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Finite/fixedtime synchronization of delayed Cliffordvalued recurrent neural networks
Advances in Difference Equations volume 2021, Article number: 276 (2021)
Abstract
This paper investigates the problem of finite/fixedtime synchronization for Cliffordvalued recurrent neural networks with timevarying delays. The considered Cliffordvalued drive and response system models are firstly decomposed into realvalued drive and response system models in order to overcome the difficulty of the noncommutativity of the multiplication of Clifford numbers. Then, suitable timedelayed feedback controllers are devised to investigate the synchronization problem in finite/fixedtime of error system. On the basis of new Lyapunov–Krasovskii functional and new computational techniques, finite/fixedtime synchronization criteria are formulated for the corresponding realvalued drive and response system models. Two numerical examples demonstrate the effectiveness of the theoretical results.
1 Introduction
Recurrent neural network (RNN) models have been used effectively to address optimization, associative memory, signal and image processing, as well as other complex problems. Recently, the dynamic analysis of RNN models has attracted immense interest from various researchers, and useful methods with respect to the stability theory of RNN models have been published [1–4]. In this respect, time delays are frequently observed in neural network (NN) models, owing to the slow speed of signal spread. Time delays are the main source of various dynamics such as chaos, poor functionality, divergence, and instability [5–9]. As such, studies on NN dynamics involving constant or timevarying delays are essential. On the other hand, quaternionvalued and complexvalued NN models are useful in many fields, including night vision color, radar images, polarized signals classification, 3D wind forecasts, and others [10–14]. Recently, many important results concerning different dynamics of the complexvalued and quaternionvalued NN models have been published [6–8, 15–18]. For example, stability analysis [8, 9, 16, 17], finitetime stability [6], stabilizability and instabilizability [7], optimization [19, 20], controllability and observability [21], multistability [18], and so on.
Clifford algebra provides a solid principle to solve geometry problems. It has been implemented in many diverse areas, e.g., neural systems, computers, robot and control problems [22–26]. Cliffordvalued NN models represent a generalization of real, complex, and quaternionvalued NN models. To address the challenges associated with highdimensional data and spatial geometric transformation, Cliffordvalued NN models are superior to real, complex, and quaternionvalued NN models [24–27]. Recently, theoretical and applied studies on Cliffordvalued NN models have become a new research subject. However, the dynamic properties of Cliffordvalued NN models are usually more complex than those of real, complex, and quaternionvalued NN models. Due to the noncommutativity of multiplication with respect to Clifford numbers [28–36], studies on Cliffordvalued NN dynamics are still limited. By using the linear matrix inequality approach, the authors in [28] derived the global exponential stability criteria for delayed Clifford RNN models. By using the decomposition process, the issue of global asymptotic stability in Cliffordvalued NN models was explored in [29]. In [31], the authors studied the presence of globally asymptotic almost automorphic synchronization to the problem of Cliffordvalued RNN models by using suitable feedback controllers. Utilizing the Banach fixed point theorem and Lyapunov functional, the global asymptotic almost periodic synchronization problems for Cliffordvalued NN models were examined in [33]. Recently, the effects of neutral delay and discrete delays have been considered in a class of Cliffordvalued NNs [37], and the associated existence, uniqueness, and global stability criteria have been obtained. By considering impulsive effects, the problem of global exponential stability of Cliffordvalued NN models with timevarying delays has been investigated in [38].
On the other hand, synchronization is an important dynamical behavior in network systems, which has been applied in many different areas, including image processing, neural computing, traffic systems, and secure communication. As such, various kinds of synchronization have already been suggested in the previous literature [39–46]. Different from the classical synchronization analysis, finitetime synchronization means that the dynamical behaviors of coupled systems achieve the same time spatial state in finite settling time. Therefore, the concept of finitetime synchronization occurs naturally. In recent decades, literature for the finitetime synchronization of NNs has been widely published [47–52]. The use of the feedback controller technique to achieve finitetime synchronization of complex dynamical networks with time delay has been investigated [47]. In [48], the authors showed a detailed study of the finitetime synchronization of stochastic coupled NNs subject to Markovian switching and input saturation. In [49], the authors explored synchronization issues in finitetime complexvalued RNN models with timevarying delays and discontinuous activation functions. With respect to finitetime distributed delays, the synchronization issue of finitetime complexvalued NN models was examined in the work [50].
