Since neural networks are widely used in computer science [1], remote sensing [2], autonomous control systems [3], and other fields, their dynamic behaviors have been extensively studied over the past several decades. It is worth noting that neural networks (NNs) only consider the dynamic level of neural activity. However, it is essential to recognize that synaptic weights between neurons change over time [4]. Consequently, $Meyer-b\ddot{a}se$ et al. [5] developed competitive neural networks (CNNs) with different time scales in 1996, which can be viewed as an extension of Hopfield neural networks and cellular networks [6,7]. CNNs are defined using two types of state variables: short-term memory (STM) describes the rapid neural activity, while long-term memory (LTM) depicts the slow, unsupervised synaptic modifications. On the other hand, coupled competitive neural networks(CCNNs) consist of several interconnected subsystems, and due to their complex dynamic behavior, they have garnered significant attention [8,9].

The activation functions of NNs are widely recognized for describing the connection between the input and output of a single neuron. They are commonly considered to be continuous. When the activation function is believed to be at the high gain limit, however, the activation function approaches discontinuity. As a result, an increasing number of scholars have been conducting considerable research on NNs with discontinuous activation functions [10,11,12,13]. On the other hand, the dynamic behaviors of NNs are frequently impacted by external disturbances, such as changs in network structure, hardware facilities and environmental noise. As far as we are aware, there are few studies take both discontinuous activation functions and external disturbances into account when discussing CCNNs. Therefore, it is both intriguing and challenging to research discontinuous activation functions and external disturbances in CCNNs.

Synchronization means that two or more dynamical systems adjust themselves to exhibit a common dynamical behavior. The synchronization problem of NNs has garnered significant attention recently due to its wide applicability in communication systems, biological sciences, mechanical engineering and other domains [14,15,16,17]. However, the synchronization of the aforementioned NNs only considers the cooperative relationships between network nodes. In many practical systems, relationships of competition and cooperation coexist. Therefore, the synchronization issue of NNs with both competitive and cooperative connections between nodes, known as bipartite synchronization, is of crucial importance and has been researched in [18,19]. On the other hand, due to inherent network constraints, complete synchronization may not be achievable, and instead, quasi-synchronization is observed. Quasi-synchronization implies that the synchronization error no longer approaches zero but rather converges to a bounded set. To our knowledge, few papers have addressed the quasi-bipartite synchronization problem of CCNNs.

In addition to asymptotic synchronization or exponential synchronization, finite-time (FET) synchronization has gained widespread attention as a more practical form of network synchronization in recent years. In FET synchronization, the synchronization time is bounded, but it is dependent on the initial state of the node. To eliminate this dependence on the initial state, fixed-time synchronization is proposed based on the fixed-time (FXT) stability [20], and its synchronization time only depends on the system or control parameters. Compared with the rich results of NNs [21,22], it is extremely scarce at present to explore the FXT synchronization of CCNNs. In [23], authors studied the FET bipartite synchronization of delayed CCNNs under quantized control. Whereas, to our knowledge, there are limited reports on FXT quasi-bipartite synchronization of CCNNs, and further research is needed.

Over the past few decades, the control problem of networks has been one of the most widely studied topics, and many useful control methods have been developed, such as adaptive control, sliding mode control, impulse control, and intermittent control, among others. Intermittent control involves alternating periods of applying control input and periods of no control input, making it a more economical choice compared to continuous control schemes. Hence, the intermittent control strategy has received extensive attention [24,25,26,27]. In [28], the FXT synchronization problem of time-delay complex networks under intermittent pinning control is studied. The author in [29] solved the FXT and predefined-time cluster lag synchronization of stochastic multi-weighted complex networks via intermittent quantized control. To our knowledge, there is currently no existing literature that addresses the challenging problem of FXT quasi-bipartite synchronization of CCNNs under intermittent control.

Motivated by the analysis provided above, the primary objective of this paper is to investigate FXT quasi-bipartite synchronization in coupled competitive neural networks. Firstly, the model under consideration incorporates time-varying delays, discontinuous activation functions, and external disturbances simultaneously, rendering it more comprehensive. Secondly, we introduce an innovative FXT aperiodic intermittent control scheme, making the pioneering endeavor to explore quasi-bipartite synchronization in CCNNs. Furthermore, some robust criterion are established for FXT quasi-bipartite synchronization based on the theory of practical FXT stability. Finally, we provide estimations for error bounds and settling times.

Below are shown the remaining contents. Section 2 gives some necessary preliminary knowledge and model description. Section 3 introduces the main theoretical conclusions. Section 4 provides numerical examples to validate the theoretical conclusions. Section 5 finishes our study and discusses future research.

