{pq}(v_{q}(t)) f_{q}(v_{q}(t)) +\sum\limits_{q = 1}^{2}b_{pq}(v_{q}(\gamma(t))) g_{q}(v_{q}(\gamma(t))) +w_{p}(t)+u_{p}(t),\; i = 1,2,\\ \end{aligned} \end{equation} (4.4)

    where \varphi_{1} = (-3, -4)^{T}, \varphi_{2} = (-1, -2)^{T} .

    Let \alpha = 1.1 , \beta = 0.5 , \mathbb{T}\le\ln(\theta)/\vartheta = 0.01 , ||A^{*}|| = 0.726, ||B^{*}||^{2} = 0.1 ||F^{*}|| = 1, ||G^{*}|| = 1 ||C|| = 0.01, ||D|| = -1 .

    Solving the following transcedental equations,

    \begin{aligned} 0.001\rho+1.772\rho(1+1.772\rho) \exp\{1.772\rho\} = 1, \end{aligned}
    \begin{aligned} &0.22(1+\mu)\exp\{0.02(1.772+0.1\mu)\} +1.1\exp\{-0.5\times0.01\} = 1. \end{aligned}

    We can obtain \bar{\rho} = 0.2506 , \tilde{\rho} = 0.3314 , let \rho = 0.2 , \rho = 0.15 , \rho = 0.1 , \rho = 0.25 , when the length of the DAs in the systems is less than the calculated upper bound, we have that MNNs (4.3) and (4.4) with the controllers is ESy. The states of the MNNs (4.3) and (4.4) are shown in Figures 48, respectively.

    Figure 5.  The state with \rho = 0.2 .
    Figure 6.  The state with \rho = 0.15 .
    Figure 7.  The state with \rho = 0.1.
    Figure 8.  The state with \rho = 0.25.

    Example 3. Consider two-dimensional MNNs with DAs and SPs.

    \begin{equation} \begin{aligned} dw_{p}(t) = &-[d_{p}w_{p}(t) +\sum\limits_{q = 1}^{2}a_{pq}(w_{q}(t)) f_{q}(w_{q}(t)) +\sum\limits_{q = 1}^{2}b_{pq}(w_{q}(\gamma(t))) g_{q}(w_{q}(\gamma(t)))\\ &+w_{p}(t)]dt+\sigma w_{p}d\omega(t).\; i = 1,2.\\ \end{aligned} \end{equation} (4.5)

    All other parameters remain the same as in Example 2. The response system is,

    \begin{equation} \begin{aligned} dv_{p}(t) = &-[d_{p}v_{p}(t) +\sum\limits_{q = 1}^{2}a_{pq}(v_{q}(t)) f_{q}(v_{q}(t)) +\sum\limits_{q = 1}^{2}b_{pq}(v_{q}(\gamma(t))) g_{q}(v_{q}(\gamma(t)))\\ &+w_{p}(t)+u_{p}(t)]dt+\sigma v_{p}d\omega(t),\; i = 1,2.\\ \end{aligned} \end{equation} (4.6)

    Solving the transcedental equations. We can obtain \bar{\rho} = 0.1567 , \tilde{\rho} = 0.4374 , \bar{\sigma} = 0.2463 , let \rho = 0.1 , \sigma = 0.04 , we have that MNNs (4.5) and (4.6) is ESy, the state trajectories are shown in Figure 9. when \rho = 0.05 , \sigma = 0.04 , the state trajectories are shown in Figure 10. When \rho = 0.9 , \sigma = 0.1 , the state trajectories are shown in Figure 11.

    Figure 9.  The state of MMNs (51) and (52) with \sigma = 0.1 and \rho = 0.04.
    Figure 10.  The state of MMNs (51) and (52) with \sigma = 0.05 and \rho = 0.04.
    Figure 11.  The state of MMNs (51) and (52) with \sigma = 0.1 and \rho = 0.9.

