Pseudo almost periodic solutions for quaternion-valued high-order Hopﬁeld neural networks with time-varying delays and leakage delays on time scales

: This paper deals with a class of quaternion-valued high-order Hopﬁeld neural networks with time-varying delays and leakage delays on time scales. Based on the Banach ﬁxed point theorem and the theory of calculus on time scales, some su ﬃ cient conditions are obtained for the existence and global exponential stability of pseudo almost periodic solutions for the considered networks. The results of this paper are completely new. Finally, an example is presented to illustrate the e ﬀ ectiveness of the obtained results.


Introduction
Because high-order Hopfield neural networks have more extensive applications than Hopfield neural networks, various dynamical behaviours of high-order Hopfield neural networks such as the existence and stability of equilibrium points [1,2], anti-periodic solutions [3], almost periodic solutions [4][5][6] and pseudo almost periodic solutions [7] have been studied by many scholars.
On the one hand, due to the limited switching speed of neurons and amplifiers, time delays are inevitably introduced into neural network models [8][9][10][11][12]. Among all kinds of time delays, the leakage delay, that is, the time delay in the leakage term, has been proved to have a great influence on the dynamics of the system. Therefore, it is significant to consider neural networks with time delays in leakage terms [13][14][15][16].
On the other hand, both continuous-time and discrete-time neural networks have equally importance in various applications. Therefore, it is necessary to consider both continuous time neural networks and discrete time neural networks. Fortunately, the theory of time scale calculus [17] can unify the study of continuous analysis and discrete analysis, so the study of neural network models on time scale can unify the study of continuous-time and discrete-time neural networks [18][19][20][21].
In addition, quaternion-valued neural networks, as an extension of real-valued neural networks and complex-valued neural networks, have been extensively applied in many fields such as robotics, satellite attitude control, computer graphics, ensemble control and so on [22][23][24]. Currently, the study quaternion-valued neural networks have received much attention of many scholars.
Moreover, although non-autonomous neural networks are more general and practical than autonomous neural networks, so far, there are still few results about the dynamic behavior of non-autonomous quaternion-valued neural networks [25][26][27][28][29]. It is well known that periodicity, almost periodicity and pseudo almost periodicity are very important dynamic behaviors of non-autonomous systems. Besides, almost periodicity is more reasonable than periodicity. Also, pseudo almost periodicity is more complex than almost periodicity [30,31]. Therefore, for non-autonomous neural networks, pseudo almost periodic oscillation is a very important dynamics [32][33][34][35][36][37].
However, up to now, there has been no paper published on the pseudo almost periodic oscillation of quaternion-valued high-order Hopfield neural networks. Besides, the pseudo almost periodic oscillation of quaternion-valued neural networks with quaternion leakage coefficients on time scales has not been reported. Consequently, it is necessary to study the pseudo almost periodic solutions of quaternion-valued high-order Hopfield neural networks on time scales whose leakage coefficients are also quaternions.
Motivated by the above statement, in this paper, we consider the following quaternion-valued highorder Hopfield neural network with time-varying delays and leakage delays on time scales: where p ∈ {1, 2, . . . , n} =: S, n is the number of neurons in layers; x p (t) denotes the activation of the pth neuron at time t; a p (t) ∈ Q represents the rate with the pth unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs at time t; b pq (t), c pq (t) ∈ Q are the delay connection weights from neuron q to neuron p at time t, respectively; T pql (t) ∈ Q denotes the second-order connection weights of the neural network; f q , g q , h q : Q → Q are the activation functions of signal transmission; u p (t) ∈ Q is the external input on the pth unit at time t; η p denotes the leakage delay at time t and satisfies t − η p (t) ∈ T; τ pq , δ pql and ϑ pql are transmission delays at time t and satisfy The skew field of quaternions is denoted by where q R , q I , q J , q K are real numbers, the three imaginary units i, j and k obey the Hamilton's multiplication rules: Also, for convenience, we introduce the following notation: The initial condition of system (1.1) is of the form Our main aim of this paper is to study the existence and stability of pseudo almost periodic solutions of (1.1). The main contributions of this paper are listed as follows. Firstly, this is the first time to consider quaternion-valued neural networks on time scales with all the coefficients are quaternions except time delays. Secondly, this is the first paper to study the pseudo almost periodic solutions for quaternion-valued high-order Hopfield neural networks with time-varying delays and leakage delays on time scales. Finally, our method of this paper can be used to study pseudo almost periodic solutions for other types of quaternion-valued neural networks on time scales. This paper is organized as follows: In Section 2, we introduce some definitions, preliminary lemmas. In Section 3, we establish some sufficient conditions for the existence and global exponential stability of pseodo almost periodic solutions of system (1.1). In Section 4, we give an example to demonstrate the feasibility of our results. This paper ends with a brief conclusion in Section 5.

