Global Asymptotic Almost Periodic Synchronization of Clifford-Valued CNNs with Discrete Delays

In this paper, we are concerned with Clifford-valued cellular neural networks (CNNs) with discrete delays. Since Clifford algebra is a unital associative algebra and its multiplication is noncommutative, to overcome the difficulty of the noncommutativity of the multiplication of Clifford numbers, we first decompose the considered Clifford-valued neural network into 2m2n real-valued systems. Second, based on the Banach fixed point theorem, we establish the existence and uniqueness of almost periodic solutions of the considered neural networks.Then, by designing a novel state-feedback controller and constructing a proper Lyapunov function, we study the global asymptotic synchronization of the considered neural networks. Finally, a numerical example is presented to show the effectiveness and feasibility of our results.


Introduction
Clifford algebras, which were invented by mathematician W. K. Clifford, are algebras generated from vector spaces with quadratic forms and are unitary associative algebras.They contain real numbers, complex numbers, quaternions, and several other hypercomplex systems as special cases.Clifford algebras have important applications in a variety of fields, including geometry, theoretical physics, and digital image processing [1].Clifford-valued neural networks generalize real-valued, complex-valued, quaternion-valued, and octonion-valued neural networks and so on.Since the Clifford-valued neural networks can use multistate activation functions to process multilevel information and require much fewer connection weight parameters of the networks, they have recently been an active research area [2][3][4][5].Due to the noncommutativity of Clifford numbers' multiplication, it brings great difficulties to the research of Clifford-valued neural networks.At present, the results of research on Clifford-valued neural networks are still rare, especially, for the nonautonomous ones.
On the one hand, it is well known that, in the design and implementation of neural networks, the existence of periodic solutions or almost periodic solutions of nonautonomous neural networks is as important as the existence of equilibrium points of autonomous neural networks.At the same time, we know that, even if all the time-varying coefficients in a neural network are periodic functions (such as connection coefficients, connection weight functions, external inputs, and time delays, etc.), if their periods are incommensurable, then, for such a neural network, it is also impossible to have a periodic solution.Therefore, studying the almost periodicity of neural networks is more practical and important than studying the periodicity of neural networks.At present, the periodicity has been extensively studied as an important dynamic property of various neural networks.However, there has been no paper published on the almost periodicity of Clifford-valued neural networks yet.
On the other hand, the synchronization of nonlinear systems has become an important research topic due to its potential applications in various fields such as secure communication, image encryption, information science, and so on.Particularly, recently, many authors have studied the synchronization problem for various neural network systems [6][7][8][9][10][11][12][13][14][15].For example, the synchronization problem for chaotic memristor-based neural networks with time-varying delays 2 Complexity was studied in [11].The global asymptotic synchronization problem of nonidentical fractional-order neural networks with Riemann-Liouville derivative was investigated in [12].The synchronization of an inertial neural network with time-varying delays was investigated in [13].The finitetime cluster synchronization of coupled fuzzy cellular neural networks with Markovian switching topology, discontinuous activation functions, proportional leakage, and time-varying unbounded delays was studied in [14].The global exponential almost periodic synchronization of quaternion-valued neural networks with time-varying delays was investigated in [15].But until now, the results of the synchronization of Cliffordvalued neural networks have not been reported.
In summary, it is meaningful to study the almost periodicity and the synchronization problem of Clifford-valued neural networks.Therefore, our main purpose of this paper is to investigate the problem of the existence of almost periodic solutions and global asymptotic synchronization of Clifford-valued CNNs with discrete delays.To the best of our knowledge, this is the first paper to study such a problem for Clifford-valued neural networks with discrete delays.Our methods of this paper can be used to study other types of Clifford-valued neural networks.
The rest of this paper is organized as follows: in Section 2, we introduce some basic concepts, notations, and lemmas and give a model description.In Section 3, we study the existence of almost periodic solutions of Clifford-valued CNNs with discrete delays.In Section 4, we investigate the global asymptotic almost periodic synchronization of Clifford-valued CNNs with discrete delays.In Section 5, we give an example to demonstrate the effectiveness and feasibility of our results.In Section 5, we give a conclusion.