Notations: R represents the set of real numbers. The n-dimensional Euclidean space is represented by the $R^n$. $R^{n\times m}$ is the set of $n\times m$ real matrices. ${\mathcal{N}}_{+}=\{1,2,...,N\}$. $I_n$ is identity matrix. ${\mathbf{0}}_n$ denotes zero matrix. $\left|P\right|=\left(\right|p_{ij}{\left|\right)}_{n\times n}$. For a symmetric matrix B, ${\lambda }_{max}\left(B\right)$ represents the maximum eigenvalue of matrix B. $diag(·)$ represents the diagonal matrix. ${\mathbf{1}}_n$ denotes that all elements of a column vector are 1. For any $p={(p_1,...,p_n)}^T\in R^n$, $sgn\left(p\right)=diag(sign\left(p_1\right),...,sign\left(p_n\right))$, $sign(·)$ represents the $sign$ function. $Sign\left(D\right)={\left(sign\left(d_{ij}\right)\right)}_{n\times n}$. The 2-norm of the vector p is denoted by ${\parallel p\parallel }_2$. For vector $\mathbf{w}>0$$(<,\ge ,\le )$, all of the components of $\mathbf{w}$ are positive (negative, non-negative, non-positive). For vectors ${\mathbf{q}}_1$ and ${\mathbf{q}}_2$, ${\mathbf{q}}_1<{\mathbf{q}}_2$$({\mathbf{q}}_1\le {\mathbf{q}}_2)$ implies ${\mathbf{q}}_1-{\mathbf{q}}_2<0$$({\mathbf{q}}_1-{\mathbf{q}}_2\le 0)$. Notation ⊗ denotes Kronecker product. $k>0$, $\mathcal{C}([-k,0],R^n)$ denotes the set of continuous function from $[-k,0]$ to $R^n$.

Consider the following competitive neural networks with time-varying delay: where $z_k\left(t\right)$ is the state variable. $m_{kl}\left(t\right)$ represents synaptic efficiency. $c_k>0$ represents the self-feedback coefficient. Connection weights and delay connection weights, respectively, are represented by $a_{kq}$ and $b_{kq}$. $f_q(·)$ is the output of neuron which is discontinuous. ${\omega }_l$ is the weight of an external stimulus. ${\overline{c}}_k$ and ${\overline{a}}_k$ are given constants. $E_k$ is the intensity of the external stimulus. $\epsilon$ is the time scale of $STM$. Where $\tau$ is a known constant, $\tau \left(t\right)$ is the time-varying delay that satisfies $0\le \tau \left(t\right)\le \tau$.

Let $s\left(t\right)={\left(s_1\left(t\right),...,s_n\left(t\right)\right)}^T$, $s_k\left(t\right)={\sum }_{l=1}^p{\omega }_l{m}_{kl}\left(t\right)$ and $\omega ={({\omega }_1,...,{\omega }_p)}^T$. Without losing generality, suppose ${\parallel \omega \parallel }_2^2=1$, then network (1) can be written as where $z\left(t\right)={\left(z_1\left(t\right),...,z_n\left(t\right)\right)}^T$, $f\left(z\left(t\right)\right)={\left(f_1\left(z_1\left(t\right)\right),...,f_n\left(z_n\left(t\right)\right)\right)}^T$, ${\displaystyle A=(\frac{a_{kq}}{\epsilon }{)}_{n\times n}}$, ${\displaystyle B=(\frac{b_{kq}}{\epsilon }{)}_{n\times n}}$, ${\displaystyle C=diag\left(\frac{c_1}{\epsilon },...,\frac{c_n}{\epsilon }\right)}$, $\overline{C}=diag({\overline{c}}_1,...,{\overline{c}}_n)$, $\overline{A}=diag({\overline{a}}_1,...,{\overline{a}}_n)$, ${\displaystyle E=diag(\frac{E_1}{\epsilon },...,}$

${\displaystyle \frac{E_n}{\epsilon })}$. The initial value of system (2) is given by $z\left(t\right)={\varphi }_z\left(t\right)\in \mathcal{C}([-\tau ,0],R^n)$ and $s\left(t\right)={\varphi }_s\left(t\right)\in \mathcal{C}([-\tau ,0],R^n)$.

A class of CCNNs with external disturbances are modeled as follows: where $Z_i\left(t\right)={\left(Z_{i1}\left(t\right),...,Z_{in}\left(t\right)\right)}^T\in R^n$ and $S_i\left(t\right)={\left(S_{i1}\left(t\right),...,S_{in}\left(t\right)\right)}^T\in R^n$ are the state variables of $STM$ and $LTM$, respectively; ${\Xi }_i\left(t\right)={\left({\Xi }_{i1}\left(t\right),...,{\Xi }_{in}\left(t\right)\right)}^T$ indicates the external disturbance vector. $D=\left(d_{ij}\right)$ and $U=\left(u_{ij}\right)\in R^{N\times N}$ are the adjacency matrix associated with the signed graph $G\left(D\right)$ and $G\left(U\right)$ of the CCNNs, satisfying $d_{ii}=0$$\left(u_{ii}=0\right)$ for $i\in {\mathcal{N}}_{+}$; for $i\ne j$, $d_{ij}\ne 0$$\left(u_{ij}\ne 0\right)$ if there is a directed communication link from node j to node i, otherwise $d_{ij}=0$$\left(u_{ij}=0\right)$. $R_i\left(t\right)$ and ${\overline{R}}_i\left(t\right)$ are controllers to be designed. The initial conditions of network (3) meet: $Z_i\left(t\right)={\phi }_i\left(t\right)\in \mathcal{C}([-\tau ,0],R^n)$ and $S_i\left(t\right)={\overline{\phi }}_i\left(t\right)\in \mathcal{C}([-\tau ,0],R^n)$.