Preliminaries
Definition 2.1. [38,39] A time scale T is an arbitrary nonempty closed subset of the real set R with the topology and ordering inherited from R. The forward and backward jump operators σ, ρ : T → T, and the forward graininess µ : T → [0, ∞) are defined, respectively, by The point t ∈ T is called left-dense, left-scattered, right-dense or right-scattered if ρ(t) = t, ρ(t) < t, σ(t) = t or σ(t) > t, respectively. Points that are right-dense and left-dense at the same time are called dense. If T has a left-scattered maximum m, define T κ = T − {m}; otherwise, set T κ = T. If T has a right-scattered maximum m, define T κ = T − {m}; otherwise, set T κ = T.
Definition 2.2. [38,39] Assume that f : T → R is a function and let t ∈ T k . Then we define f ∆ (t) to be the number (provided it exists) with the property that given any ε > 0, there is a neighborhood U of t (i.e, U = (t − δ, t + δ) ∩ T for some δ > 0) such that for all s ∈ U. We call f ∆ (t) the delta (or Hilger) derivative of f at t. Moreover, we say that f is delta (or Hilger) differentiable (or in short: differentiable) on T k provided f ∆ (t) exists for all t ∈ T k . The function f ∆ : T k → R is then called the (delta) derivative of f on T k .

Definition 2.6. [18] A time scale T is called an almost periodic time scale if
We denote by BC(T, Q n ) the set of all bounded continuous functions from T to Q n . Similar to Definition in [18], we give the following definition.
is a relatively dense set in R for all ε > 0; that is, for any given ε > 0, there exists a constant l(ε) > 0 such that each interval of length l(ε) contains at least one τ(ε) ∈ T (ε, f ) such that We denote by AP(T, Q n ) the set of all almost periodic functions defined on T. Define the class of functions PAP 0 (T, Q n ) as follows: Similar to Definition in [35], we give the following definition.
We denote by PAP(T, Q n ) the set of all pseudo almost periodic functions from T to Q n . Similar to the proofs in [21], it is not difficult to prove the following lemmas.
Throughout this paper, we assume that the following conditions hold:

Main results
is a solution of the following system: where p ∈ S, t ∈ T, and vice versa.
On the other hand, let x be a bounded solution of (1.1), then for p ∈ S, Multiply both sides of (3.2) by e −a R p (t 0 , σ(t)), we can get where t ≥ t 0 , t 0 ∈ T. Integrating both sides of (3.3) from t 0 to t, we obtain Letting t 0 → −∞, then we obtain that (1.1) holds. The proof is complete.
First, we will prove that Φ maps E into itself. To this end, let ×h l (φ l (s − ϑ pql (s))) + u p (s), p ∈ S.
Then, by Lemmas 2.2 and 2.3, we find that F p (s) ∈ PAP(T, Q). So, for all p ∈ S, we can set where Ω 1 p ∈ AP(T, Q) and Ω 0 p ∈ PAP 0 (T, Q). In fact, since a R p ∈ AP(T, R + ) and F 1 p ∈ AP(T, Q), for every ε > 0, there exists l > 0 such that every interval of length l contains a number τ ∈ Π satisfying Consequently, we have which implies that Ω 1 p ∈ AP(T, Q). Then, we will prove that Ω 0 p ∈ PAP 0 (T, Q). In addition, from which implies that Ω 0 p ∈ PAP 0 (T, Q). Therefore, x φ p ∈ PAP(T, Q), that is, Φ maps E into PAP T, Q n . Next, we will show that Φ is a self-mapping from E * to E * . In fact, for each φ ∈ E * , we have Thus, we have On the other hand, we have Noting the fact that for φ ∈ E * , we have

It follows from (3.4)-(3.5), and (H 3 ) that
thus, we have Φφ ∈ E * . Finally, we will show that Φ is a contraction mapping in E * . For any φ, ψ ∈ E * , we can get It follows from (3.6) that On the other hand, we can derive that

By (3.7), (3.8) and (H 3 ), we have
Φφ − Φψ E ≤ ρ φ − ψ E , in view of the definition of ρ, which implies that Φ is a contraction mapping. Therefore, Φ has a unique fixed point in E * , that is, (1.1) has a unique pseudo almost periodic solution in E * . The proof is complete.
Theorem 3.2. Assume that (H 1 )-(H 3 ) hold, then system (1.1) has a unique pseudo almost periodic solution that is globally exponentially stable.
If T = Z, we take By a simple calculation, we have When T = R, we have Hence, whether T = R or T = Z, all the conditions of Theorems 3.1 and 3.2 are satisfied. Consequently, we know that system (1.1) has a pseudo almost periodic solution, which is globally exponentially stable. Simulated by Matlab, when T = R and T = Z, Figures 1 and 2 show the time responses of the variables of system (1.1). Figure 1 has initial values        x 1 R (n) x 2 R (n) Figure 2. T = Z. Transient states of the solutions x R p (n), x I p (n), x J p (n) and x K p (n), where p = 1, 2.

Conclusions
In this paper, we have established the existence and global exponential stability of pseudo almost periodic solutions for quaternion-valued high-order Hopfield neural networks with time-varying delays and leakage delays on time scales. The results of this paper are essentially new. In addition, we expect to extend this work to study other types of quaternion-valued neural networks on time scales.