Pseudo almost periodic synchronization of Clifford-valued fuzzy cellular neural networks with time-varying delays on time scales

At present, the research on discrete-time Clifford-valued neural networks is rarely reported. However, the discrete-time neural networks are an important part of the neural network theory. Because the time scale theory can unify the study of discrete- and continuous-time problems, it is not necessary to separately study continuous- and discrete-time systems. Therefore, to simultaneously study the pseudo almost periodic oscillation and synchronization of continuous- and discrete-time Clifford-valued neural networks, in this paper, we consider a class of Clifford-valued fuzzy cellular neural networks on time scales. Based on the theory of calculus on time scales and the contraction fixed point theorem, we first establish the existence of pseudo almost periodic solutions of neural networks. Then, under the condition that the considered network has pseudo almost periodic solutions, by designing a novel state-feedback controller and using reduction to absurdity, we obtain that the drive-response structure of Clifford-valued fuzzy cellular neural networks on time scales with pseudo almost periodic coefficients can realize the global exponential synchronization. Finally, we give a numerical example to illustrate the feasibility of our results.


Introduction
Fuzzy cellular neural networks, introduced into the field of artificial neural networks in 1996 by Yang and Yang [1,2], are a combination of fuzzy operations (fuzzy AND and fuzzy OR) and cellular neural networks. They combine the advantages of neural network and fuzzy theory and integrate learning, association, recognition, and information processing. Because fuzzy neural networks are based on uncertainty, which is a common problem in the study of brain model, they are closer to human brain than the general neural networks. Therefore the fuzzy cellular neural networks are widely used in the fields such as pattern recognition, computer science, artificial intelligence, optimal control, equation solving, robotics, military science, and so on. Because the application of neural networks in these fields is related to their long-term behaviors and the time delay is inevitable in real neural where e A = e h 1 e h 2 · · · e h ν with A = {h 1 , h 2 , . . . , h ν }, 1 ≤ h 1 < h 2 < · · · < h ν ≤ m. Moreover, e ∅ = e 0 = 1 and e {h} = e h , h = 1, 2, . . . , m, are called Clifford generators, which satisfy the relations Let Q = {∅, 1, 2, . . . , A, . . . , 12 · · · m}. Then it is easy to see that where A is a shorthand for A∈Q . For u = A u A e A ∈ A and v = (v 1 , v 2 , . . . , v n ) T ∈ A n , the norms of u and v are defined as u A = A (u A ) 2 and v A n = max 1≤p≤n v p A , respectively; for v = (v 1 , v 2 , . . . , v n ) T ∈ A n , the norm of v is defined as v A n = max 1≤p≤n v p A . For information on the Clifford algebra, we refer the reader to [50]. Definition 2.1 ([8]) For x, y ∈ R, we denote x ∧ y = min{x, y} and x ∨ y = max{x, y}.
Let T be a time scale, that is, an arbitrary nonempty closed subset of the real set R with the topology and ordering inherited from R, let σ and η denote the forward jump operator and the graininess function, respectively, and let R denote the set of regressive functions on T. We define the set R + = {r ∈ R : 1 + μ(t)r(t) > 0, ∀t ∈ T}. For the time scale theory, we refer the reader to [51].
From now on, we assume that T is an almost periodic time scale. Definition 2.4 A function f ∈ C(T, A n ) is called almost periodic if for every ε > 0, there exists a constant l(ε) > 0 such that each interval of length l(ε) contains τ ∈ such that We denote by AP(T, A n ) the set of all almost periodic functions defined on T and by BC(T, A n ) the set of all bounded continuous functions from T to A n .
Inspired by Definition 3 in [11], we introduce the following definition.
We denote by PAP(T, A n ) the class of all pseudo almost periodic functions defined on T. From the above definition, similarly to the proofs of Lemmas 2.5 and 2.6 in [11], it is not difficult to prove the following two lemmas.
The following lemma can be proved by the same proof method as that for Lemma 6 in [53].
Similarly to the proof of Corollary 1 in [1], we can prove the following: Similarly to the proof of Lemma 2.2 in [28], we can easily prove the following: In the paper, we consider the following Clifford-valued fuzzy cellular neural network with time-varying delays on time scale T: where n is the number of neurons in layers; x i (t) ∈ A, μ j (t) ∈ A, and I i (t) ∈ A express the state, input, and bias of the ith neuron, respectively, where A is a Clifford algebra; a i (t) > 0 is the rate at which the ith neuron resets its potential to the resting state in isolation when they are disconnected from the network and the external inputs at time t; α ij (t) ∈ A represents an element of the fuzzy feedback MIN template;α ij (t) ∈ A is an element of the fuzzy feedback MAX template; T ij (t) ∈ A and S ij (t) ∈ A are fuzzy feed forward MIN template and fuzzy feed forward MAX template, respectively; b ij (t) ∈ A is an element of the feedback template; d ij (t) ∈ A is the feed forward template; and denote the fuzzy AND and fuzzy OR operations, respectively; f j , g j , andg j : A → A are the activation functions; η ij (t), τ ij (t), andτ ij (t) correspond to the transmission delays at time t and satisfy Throughout the rest of the paper, we adopt the following notation: The initial values of system (1) are as follows: To obtain our primary results, we need the following assumptions:

The existence of pseudo almost periodic solutions
In this section, we state and prove sufficient conditions for the existence of pseudo almost periodic solutions of (1). Proof Firstly, it is easy to check that if x ∈ BC(T, A n ) is a solution of the integral equation then x is also a solution of system (1). Secondly, we define the operator ϒ : Y → BC(T, A n ) by where ϕ ∈ Y, We will show that ϒ : Y → Y is well defined. In fact, by Lemmas 2.1, 2.2, and 2.5 for any Thirdly, we will prove that ϒ(Y 0 ) ⊂ Y 0 . In fact, for every ϕ ∈ Y 0 , we have that and so we deduce that Finally, we will prove that ϒ is a contraction. Noting that, for any ϕ, ψ ∈ Y 0 , Hence ϒ is a contraction mapping. Consequently, ϒ has a unique fixed point in Y 0 , that is, system (1) possesses a unique pseudo almost periodic solution in Y 0 . This completes the proof of Theorem 3.1.

Pseudo almost periodic synchronization
In this section, we take (1) as the driving system to study the global exponential synchronization of the drive-response structure of system (1). For this purpose, we take the following system as the response system: where t ∈ T, i ∈ I, y i (t) ∈ A denotes the state of the response system, θ i (t) is a statefeedback controller, the remaining notations are the same as those in system (2), and the initial condition is as follows: . Subtracting (1) from (2) yields the following error system: To achieve the global exponential synchronization of the drive-response systems, we design the following state-feedback controller: Definition 4.1 The response system (2) and the drive system (1) are said to be globally exponentially synchronized if for every solution z of the error system (3), there exist positive constants ξ with ξ ∈ R + and M > 1 such that  A), and there exist positive constant numbers L h j , Lh j such that for any u, v ∈ A, ij Lh j . Then the response system (1) and the drive system (2) are globally exponentially synchronized.
Proof Multiplying (3) by e -(a i +c i ) (t 0 , σ (t)) and then integrating it over the interval [t 0 , t] T , where t 0 ∈ [-ζ , 0] T , we get that Set Then by (S 6 ), for i ∈ I, we find Because of the continuity of i and the fact that i (ω) → -∞ as ω → +∞, we see that there exist θ i such that i (θ i ) = 0 and i (ω) > 0 for ω ∈ (0, θ i ), i ∈ I. Obviously, we have i (e) ≥ 0, i ∈ I, where e = min i∈I {θ i }. So, we can choose a positive constant For e ξ (t, t 0 ) > 1, where t ∈ [-ζ , t 0 ] T , it is evident that Further, we will prove the following inequality: To this end, we first prove that for any ς > 1, which implies that for all i ∈ I, Otherwise, if (7) is not true, then there exist i 0 ∈ I andt ∈ (t 0 , +∞) T such that and Therefore there must exist a constant C ≥ 1 such that and In view of (8), (9), (4), and M > 1, we have
If T = R, then we take and if T = Z, then we take Obviously, (S 1 ) and (S 4 ) hold. By calculating we have L  (1) and (2)

Conclusion
In this paper, we study pseudo almost periodic synchronization of Clifford-valued fuzzy cellular neural networks with time-varying delays on time scales by a direct method. That is to say, we do not decompose the Clifford-valued systems into real-valued systems, but directly study the Clifford systems. This is the first paper to study the pseudo almost periodic synchronization of Clifford-valued neural networks on time scales. The results of this paper are brand-new, and the proposed approach can be used to study the periodic, almost periodic, and almost automorphic synchronization for other types of neural networks on time scales. Studying the dynamics of Clifford-valued neural networks on time scales can not only unify the research of discrete-and continuous-time neural networks, but also unify the research of real-valued, complex-valued, and quaternion-valued neural networks.