For the convenience of discussion, define ${\mathbf{x}}_i\left(t\right)=\left(Z_i^T\left(t\right),S_i^T\left(t\right)\right)$, $A_1={\left(A^T,{\overline{A}}^T\right)}^T$, $B_1={\left(B^T,{\mathbf{0}}_n\right)}^T$, $W_i\left(t\right)={\left({\Xi }_i^T\left(t\right),{\Xi }_i^T\left(t\right)\right)}^T$, $K_i\left(t\right)={\left(R_i^T\left(t\right),{\overline{R}}_i^T\left(t\right)\right)}^T$, $\mathbf{I}=\left(I_n,{\mathbf{0}}_n\right)$, $C_1=\left[\begin{array}{ccc}C& & -E\\ 0_n& & \overline{C}\end{array}\right]$, ${\mathbf{d}}_{ij}=\left[\begin{array}{ccc}{d}_{ij}& & 0\\ 0& & u_{ij}\end{array}\right]\otimes I_n$ and $D_1={\left({\mathbf{d}}_{ij}\right)}_{2nN\times 2nN}$.

Therefore, the coupled competitive neural networks (3) become:

From (2), the tracking target can be described as follows: where $\mathbf{y}\left(t\right)={\left(z^T\left(t\right),s^T\left(t\right)\right)}^T$ is the state vector.

The necessary definitions, lemmas and assumptions are given below.

For each $q=1,...,n$, the following assumptions are introduced:

Assumption 3 implies that there exists a diagonal matrix $\omega =diag(w_1,...,w_N)$, where $w_i=1$ if node $v_i\in {\mathcal{V}}_1$, otherwise $w_i=-1$. To achieve the FXT quasi-bipartite synchronization of CCNNs, the intermittent controller is designed as follows: where $i\in {\mathcal{N}}_{+}$, $0<\alpha <1$, $\beta >1$, $\lambda$, ${\sigma }_1$, ${\sigma }_2$ are positive constants.

Combined with controller (8), when $t\in [{\zeta }_k,{\mu }_k)$, the CCNNs (4) is rewritten as follows: where ${\overline{D}}_1={\left({\overline{\mathbf{d}}}_{ij}\right)}_{2nN\times 2nN}$, ${\overline{\mathbf{d}}}_{ij}={\mathbf{d}}_{ij}$ for $i\ne j$ and ${\overline{\mathbf{d}}}_{ii}=-{\sum }_{j=1,j\ne i}^N\left|{\mathbf{d}}_{ij}\right|$. Then, according to Assumption 3, the CCNNs (9) can be transformed into: where ${\tilde{\mathbf{x}}}_i\left(t\right)=w_i{\mathbf{x}}_i\left(t\right)$, ${\tilde{\mathbf{d}}}_{ij}=w_i{\overline{\mathbf{d}}}_{ij}{w}_j=\left|{\mathbf{d}}_{ij}\right|$ for $i\ne j$ and ${\tilde{\mathbf{d}}}_{ii}=-{\sum }_{j=1,j\ne i}^N\left|{\mathbf{d}}_{ij}\right|$, $i\in {\mathcal{N}}_{+}$.

Similar analysis, for $t\in [{\mu }_k,{\zeta }_{k+1})$, one has

By definition 1, there is $\varpi \left(t\right)={\left({\varpi }_1\left(t\right),...,{\varpi }_n\left(t\right)\right)}^T\in K\left[f\right(\mathbf{I}\mathbf{y}\left)\right]$ such that

Similarly, there exists at least one measurable function ${\hslash }_i\left(t\right)=({\hslash }_{i1}\left(t\right),...,$ ${\hslash }_{in}\left(t\right){)}^T\in K\left[f\left(\mathbf{I}{\tilde{\mathbf{x}}}_i\right)\right]$ such that

In this part, we will consider the case of $W_i\left(t\right)=0$ and $W_i\left(t\right)\ne 0$. The Fixed-time quasi-bipartite synchronization criterion is derived by designing intermittent controller.

Define the synchronization error $e_i\left(t\right)={\tilde{\mathbf{x}}}_i\left(t\right)-\mathbf{y}\left(t\right)$, then the error dynamical system is: where $g_i\left(t\right)={\hslash }_i\left(t\right)-\varpi \left(t\right)$, $g_i(t-\tau \left(t\right))={\hslash }_i(t-\tau \left(t\right))-\varpi (t-\tau \left(t\right))$.