Preliminaries and Model Description
The real Clifford algebra over R  is defined as where For simplicity, when one element is the product of multiple Clifford generators, we will write its subscripts together.For example,  4  5  7  9 =  4579 .Denote Λ = {0, 1, 2, . . ., , . . ., 12 ⋅ ⋅ ⋅ }, and then we have where ∑  is short for ∑ ∈Λ and A is isomorphic to R 2  .For any  = ∑    ∈ A, the involution of  is defined as where In view of     = (−1) and then    A = (−1) [⋅]   .In addition, for any  ∈ A, we can find a unique   satisfying  ⋅ = (−1) [⋅]   for     = (−1) [⋅]   .Therefore, and Throughout this paper, A  , R × , A × represent the dimensional real Clifford vector space, the set of all  ×  real matrices, and the set of all  ×  real Clifford matrices.We define the norm of In this paper, we consider the following Clifford-valued cellular neural network with discrete delays: where  = 1, 2, . . ., , and  corresponds to the number of units in the neural network;   () ∈ A denotes the activation of the th neuron at time ;   () > 0 represents the rate with which the th unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs at time ;   (),   () ∈ A represent the strengths of connectivity without and with time delay between cells  and  at time , respectively;   ,   : A → A are the activation functions of the signal transmission;   () ∈ A is an external input on the th unit at time ;   () ≥ 0 is the transmission delay at time .
The initial values of (8) are where 8) can be transformed into the following equation: with the initial value: Definition (see [16]).A function  ∈ (R, R  ) is said to be almost periodic on R, if, for any  > 0, it is possible to find a real number  = () > 0 such that, for any interval with length (), there is a number  = () in this interval such that |( + ) − ()| < .
Denote by (R, R  ) the set of all almost periodic functions on R; then (R, R  ) is a Banach space with the supremum norm ‖‖ = ∑  =1 sup ∈R |  ()|.
Denote by (R, A  ) the set of all such functions.Consider the following linear system where According to [16], one can easily get the following lemma.

The Existence of Almost Periodic Solutions
In this section, we study the existence and uniqueness of almost periodic solutions by the contracting mapping principle.
In order to obtain our results, we introduce the following assumptions.
(H 1 ) For ,  = 1, 2, . . ., ,   ,   ∈ (R, R + ),   ,   ,   ∈ (R, A) and for each 1 en system of ( ) has a unique almost periodic solution in the region Proof.For ∀Φ ∈ X, we consider the following system: Since ( 1 ), it follows from Lemma 3 that system (24) has a unique almost periodic solution We define a mapping  : X 0 → X as follows: We first prove that (X 0 ) ⊂ X 0 .To this end, for each Φ ∈ X 0 , we have which implies Φ ∈ X 0 .Hence, (X 0 ) ⊂ X 0 .Then, we prove that  is a contracting mapping.In fact, for every Φ, Ψ ∈ X 0 , we have Hence,  : X 0 → X 0 is a contracting mapping.Therefore, by the Banach fixed point theorem, there exists a unique point Φ * ∈ X 0 such that Φ * = Φ * ; that is, system (14) has a unique almost periodic solution.In view of Remark 4, we know that system (8) has a unique almost periodic solution.
The proof is complete.

Almost Periodic Synchronization
In this section, we will investigate the global asymptotic synchronization problem of Clifford-valued CNNs with discrete delays.To this end, we consider the system (8) as the drive system and design the response system as where   ∈ ([−, 0], A),   () ∈ A represents the state of the response system,   () is a state-feedback controller, and the rest notation is the same as those in system (8).
Let    () =    () −    (); then, from ( 8) and ( 29), we obtain the following error system: where   () =   () −   ().To realize the global asymptotic synchronization of the drive-response system, we choose the following state-feedback controller where  = 1, 2, . . ., ,   : R → R + ,   : R → A, ℎ  : A → A. By using a similar approach of transforming system (8) into system ( 14) and adopting the similar notation there, system (30) can be transformed into the following real-valued system: > 0, Remark .Under the premise that system (8) has an almost periodic solution, the synchronization of almost periodic system (8) and almost periodic system (29) is called the almost periodic synchronization.
(34) en the drive system ( ) and the response system ( ) implement global asymptotical almost periodic synchronization.
Proof.By Theorem 6, system (8) has an almost periodic solution.In order to show that system (8) and system (29) are globally asymptotically synchronized, we consider a Lyapunov function where Calculating the derivatives of  1 and  2 along the solutions of system (32), we have and Therefore, lim →+∞  1 () = 0.By the definition of  1 (), we obtain that lim →+∞ |   () −    ()| = 0, for all  ∈ Λ.Hence, the drive system (8) and the response system (29) are globally asymptotically synchronized.The proof is complete.

A Numerical Example
In this section, we present an example to illustrate our results.

Conclusion
In 2 (Ｎ) Ｔ 12  1 (Ｎ) Ｔ 12  2 (Ｎ) neural networks into 2  2  real-valued systems and by designing a new state-feedback controller.Our results are new and our methods can be used to study other types of Cliffordvalued neural networks.