    In this paper, the robustness analysis of MNNs exponential synchronization problem with DAs and SPs is studied by using the Granwall inequalities and inequality techniques, and a method different from Linear Matrix Inequality method (LMI) and Lyapunov theory is used to solve the synchronization robustness of MNNs.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

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



    [1] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Set. Syst., 20 (1986), 1–6.
    [2] L. A. Zadeh, Fuzzy sets, Inf. Control, 8 (1965), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X doi: 10.1016/S0019-9958(65)90241-X
    [3] E. Szmidt, J. Kacprzyk, Intuitionistic fuzzy sets in some medical applications, In: International conference on computational intelligence, 2206 (2001), 148–151. https://doi.org/10.1007/3-540-45493-4_19
    [4] Z. Wang, F. Xiao, W. Ding, Interval-valued intuitionistic fuzzy jenson-shannon divergence and its application in multi-attribute decision making, Appl. Intell., 2022, 16168–16184. https://doi.org/10.1007/s10489-022-03347-0 doi: 10.1007/s10489-022-03347-0
    [5] J. Liu, J. Mai, H. Li, B. Huang, Y. Liu, On three perspectives for deriving three-way decision with linguistic intuitionistic fuzzy information, Inform. Sci., 588 (2022), 350–380. https://doi.org/10.1016/j.ins.2021.12.072 doi: 10.1016/j.ins.2021.12.072
    [6] F. Bilgili, F. Zarali, M. F. Ilgün, C. Dumrul, Y. Dumrul, The evaluation of renewable energy alternatives for sustainable development in Turkey using intuitionistic fuzzy-TOPSIS method, Renew. Energ., 189 (2022), 1443–1458. https://doi.org/10.1016/j.renene.2022.03.058 doi: 10.1016/j.renene.2022.03.058
    [7] S. Zeng, J. Zhou, C. Zhang, J. M. Merigó, Intuitionistic fuzzy social network hybrid MCDM model for an assessment of digital reforms of manufacturing industry in China, Technol. Forecast. Soc., 176 (2022), 121435. https://doi.org/10.1016/j.techfore.2021.121435 doi: 10.1016/j.techfore.2021.121435
    [8] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl., 35 (1971), 512–517.
    [9] W. J. Liu, 1982. Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Set. Syst., 8 (1982), 133–139.
    [10] E. Yetkin, N. Olgun, Direct product of fuzzy groups and fuzzy rings, Int. Math. Forum., 6 (2011), 1005–1015.
    [11] F. A. Azam, A. A. Mamun, F. Nasrin, Anti-fuzzy ideal of ring, Annal, Fuzzy Math. Inform., 5 (2013), 349–360.
    [12] R. Kellil, Sum and product of fuzzy ideals of ring, Int. J. Math. Comput. Sci., 13 (2018), 187–205.
    [13] J. G. Kim, Fuzzy orders relative to fuzzy subgroups, Inform. Sci., 80 (1994), 341–348. https://doi.org/10.1016/0020-0255(94)90084-1 doi: 10.1016/0020-0255(94)90084-1
    [14] N. Ajmal, Homomorphism of fuzzy groups, correspondence theorem and fuzzy quotient groups, Fuzzy Set. Syst., 61 (1994), 329–339.
    [15] A. K. Ray, On product of fuzzy subgroups, Fuzzy Set. Syst., 105 (1999), 181–183.
    [16] R. Biswas, Intuitionistic fuzzy subgroups, Math. Fortum., 10 (1989), 37–46.
    [17] P. K. Sharma, (α, β)-cut of intuitionistic fuzzy groups, Int. Math. Forum., 6 (2011), 2605–2614.
    [18] P. K. Sharma, On intuitionistic anti-fuzzy subgroup of a group, Int. J. Math. Appl. Statist., 3 (2012), 147–153.
    [19] P. K. Sharma, t- intuitionistic fuzzy subgroups, Int. J. Fuzzy Math. Syst., 2 (2012), 233–243.