However, in practical networks, it is difficult to obtain initial values in advance. Therefore, finitetime synchronization of such networks is impossible. Fixedtime synchronization would be a reasonable choice for scientists and engineers in this case as well. The settling time for fixedtime synchronization is calculated using known parameters and does not depend on the initial values, so it is widely accepted when compared to classical finitetime synchronization [53–56]. Furthermore, in realistic engineering problems, it is desirable that the systems can be synchronized for any initial conditions within a fixedtime period. The definition of fixedtime stability was suggested in [53]. Thereafter, the principle of fixedtime stability in various areas, including rigid spacecraft, secure communication, and power systems, was successfully utilized [57–59]. In the papers [54, 55], during fixedtime synchronization of the NNs being addressed, the controllers were designed so as to achieve synchronization criteria within a fixed settling time interval. In [56], a unified control framework was designed for finitetime and fixedtime synchronization of discontinuous complex networks. In [57], fixedtime synchronization of quaternionvalued NNs was obtained by the feedback control principle and the Lyapunov functional method.
However, to the best of authors’ knowledge, the problem of finite/fixedtime synchronization of Cliffordvalued RNN with timevarying delays has not yet been considered by any researcher. Indeed, this interesting topic is still an open challenge. Therefore, we attempt to derive sufficient conditions pertaining to finite/fixedtime synchronization of Cliffordvalued RNN models in this paper. The main contributions of this paper are as follows:

(1)
The finitetime and fixedtime synchronization of Cliffordvalued RNNs with timevarying delays is investigated for the first time.

(2)
By considering appropriate feedback controllers, Lyapunov functional, and new computational methods, some sufficient conditions that ascertain the finite/fixedtime synchronization of Cliffordvalued RNN models are derived by decomposing the Cliffordvalued RNN model into realvalued models.

(3)
When Cliffordvalued NN model is reduced to real, complex, and quaternionvalued ones, the results obtained in this paper are valid as special cases.

(4)
Two numerical examples with simulations are given to support the effectiveness and merits of the theoretical results.
We organize this paper as follows. The proposed Cliffordvalued RNN model is formally defined in Sect. 2. The finite/fixedtime synchronization criteria are presented in Sect. 3, while two numerical examples are given in Sect. 4. The conclusion of this paper is given in Sect. 5.
2 Mathematical fundamentals and problem formulation
2.1 Notations
The superscripts T and ∗ indicate the matrix transposition and matrix involution transposition, respectively. Any matrix (<0) denotes a positive (negative) definite matrix, while \(\mathbb{A}\) is defined as the Clifford algebra which has m generators over real number \(\mathbb{R}\). In addition, \(\mathbb{R}^{n}\) and \(\mathbb{A}^{n}\) denote the ndimensional real vector space and ndimensional real Clifford vector space, respectively; while \(\mathbb{R}^{n \times n}\) and \(\mathbb{A}^{n \times n}\) denote the set of all \(n \times n\) real matrices and the set of all \(n \times n\) real Clifford matrices, respectively. We define the norm of \(\mathbb{R}^{n}\) as \(\r\=\sum_{i=1}^{n}r_{i}\), and for \(A=(a_{ij})_{n\times n}\in \mathbb{R}^{n\times n}\), denote \(\A\=\max_{1\leq i\leq n} \{\sum_{j=1}^{n}a_{ij} \}\). While \(r=\sum_{A}r^{A}e_{A}\in \mathbb{A}\), denote \(r_{\mathbb{A}}=\sum_{A}r^{A}\), and for \(A=(a_{ij})_{n\times n}\in \mathbb{A}^{n\times n}\), denote \(\A\_{\mathbb{A}}=\max_{1\leq i\leq n} \{\sum_{j=1}^{n}a_{ij}_{ \mathbb{A}} \}\). For \(\varphi \in \mathcal{C}([\tau , 0], \mathbb{A}^{n})\), \(\\varphi \_{\tau }\leq \sup_{\tau \leq s\leq 0}\\varphi (t+s) \\).
2.2 Clifford algebra
In this subsection, we recall some definitions, notations, and basic results of Clifford algebra. The Clifford real algebra over \(\mathbb{R}^{m}\) is defined as
where \(e_{A}=e_{l_{1}}e_{l_{2}},\ldots, e_{l_{\nu }}\) with \(A=\{l_{1},l_{2},\ldots ,l_{\nu }\}\), \(1 \leq l_{1} < l_{2} <\cdots < l_{\nu }\leq m\). Moreover, \(e_{\emptyset }=e_{0}=1\) and \(e_{l}=e_{\{l\}}\), \(l=1,2,\ldots ,m\), are denoted as the Clifford generators, and they satisfy \(e_{i}e_{j} + e_{j}e_{i} = 0\), \(i \neq j\), \(i,j={1,2,\ldots ,m}\), \(e_{i}^{2} =1\), \(i = 1,2,\ldots ,m\).
Remark 2.1
When one element is the product of multiple Clifford generators, we will write its subscripts together: \(e_{4}e_{5}e_{6}e_{7}=e_{4567}\).
Let \(\Lambda =\{\emptyset ,1,2,\ldots , A,\ldots , 12,\dots, m\}\), and we have
where \(\sum_{A}\) denotes \(\sum_{A\in \Lambda }\) and \(\dim \mathbb{A}=\sum_{k=0}^{m}\binom{m}{k}=2^{m}\). For \(r=\sum_{A}r^{A}e_{A}:\mathbb{R}\rightarrow \mathbb{A}\), where \(r^{A}:\mathbb{R}\rightarrow \mathbb{R}\), \(A\in \Lambda \), its derivative is represented by \(\frac{dr(t)}{dt}=\sum_{A}\frac{dr^{A}(t)}{dt}e_{A}\). For more knowledge about Clifford algebra, we refer the reader to [28–30].
2.3 Problem definition
Consider the following Cliffordvalued RNN model with timevarying delays:
where \(i\in N\), \(j\in N\) (\(N=1,2,\ldots,n\)), and n corresponds to the number of neurons; \(r_{i}(t)\in \mathbb{A}\) represents the state vector of the ith unit; \(d_{i}\in \mathbb{R}^{+}\) indicates the rate with which the ith unit will reset its potential to the resting state in isolation when it is disconnected from the network and external inputs; \(a_{ij}, b_{ij}\in \mathbb{A}\) indicate the strengths of connection weights without and with timevarying delays between cells i and j, respectively; \(k_{i}\in \mathbb{A}\) is an external input on the ith unit; \(h_{j}(\cdot ):\mathbb{A}^{n} \rightarrow \mathbb{A}^{n}\) is the activation function of signal transmission; \(\tau _{j}(t)\) corresponds to the transmission delay which satisfies \(0\leq \tau _{j}(t)\leq \tau \), where τ is a positive constant, and \(\tau =\max_{1\leq j\leq n}\{\tau _{j}(t)\}\). Furthermore, \(\varphi _{i}\in \mathcal{C}([\tau , 0],\mathbb{A}^{n})\) is the initial condition with respect to model (1).
Remark 2.2
It is clear that NN model (1) includes realvalued, complexvalued, and quaternionvalued NN models. These mean that the proposed NN model is more general than the corresponding one in the existing articles. For example, when we consider \(m=0\) in NN model (1), then the model can be reduced to the realvalued NN model proposed in [5]. If we take \(m=1\) in NN model (1), then the model can be reduced to the complexvalued NN model proposed in [9]. If we choose \(m=2\) in NN model (1), then the model can be reduced to the quaternionvalued NN model proposed in [17].
The corresponding response system is defined by
where \(i\in N\), \(j\in N\) (\(N=1,2,\ldots,n\)), and n corresponds to the number of neurons; \(s_{i}(t)\in \mathbb{A}\) represents the state vector of the ith unit; \(\phi _{i}\in \mathcal{C}([\tau , 0],\mathbb{A}^{n})\) is the initial condition for model (3). In addition, \(u_{i}(t)\) is statefeedback controller, while other notations associated with (3) and (4) are the same as those in (1) and (2).

(A1)
Function \(h_{j}(\cdot )\) fulfills the Lipschitz continuity condition with respect to the ndimensional Clifford vector. For each \(j\in N\), there exists a positive constant \(l_{j}\) such that, for any \(x, y\in \mathbb{A}\),
$$\begin{aligned} \bigl\vert h_{j}(x)h_{j}(y) \bigr\vert _{\mathbb{A}}\leq l_{j} \vert xy \vert _{\mathbb{A}}, \quad j \in N, \end{aligned}$$(5)where \(l_{j}\) (\(j\in N\)) is known as the Lipschitz constant and \(h_{j}(0)=0\). In addition, there exists a positive constant \(l_{j}\) such that \(h_{j}(x)_{\mathbb{A}}\leq l_{j}\) for any \(x\in \mathbb{A}\).
3 Main results
To address the issue of noncommutativity of multiplication of Clifford numbers, we transform the original Cliffordvalued models into multidimensional realvalued models. This can be achieved with the help of \(e_{A}\bar{e}_{A}=\bar{e}_{A}e_{A}=1\) and \(e_{B}\bar{e}_{A}e_{A}=e_{B}\). Given any \(\mathscr{G}\in \mathbb{A}\), unique \(\mathscr{G}^{C}\) that is able to satisfy \(\mathscr{G}^{C}e_{C}h^{A}e_{A}=(1)^{\sigma [B.\bar{A}]}\mathscr{G}^{C}h^{A}e_{B}= \mathscr{G}^{B.\bar{A}}h^{A}e_{B}\) can be identified. By decomposing (1) and (2) into \(\dot{r}=\sum_{A}\dot{r}^{A}e_{A}\), we have the following realvalued models:
where
Remark 3.1
If \(r=(r_{1}^{0}, r_{1}^{1}, r_{1}^{2}, \ldots , r_{1}^{1,2, \ldots,m}, r_{2}^{0}, r_{2}^{1}, r_{2}^{2}, \ldots , r_{2}^{1,2, \ldots,m}, \ldots , r_{n}^{0}, r_{n}^{0}, r_{n}^{2}, \ldots , r_{n}^{1,2, \ldots,m})^{T}=\{r_{i}^{A}\}\) is a solution to system (6), then \(r=(r_{1}, r_{2}, \ldots , r_{n})^{T}\) must be a solution to system (1), where \(r_{i}=\sum_{A}r_{i}^{A}e^{A}\), \(i=1,2,\ldots,n\), \(A\in \Lambda \).
Also, we can use the same method to transform (3) and (4) into the following realvalued models:
where
Note that the remaining notations of (8) and (9) are the same as those in (6) and (7).
Define the error vector between the realvalued drive models (6), (7) and the realvalued response models (8), (9) as and \(\psi _{i}^{A}(t)=\phi _{i}^{A}(t)\varphi _{i}^{A}(t)\); then, from (6)–(9), the following error models are produced:
Remark 3.2
As we all know, the multiplication of Clifford numbers does not satisfy the commutative law, which complicates the investigation of Cliffordvalued NNs dynamics. Therefore, the known results on Cliffordvalued neural networks are limited. On the other hand, the decomposition approach is very effective to overcome the difficulty of the noncommutativity of the multiplication of Clifford numbers. Therefore, it is highly meaningful to use the decomposition method to study Cliffordvalued NNs. Recently, most existing results have been derived by decomposing Clifford valued NNs into realvalued NNs [28, 29, 33, 37, 38].
The following lemmas, definitions are used as the main techniques in the paper.
Definition 3.3
([58])
The origin of a nonlinear dynamical model is said to be globally finitetime stable if, for any solution \(r(t, r_{0})\), the following are true:

(1)
Lyapunov stability: For any \(\epsilon >0\), there is \(\delta =\delta (\epsilon )>0\) such that \(\r(t, r_{0})\<\epsilon \) for any \(\r_{0}\<\delta \) and \(t\geq t_{0}\).

(2)
Finitetime convergence: There exists a function \(T: \mathbb{R}^{n}\setminus \{0\}\rightarrow (0, +\infty )\), denoted as the settling time function, such that \(\lim_{t\rightarrow T(r_{0})}=0\) and \(r(t, r_{0})=0\) for all \(t\geq T(r_{0})\).
Definition 3.4
([59])
The origin of the nonlinear dynamical model is said to be fixedtime stable if it is globally finitetime stable and the settling time function \(T(r_{0})\) is bounded for any \(r_{0}\in \mathbb{R}^{n}\), i.e., there exists \(T_{\max }>0\) such that \(T(r_{0})\leq T_{\max }\) for all \(r_{0}\in \mathbb{R}^{n}\).
Definition 3.5
([60])
Consider driveresponse models (6) and (7) as well as (8) and (9). If, for a suitable controller \(u_{i}(t)=\sum_{A}u^{A}_{i}(t)e_{A}\), there exists a function \(T=T(\psi _{i})>0\), depending on the initial value \(\psi _{i}\), such that
and for \(t>T\), \(i\in N\), \(A\in \Lambda \), then drive models (6) and (7) and response models (8) and (9) achieve synchronization in finitetime.
Definition 3.6
([61])
A function is Cregular if it is

(1)
regular in \(\mathbb{R}^{n}\),

(2)
positive definite, i.e., for \(r\neq 0\) and ,

(3)
radially unbounded, i.e., as \(\r\\rightarrow \infty \).
Definition 3.7
([57])
Driveresponse models (6) and (7) as well as (8) and (9) are said to achieve fixedtime synchronization if, for any initial condition, there exist time \(T_{\max }\) and a settling time function such that
and , where , , \(i\in N\), \(A\in \Lambda \).
Lemma 3.8
([62])
Suppose that is continuous, differentiable, positive definite, and it satisfies the following differential inequality:
where and \(0<\upsilon <1\) are constants. Then, for any given \(t_{0}\), satisfies the following inequality:
and
and the settling time \(T^{+}\) is given by
Lemma 3.9
([59])
If there exists a continuous radically unbounded function such that

(1)
,

(2)
If there exist , , , and such that
(12)
where V̇ denotes the derivative of .
Then , \(\forall t\geq T_{\max }^{+}\) with the settling time \(T_{\max }^{+}\) given by
Lemma 3.10
([63])
If \(\xi _{1}, \xi _{2},\ldots, \xi _{n}\) are the positive constants and \(0<\alpha _{1}<\alpha _{2}\), then
3.1 Finitetime synchronization
In this subsection, the finitetime synchronization criterion for the error system models (10) and (11) is derived. The statefeedback controllers for models (10) and (11) are chosen as follows:
where \(i\in N\), \(j\in N\), \(A\in \Lambda \), and \(0<\alpha <1\) and \(\lambda _{1i}\), \(\lambda _{2i}\), \(\lambda _{3i}\) are the parameters that will be determined.
Theorem 3.11
Based on Assumption (A1) and a proper selection of the parameters to satisfy the following conditions:
Then the error system models (10) and (11) can achieve the finitetime synchronization with controller (14). Moreover, the settling time of synchronization \(T^{+}\) satisfies
Proof
Consider the following Lyapunov function which is positive definite and radially unbounded:
The Dini derivative is computed with model (10). We derive
By combining similar terms, we have
From Lemma (3.10), we obtain
Replacing (22) in (21), we have
According to Lemma (3.8), response model (8) with controller (14) can be synchronized in finitetime with drive model (6). Furthermore, the settling time of synchronization \(T^{+}\) is given by (18). Thus, Theorem 3.11 is proved. □
3.2 Fixedtime synchronization
The fixedtime synchronization criterion for driveresponse models (10) and (11) is derived. The timedelay feedback controller for model (10) is chosen as follows:
where \(i\in N\), \(j\in N\), \(A\in \Lambda \), and \(0<\alpha <1\), \(\beta >1\) and \(\lambda _{1i}\), \(\lambda _{2i}\), \(\lambda _{3i}\) \(\lambda _{4i}\) are the parameters that will be determined.
Theorem 3.12
Based on Assumption (A1) and a proper selection of the parameters to satisfy the following conditions:
The error system models (10) and (11) can achieve the fixedtime synchronization with controller (24). Moreover, the settling time of synchronization \(T^{+}\) satisfies
Proof
Consider the same Lyapunov function defined in Theorem 3.11:
The Dini derivative is computed with model (10). We derive
By combining similar terms, we have
From Lemma (3.10), we obtain
Replacing (32) and (33) in (31), we have
According to Lemma (3.9), response model (8) can be synchronized in fixed time with drive model (6) under controller (24). The settling time of synchronization \(T_{\max }^{+}\) max is given by (28) with \(\Pi _{1}=[\min_{i\in N}(\lambda _{3i})]^{\beta }\), \(\Pi _{2}=[\min_{i\in N}(\lambda _{2i})]^{\alpha }\). Thus, Theorem 3.12 is proved. □
Remark 3.13
In Theorem 3.11 and Theorem 3.12, by decomposing the original ndimensional Cliffordvalued system into a multidimensional realvalued system, several sufficient conditions are derived to show that the considered system model is finitetime and fixedtime synchronized, but the result we get is really about Cliffordvalued systems themselves.
Remark 3.14
As everyone knows, Cliffordvalued neural networks aim to investigate new capabilities and better accuracy in order to solve problems that cannot be solved with realvalued, complexvalued, quaternionvalued counterparts. For example, the results of finitetime and fixedtime synchronization of complexvalued NNs [49], fixedtime synchronization of quaternionvalued NNs [57] can then be summarized as a special case of the results of this paper.
Remark 3.15
In [30], fuzzy operations are incorporated into the Cliffordvalued cellular NN model to investigate its \(S^{p}\)almost periodic solutions. The effects of discrete delays in Cliffordvalued recurrent NNs are considered in [31], and the associated globally asymptotic almost automorphic synchronization criteria are obtained. The leakage delay is introduced into Cliffordvalued highorder Hopfield NN models in [32] to explore its existence and global exponential stability of almost automorphic solutions. However, any work on the topic of finite/fixedtime synchronization of Cliffordvalued RNN with timevarying delays has not yet been reported. Therefore, trying to fill such gaps, we for the first time derived new sufficient conditions to ensure the finite/fixedtime synchronization of Cliffordvalued RNN models with time delays. Therefore, the main results of this paper are new and different compared with those in the existing literature.
Remark 3.16
The computational complexity depends primarily on the maximum number of LMI decision variables. As everyone knows, the number of decision variables increases when using the augmented Lyapunov–Krasovskii functionals and the freeweightingmatrix method; while when the delay subintervals number becomes higher, it might prompt the complexity and the computational burden of the main results. In order to handle this issue easily, we utilized standard Lyapunov functional and estimated its time derivative without any integral inequalities and delaydecomposition approach. Hence, the proposed results in this paper may provide a smaller computational burden.
Remark 3.17
Compared with many controllers derived by early works [45, 46], the feedback controller methods are economic and easy to implement, they possess high value in real industry processes and applications.
4 Numerical examples
In this section, we present two numerical examples to demonstrate the feasibility and effectiveness of the results established in Sect. 3.
Example 1
For \(m=2\) and \(n=2\), the following twoneuron drive model (1) is considered:
The corresponding response model (3) is
The multiplication generators are: \(e_{1}^{2}=e_{2}^{2}=e_{12}^{2}=e_{1}e_{2}e_{12}=1\), \(e_{1}e_{2}=e_{2}e_{1}=e_{12}\), \(e_{1}e_{12}=e_{12}e_{1}=e_{2}\), \(e_{2}e_{12}=e_{12}e_{2}=e_{1}\), \(r_{1}=r^{0}_{1}e_{0}+r^{1}_{1}e_{1}+r^{2}_{1}e_{2}+r^{12}_{1}e_{12}\), \(r_{2}=r^{0}_{2}e_{0}+r^{1}_{2}e_{1}+r^{2}_{2}e_{2}+r^{12}_{2}e_{12}\), \(s_{1}=s^{0}_{1}e_{0}+s^{1}_{1}e_{1}+s^{2}_{1}e_{2}+s^{12}_{1}e_{12}\), \(s_{2}=s^{0}_{2}e_{0}+s^{1}_{2}e_{1}+s^{2}_{2}e_{2}+s^{12}_{2}e_{12}\).
Furthermore, we take
in which and , \(i=1,2\). The timevarying delays are considered as \(\tau _{1}(t)=\tau _{2}(t)=0.4\cos (t)+0.03\) with \(\tau _{1}=\tau _{2}=0.43\). Furthermore, the activation function satisfies Assumption (A1) with \(l_{1}=l_{2}=0.5\). By selecting \(\lambda _{11}=2.5\), \(\lambda _{12}=2.6\), \(\lambda _{21}=3.5\), \(\lambda _{22}=3.8\), \(\lambda _{31}=1.95\), \(\lambda _{32}=2\), and \(\alpha =0.5\).
Besides it is easy to obtain \(d_{1}=2.2\), \(d_{2}=2.4\), \(a^{A.\bar{B}}_{11}= 0.6\), \(a^{A.\bar{B}}_{12}= 0.9\), \(a^{A.\bar{B}}_{21}= 0.7\), \(a^{A.\bar{B}}_{22}= 0.9\), \(b^{A.\bar{B}}_{11}= 0.8\), \(b^{A.\bar{B}}_{12}=0.8\), \(b^{A.\bar{B}}_{21}= 0.7\), \(b^{A.\bar{B}}_{22}= 0.5\). The initial conditions of driveresponse systems (1) and (3) are taken as \(\varphi _{1}(t)=1.5e_{0}1.2e_{1}0.9e_{2}+2e_{12}\) for \(t\in [0.43, 0]\), \(\varphi _{2}(t)=1.6e_{0}+2.5e_{1}+2.2e_{2}1.4e_{12}\) for \(t\in [0.43, 0]\), \(\phi _{1}(t)=2.5e_{0}+1.1e_{1}+2.2e_{2}1.5e_{12}\) for \(t\in [0.43, 0]\), and \(\phi _{2}(t)=2.6e_{0}2.1e_{1}2.2e_{2}+e_{12}\) for \(t\in [0.43, 0]\). By simple calculation, we have
Moreover, the settling time of synchronization \(T^{+}\) satisfies
Clearly, all conditions of Theorem 3.11 are satisfied. Driveresponse models (1) and (3) can achieve synchronization in finitetime with controller (14). Figures 1, 2, 4, 5, 7, 8, 10, and 11, respectively, show the time responses of the states of driveresponse models (1) and (3). Besides, Figures 3, 6, 9, and 12 disclose the time responses of the states of error systems (10). From Figures 3, 6, 9, and 12, it can be seen that model (6) synchronizes with model (8) in finitetime through the controller (14) with the given initial values.
Example 2
For \(m=2\) and \(n=2\), the following twoneuron drive model (1) is considered:
The corresponding response model (3) is
The multiplication generators are: \(e_{1}^{2}=e_{2}^{2}=e_{12}^{2}=e_{1}e_{2}e_{12}=1\), \(e_{1}e_{2}=e_{2}e_{1}=e_{12}\), \(e_{1}e_{12}=e_{12}e_{1}=e_{2}\), \(e_{2}e_{12}=e_{12}e_{2}=e_{1}\), \(r_{1}=r^{0}_{1}e_{0}+r^{1}_{1}e_{1}+r^{2}_{1}e_{2}+r^{12}_{1}e_{12}\), \(r_{2}=r^{0}_{2}e_{0}+r^{1}_{2}e_{1}+r^{2}_{2}e_{2}+r^{12}_{2}e_{12}\), \(s_{1}=s^{0}_{1}e_{0}+s^{1}_{1}e_{1}+s^{2}_{1}e_{2}+s^{12}_{1}e_{12}\), \(s_{2}=s^{0}_{2}e_{0}+s^{1}_{2}e_{1}+s^{2}_{2}e_{2}+s^{12}_{2}e_{12}\).
