System Decomposition Method-Based Exponential Stability of Clifford-Valued BAM Delayed Neural Networks

This study explores new theoretical results for the global exponential stability of bidirectional associative memory delayed neural networks in the Clifford domain. By considering time-varying delays, a general class of Clifford-valued bidirectional associative memory neural networks is formulated, which encompasses real-, complex-, and quaternion-valued neural network models as special cases. To analyze the global exponential stability, we first decompose the considered <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-dimensional Clifford-valued networks into <inline-formula> <tex-math notation="LaTeX">$2^{m}n$ </tex-math></inline-formula>-dimensional real-valued networks, which avoids the inconvenience caused by the non-commutativity of the multiplication of Clifford numbers. Subsequently, we establish new sufficient conditions to guarantee the existence, uniqueness, and global exponential stability of equilibrium points for the considered networks by constructing a new Lyapunov functional and applying homeomorphism theory. Finally, we provide a numerical example accompanied by simulation results to illustrate the validity of the obtained theoretical results. The present results remain valid even when the considered neural networks degenerate into real-, complex-, and quaternion-valued networks.

The associate editor coordinating the review of this manuscript and approving it for publication was Frederico Guimarães .
Clifford numbers, and they have more complex algebraic algorithms. It has been used in a wide range of applications, including medical imaging, robotics, and natural language processing [27], [28]. Moreover, Clifford-valued NNs offer several advantages in that they process data that are not solved by real-, complex-, and quaternion-valued NNs with high accuracy, and they can also handle multidimensional data. Consequently, Clifford-valued NNs have emerged as an important research field. In [29], the problem of global asymptotic stability in Clifford-valued NNs was examined using a decomposition method. In [32], the authors derived a globally asymptotic almost automorphic synchronization of Clifford-valued recurrent NNs networks with delays. In [34], the authors considered a class of Clifford-valued neutral high-order Hopfield NNs with leakage delays and studied their existence and global stability analysis. Using the Lyapunov function method, the global exponential stability of an anti-periodic solution for Clifford-valued inertial Cohen-Grossberg NNs was investigated in [37]. Other results of Clifford-valued NNs have been reported in earlier works [35], [36], [38], [39].
In reality, time delays inevitably exist in biological and artificial NNs and cannot be neglected [5], [10], [17], [21], [40], [41]. On the other hand, the appearance of discrete time delays in NNs are usual because of the limited switching speed of neurons and amplifiers. Moreover, NNs are typically spatial in nature and the propagation velocity distribution along these paths results in a delayed propagation distribution owing to the presence of significant parallel paths with different axon sizes and lengths [25], [31], [33]. Numerous studies have demonstrated that the existence of time delays has an impact on NNs and can result in complex dynamic behaviors such as oscillations, divergence, or instability of NNs, all of which can be detrimental to the system [42], [43], [44], [45]. Consequently, incorporating mixed delays in network modeling boosts the value of the considered NNs in theory and practice.
To the best of our knowledge, no studies have been published on the existence, uniqueness, and global exponential stability of BAMNNs with time-varying delays in the Clifford domain. To fill this gap, we aim to investigate the global exponential stability of Clifford-valued BAMNNs using the system decomposition method. In recent years, a number of studies have been published on the stability of Clifford-valued NNs; however, Clifford-valued BAMNNs have not been fully explored, which motivated us to investigate this topic. The main aspects of this paper can be summarized as follows: (1) A general form of Clifford-valued BAMNNs with time-varying delays was presented to derive more realistic Clifford-valued NNs dynamics. (2) The system decomposition method was used to investigate the global exponential stability of Clifford-valued delayed BAMNNs. (3) Lyapunov stability theory, homeomorphism theory, and inequality techniques were applied to Clifford-valued delayed BAMNNs to determine the enhanced stability conditions. (4) The effectiveness of the main results was illustrated using a numerical example and simulation results.
The remainder of this paper is structured as follows: Section II provides the basic concepts of Clifford algebra, problem model, definitions, and useful lemmas. The main results of this study are presented in Section III, Theorem (3.1) presents sufficient criteria for the existence of the equilibrium point and the global exponential stability of the considered NNs. In Corollary (3.2), the results of the stability criteria are discussed for a special case. Section IV presents a numerical case study that demonstrates the feasibility of the derived results. Section V presents the conclusions of this study.

II. MATHEMATICAL FORMULATION AND PROBLEM DEFINITION A. NOTATIONS
In the remainder of this paper, the n-space real vectors, nspace real Clifford vectors, set of all n × m real matrices, and set of all n × m real Clifford matrices are denoted by R n , A n , R n×m , and A n×m , respectively. The transposition and involution transposition of the matrices are denoted by T and * , respectively. The Clifford algebra with m generators over a real number is denoted by A. The norm of R n is defined For all 1, n = 1, 2, . . . , n and 1, m = 1, 2, . . . , m.

B. CLIFFORD ALGEBRA
The Clifford real algebra over R m is given by where e A = e w 1 e w 2 . . . e w η with A = w 1 , w 2 , . . . , w η , 1 ≤ w 1 < w 2 < . . . < w η ≤ m. Furthermore, e ∅ = e 0 = 1 and e w , w = 1, m denote the Clifford generators that satisfy the following conditions: (i) e i e j + e j e i = 0, Ā] e C = S C e C , and S = C S C e C ∈ A.

C. PROBLEM FORMULATION
In this section, we consider a class of Clifford-valued BAMNNs with time-varying delays as follows: where for all t ≥ 0; p = 1, n, q = 1, m; x p (t) ∈ A and y q (t) ∈ A denote the state variables; 0 < d p ∈ R + and 0 < c q ∈ R + denote the self-feedback connection weights; a qp ∈ A, b pq ∈ A denote the interconnection weights; f p (·) : A → A and g q (·) : A → A denote the neuron activation functions; u p ∈ A and v q ∈ A denote the external inputs; τ pq (t) ∈ R + and σ qp (t) ∈ R + denote the transmission delays.
To prove the main results of this paper, the following definitions and lemmas will be used: Lemma 2.4: [44] Assume that a ∈ R + , b ∈ R + , 1 < p, and 1 p + 1 q = 1, then the following condition holds: Lemma 2.5: [45] Let H(x, y) : R 2 m (n+m) → R 2 m (n+m) continuous. If H(x, y) satisfies the following conditions:

III. MAIN RESULTS
This section presents the delay-independent criteria for the existence, uniqueness, and global exponential stability of the equilibrium point for NNs (1) using homeomorphism theory and Lyapunov functions.
Based on the previous discussion about Clifford algebra, we can decompose the Clifford-valued function into the realvalued function. For example, for the second term in NNs (1), we have m q=1 a qp g q (y q (t − σ qp (t))) VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
According to the above discussion, Clifford-valued NNs (1) can be decomposed into equivalent real-valued NNs (7) to overcome the non-commutativity problem: The initial conditions of NNs (7) are given by where (H3) By hypothesis (H2), we have The equilibrium point of NNs (1) is also the equilibrium point of NNs (7), and the stability of NNs (1) is the same as that of NNs (7). Thus, we investigate NNs (7) to obtain sufficient criteria to establish that the equilibrium point of NNs (7) is globally exponentially stable.
then the equilibrium point of NNs (7) is globally exponentially stable. Proof: The proof of this theorem involves two steps: The first step is to prove the existence and uniqueness of the equilibrium point. The point (x * , y * ) T is an equilibrium point of NNs (7) if and only if it is a solution of the following equation: where for all p = 1, n and q = 1, m. VOLUME 11, 2023 The following proves ϒ(x A , y A ) is a homeomorphism. We claim that in the first step ϒ(x A , y A ) is injective to Similarly, we get Using (14), (15) and (11), we obtain for all p = 1, n, q = 1, m which is a contradiction. Therefore, Using (17) and (18), we can get where α = min{α 1 , α 2 }, Thus, using Hölder inequality, we get that is, According to Lemma (2.5), ϒ(x A , y A ) are homeomorphisms of R 2 m (n+m) . Thus, NNs (7) has a unique solution (x * , y * ) T which is the unique equilibrium point.
The second step is to show that the equilibrium point of the NNs (7) is globally exponentially stable. Letx A . Then, NNs (7) can be reduced to the following model Choose the Lyapunov functional V(t) which is given as follows: VOLUME 11, 2023 77329 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
Based on condition (11), we can select a small constant ϵ > 0, such that Therefore, we have Furthermore, V(t) ≤ V(0) for t ≥ 0. Hence Moreover, from (24), we have where This means that the equilibrium point of NNs (7) is globally exponentially stable. This completes this proof. When we consider r = 2, Corollary (3.2) can be derived by using Theorem (3.1).
Corollary 3.2: Let (H1)-(H3) hold, there exist constants λ p ∈ R + , λ n+q ∈ R + (p = 1, n; q = 1, m) such that then the equilibrium point of NNs (7) is globally exponentially stable. Remark 3.3: Theorem (3.1) examines the global exponential stability criteria for Clifford-valued NNs by dividing the original Clifford-valued NNs into multidimensional realvalued NNs. It should be noted that the main results of this study are related to Clifford-valued NNs.
Remark 3.4: In [30], the authors investigated the S palmost periodic solutions of a fuzzy Clifford-valued cellular NN model with time-varying delays. In [31], the authors examined asymptotic almost automorphic synchronization criteria for neutral type fuzzy cellular NNs Clifford-valued recurrent NNs with time delays. In [38], the authors analyzed the global asymptotic stability criteria for Clifford-valued NNs incorporating impulsive effects and time-varying delays. However, no studies have investigated the stability of Clifford-valued BAMNNs with time delays using the decomposition method. Therefore, we investigated the global exponential stability of Clifford-valued BAMNNs with time delays using Lyapunov stability and system decomposition method. In addition, the results proposed in this study are new and differ from those in the existing literature [30], [31], [32], [33], [34], [35], [36], [37], [38], [39].
Remark 3.5: In this study, the proposed NNs (1) is more general than those presented in previous studies; therefore, there are significant differences between them. For example, by setting the Clifford generators m as 0, 1, and 2, the NNs (1) becomes a real-, complex-, and quaternion-valued NNs, respectively.

IV. NUMERICAL EXAMPLES
This section presents an example of the effectiveness and feasibility of the proposed method.

V. CONCLUSION
This study investigated the global exponential stability problem for a class of Clifford-valued BAMNNs with timevarying delays. We first decomposed the n-dimensional Clifford-valued NNs into 2 m n-dimensional real-valued NNs VOLUME 11, 2023 to avoid the inconvenience caused by the non-commutativity of Clifford number multiplication. We then established new sufficient conditions for the existence, uniqueness, and global exponential stability of the equilibrium points for the considered networks using Lyapunov functions, homeomorphism theory, and inequality techniques. Finally, the results presented in this paper are illustrated using a numerical example accompanied by the simulation results.