    [20] P. K.Sharma, t-intuitionitic fuzzy quotient group, Adv. Fuzzy Math., 7 (2012), 1–9.
    [21] P. K. Sharma, On intuitionistic fuzzy abelian subgroups, Adv. Fuzzy Set. Syst., 12 (2012), 1–16.
    [22] P. K.Sharma, Relationship between alpha-(Anti) fuzzy subgroups and (α, β)-(Anti) fuzzy subgroups, Int. Rev. Pure Appl. Math., 8 (2012), 133–140.
    [23] P. K. Sharma, A. Duggal, Intuitionistic fuzzy Bi-ideals in a rings, P. Int. Confer. Sci. Comput., 7 (2013), 816–822.
    [24] K. A. Dib, On fuzzy spaces and fuzzy group theory, Inform. Sci., 80 (1994), 253–282. https://doi.org/10.1016/0020-0255(94)90079-5 doi: 10.1016/0020-0255(94)90079-5
    [25] K. A. Dib, N. L. Youssef, Fuzzy Cartesian product, fuzzy relations and fuzzy functions, Fuzzy Set. Syst., 41 (1991), 299–315.
    [26] K. A. Dib, A. A. M. Hassan, The fuzzy normal subgroup, Fuzzy Set. Syst., 98 (1998), 393–402.
    [27] M. F. Marashdeh, A. R. Salleh, Intuitionistic fuzzy groups, Asian J. Algebra, 2 (2009), 1–10.
    [28] M. F. Marashdeh, A. R. Salleh, The intuitionistic fuzzy normal subgroup, Int. J. Fuzzy Logic Intell. Syst., 10 (2010), 82–88. https://doi.org/10.5391/IJFIS.2010.10.1.082 doi: 10.5391/IJFIS.2010.10.1.082
    [29] D. Ramot, R. Milo, M. Friedman, A. Kandel, Complex fuzzy sets, IEEE Trans. Fuzzy Syst., 10 (2002), 450–461. https://doi.org/10.1109/91.995119 doi: 10.1109/91.995119
    [30] D. Ramot, M. Friedman, G. Langholz, A. Kandel, Complex fuzzy logic, IEEE Trans. Fuzzy Syst., 11 (2003), 171–186. https://doi.org/10.1109/TFUZZ.2003.814832 doi: 10.1109/TFUZZ.2003.814832
    [31] A. S. M. Alkouri, A. R. Salleh, Complex Atanassov's intuitionistic fuzzy sets, AIP Confer. P., 1482 (2012), 464–470, https://doi.org/10.1063/1.4757515 doi: 10.1063/1.4757515
    [32] A. S. M. Alkouri, A. R. Salleh, Complex Atanassov's intuitionistic fuzzy relation, Abstr. Appl. Anal., 2013 (2013), 287382. https://doi.org/10.1155/2013/287382 doi: 10.1155/2013/287382
    [33] A. S. M. Alkouri, A. R. Salleh, Some operations on complex Atanassov's intuitionistic fuzzy sets, AIP Confer. P., 1571 (2013), 987. https://doi.org/10.1063/1.4858782 doi: 10.1063/1.4858782
    [34] T. Mahmood, Z. Ali, S. Baupradist, R. Chinram, Analysis and applications of Bonferroni mean operators and TOPSIS method in complete cubic intuitionistic complex fuzzy information systems, Symmetry, 14 (2022), 533. https://doi.org/10.3390/sym14030533 doi: 10.3390/sym14030533
    [35] M. Azam, M. S. A. Khan, S. Yang, A decision-making approach for the evaluation of information security management under complex intuitionistic fuzzy set environment, J. Math., 2022 (2022), 9704466. https://doi.org/10.1155/2022/9704466 doi: 10.1155/2022/9704466
    [36] R. Nandhinii, D. Amsaveni, On bipolar complex intuitionistic fuzzy graphs, TWMS J. Appl. Eng. Math., 12 (2022), 92–106.
    [37] N. Yaqoob, M. Gulistan, S. Kadry, H. A. Wahab, Complex intuitionistic fuzzy graphs with application in cellular network provider companies, Mathematics, 7 (2019), 35. https://doi.org/10.3390/math7010035 doi: 10.3390/math7010035
    [38] G. Huang, L. Xiao, W. Pedrycz, D. Pamucar, G. Zhang, L. Martínez, Design alternative assessment and selection: A novel Z-cloud rough number-based BWM-MABAC model, Inform. Sci., 603 (2022), 149–189. https://doi.org/10.1016/j.ins.2022.04.040 doi: 10.1016/j.ins.2022.04.040