Furthermore, we take
in which and , \(i=1,2\). By selecting \(\lambda _{11}=2.5\), \(\lambda _{12}=2.6\), \(\lambda _{21}=3.5\), \(\lambda _{22}=3.8\), \(\lambda _{31}=4.2\), \(\lambda _{32}=4.4\), \(\lambda _{41}=1.95\), \(\lambda _{42}=2\), \(\alpha =0.5\), and \(\beta =1.5\)
The timevarying delays are considered as \(\tau _{1}(t)=\tau _{2}(t)=0.5\cos (t)+0.02\) with \(\tau _{1}=\tau _{2}=0.52\). Furthermore, the activation function satisfies Assumption (A1) with \(l_{1}=l_{2}=0.5\). Besides it is easy to obtain \(d_{1}=2.6\), \(d_{2}=2.8\), \(a^{A.\bar{B}}_{11}= 0.7\), \(a^{A.\bar{B}}_{12}= 0.8\), \(a^{A.\bar{B}}_{21}= 0.5\), \(a^{A.\bar{B}}_{22}= 0.8\), \(b^{A.\bar{B}}_{11}= 0.5\), \(b^{A.\bar{B}}_{12}=0.6\), \(b^{A.\bar{B}}_{21}= 0.5\), \(b^{A.\bar{B}}_{22}= 0.8\). The initial conditions of driveresponse models (1) and (3) are taken as \(\varphi _{1}(t)=1.5e_{0}+1.8e_{1}0.9e_{2}+2e_{12}\) for \(t\in [0.52, 0]\), \(\varphi _{2}(t)=1.6e_{0}2e_{1}+2.2e_{2}1.4e_{12}\) for \(t\in [0.52, 0]\), \(\phi _{1}(t)=2.5e_{0}+1.7e_{1}+2.2e_{2}1.5e_{12}\) for \(t\in [0.52, 0]\), and \(\phi _{2}(t)=2.6e_{0}+1.2e_{1}2.2e_{2}+e_{12}\) for \(t\in [0.52, 0]\). By simple computation, we have
Clearly, all conditions of Theorem 3.12 are satisfied. Therefore, driveresponse models (6) and (8) achieve fixedtime synchronization with controller (24). Moreover, the settling time of synchronization \(T_{\max }^{+}\) satisfies
5 Conclusion
In this article, we have studied the finite/fixedtime synchronization for Cliffordvalued RNN models with timevarying delays. In order to overcome the difficulty of the noncommutativity of the multiplication of Clifford numbers, we first decomposed the considered Cliffordvalued drive and response models into realvalued drive and response models. Besides, suitable timedelayed feedback controllers have been constructed to examine the synchronization problem associated with the finite/fixedtime error models. By utilizing the finite/fixedtime stability concepts, some computational techniques, new synchronization criteria have been derived through appropriate Lyapunov functions to guarantee that the driveresponse models achieve synchronization in finite/fixedtime. Finally, we have also presented numerical examples to illustrate the effectiveness of the results. The results obtained in this paper can be further extended to other complex systems. We would like to extend our results to more general Cliffordvalued NN models, such as Cohen–Grossberg Cliffordvalued NNs, Cliffordvalued inertial NNs, Cliffordvalued highorder Hopfield NNs, and fuzzy Cliffordvalued NNs. Moreover, we will focus on the problem of global stabilization analysis of Cliffordvalued NN models with the help of various control systems. The corresponding results will be carried out in the near future.
Availability of data and materials
Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.
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Acknowledgements
This research is made possible through financial support from the Rajamangala University of Technology Suvarnabhumi, Thailand. The authors are grateful to the Rajamangala University of Technology Suvarnabhumi, Thailand for supporting this research.
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The research is supported by the Rajamangala University of Technology Suvarnabhumi, Thailand.
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Funding acquisition, NB; Conceptualization, GR; Software, GR and NB; Formal analysis, GR and NB; Methodology, GR and NB; Supervision, RS, PA, GR and CPL; Writing–original draft, GR; Validation, GR; Writing–review and editing, GR and NB. All authors have read and agreed to the published version of the manuscript.
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Boonsatit, N., Rajchakit, G., Sriraman, R. et al. Finite/fixedtime synchronization of delayed Cliffordvalued recurrent neural networks. Adv Differ Equ 2021, 276 (2021). https://doi.org/10.1186/s13662021034381
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DOI: https://doi.org/10.1186/s13662021034381
Keywords
 Cliffordvalued neural network
 Synchronization
 Finitetime
 Fixedtime
 Lyapunov–Krasovskii functional