    [39] L. Xiao, G. Huang, W. Pedrycz, D. Pamucar, L. Martínez, G. Zhang, A q-rung orthopair fuzzy decision-making model with new score function and best-worst method for manufacturer selection, Inform. Sci., 608 (2022), 153–177. https://doi.org/10.1016/j.ins.2022.06.061 doi: 10.1016/j.ins.2022.06.061
    [40] G. Huang, L. Xiao, W. Pedrycz, G. Zhang, L. Martinez, Failure mode and effect analysis using T-Spherical fuzzy maximizing deviation and combined comparison solution methods, IEEE T. Reliab., 2022, 1–22. https://doi.org/10.1109/TR.2022.3194057 doi: 10.1109/TR.2022.3194057
    [41] A. Al-Husban, A. R. Salleh, Complex fuzzy group based on complex fuzzy space, Glob. J. Pure Appl. Math., 12 (2016), 1433–1450.
    [42] M. O. Alsarahead, A. G. Ahmad, Complex fuzzy subgroups, Appl. Math. Sci., 11 (2017), 2011–2021.
    [43] M. O. Alsarahead, A. G. Ahmad, Complex fuzzy subrings, Int. J. Pure Appl. Math., 117 (2017), 563–577.
    [44] M. O. Alsarahead, A. G. Ahmad, Complex fuzzy soft subgroups, J. Qual. Meas. Anal., 13 (2017), 17–28.
    [45] W. J. Liu, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Set. Syst., 8 (1982), 133–139.
    [46] H. Alolaiyan, H. A. Alshehri, M. H. Mateen, D. Pamucar, M. Gulzar, A novel Algebraic structure of (α, β)-complex fuzzy subgroups, Entropy, 23 (2021), 992. https://doi.org/10.3390/e23080992 doi: 10.3390/e23080992
    [47] R. Al-Husbana, A. R. Salleh, A. G. Ahmad, Complex intuitionistic fuzzy group subrings, AIP Confer. P., 1784 (2016), 05006. https://doi.org/10.1063/1.4966825 doi: 10.1063/1.4966825
    [48] R. Al-Husbana, A. R. Salleh, A. G. Ahmad, Complex intuitionistic fuzzy normal subgroup, Int. J. Pure Appl. Math., 115 (2017), 455–466.
    [49] F. Xiao, W. Pedrycz, Negation of the quantum mass function for multisource quantum information fusion with its application to pattern classification, IEEE T. Pattern Anal. Mach. Intell., 2022, 35420983. https://doi.org/10.1109/TPAMI.2022.3167045 doi: 10.1109/TPAMI.2022.3167045
    [50] F. Xiao, Z. Cao, C. Lin, A complex weighted discounting multisource information fusion with its application in pattern classification, IEEE T. Knowl. Data Eng., 2022, 1–16. https://doi.org/10.1109/TKDE.2022.3206871 doi: 10.1109/TKDE.2022.3206871
    [51] F. Xiao, Generalized quantum evidence theory, Appl. Intell., 2022 (2022). https://doi.org/10.1007/s10489-022-04181-0 doi: 10.1007/s10489-022-04181-0
    [52] F. Xiao, CEQD: a complex mass function to predict interference effects, IEEE T. Cybernetics, 52 (2022), 7402–7414. https://doi.org/10.1109/TCYB.2020.3040770 doi: 10.1109/TCYB.2020.3040770
    [53] G. Huang, L. Xiao, G. Zhang, Assessment and prioritization method of key engineering characteristics for complex products based on cloud rough numbers, Adv. Eng. Inform., 49 (2021), 101309. https://doi.org/10.1016/j.aei.2021.101309 doi: 10.1016/j.aei.2021.101309
    [54] G. Huang, L. Xiao, G. Zhang, Decision-making model of machine tool remanufacturing alternatives based on dual interval rough number clouds, Eng. Appl. Artif. Intell., 104 (2021), 104392. https://doi.org/10.1016/j.engappai.2021.104392 doi: 10.1016/j.engappai.2021.104392
  • Reader Comments
  • © 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2486) PDF downloads(145) Cited by(2)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog