Almost Periodicity in Shifts Delta(+/-) on Time Scales and Its Application to Hopfield Neural Networks

Engineering Letters, 29:3, EL_29_3_06 ______

Almost Periodicity in Shifts Delta(+/-) on Time Scales and Its Application to Hopﬁeld Neural Networks Meng Hu, and Lili Wang

Abstract—In this paper, we first give three definitions with the definitions and the properties of shifts δ±, some basic results aid of the shift operators δ, that is, almost periodicity in shifts on almost periodic differential equations in shifts δ± on time δ, almost periodic function in shifts δ and ∆-almost periodic scales are established. Moreover, we applying the obtained function in shifts δ, then by using the theory of calculus on time scales and some mathematical methods, the existence and results to study a class of Hopfield neural networks on time scales as follows uniqueness theorem of solution of linear dynamic system on  almost periodic time scale in shifts δ is obtained. Finally, ∑n  ∆ − we applying the obtained results to study the existence and  yi (t) = ai(t)yi(t) + cij(t)gj(yj(t)) exponential stability of the almost periodic solution in shifts  j=1 ∑n δ of a class of Hopfield neural networks with delays, several (1)  + dij(t)fj(yj(δ−(τij, t))) + Ii(t), examples and numerical simulations are given to illustrate and  j=1 reinforce the main results.  yi(s) = ϕi(s), s ∈ [δ−(ˆτ, t0), t0]T, i ∈ N, Index Terms—Almost periodic time scale in shifts δ; Expo- nential dichotomy; Hopfield neural networks; Almost periodic where t ∈ T, T is an almost periodic time scale in shifts solution in shifts δ; Global exponential stability. δ±, N = {1, 2, . . . , n}, the integer n corresponds to the number of units in (1); yi(t) corresponds to the state of the I.INTRODUCTION ith unit at time t; ai(t) > 0 represents the passive decay rate; c and d weight the strength of jth unit on the ith T is well known that almost periodicity plays an im- ij ij unit at time t; I (t) is the input to the ith unit at time t from portant role in dynamic systems both theoretical study i I outside the networks; g and f denote activation functions and practical applications, see [1-6]. In recent years, with i i of transmission; the function δ−(·, t) is a delay function the development of the theory of time scales (see [7,8]), generated by the backward shift operator δ− on time scale the existence and stability of almost periodic solutions of T, τ corresponds to the signal transmission delay along the dynamic equations on time scales received many researchers’ ij axon of the jth unit which is nonnegative and bounded, and special attention; see, for example, [9-12]. In these works, τˆ = max{τij}. the almost periodic time scale T satisfies the condition i,j∈N “t ± s ∈ T, ∀t ∈ T, s is the translation constant”. Under this condition all almost periodic time scales are unbounded II.ALMOSTPERIODICITYINSHIFTS δ± below and above. However, there are many time scales such T R as qZ = {qn : q > 1 is constant and n ∈ Z} ∪ {0} which Let is a nonempty closed subset (time scale) of . The σ, ρ : T → T is neither closed under the operation t ± s nor unbounded forward and backward jump operators and the µ T → R+ below. Therefore, almost periodicity on time scales need to graininess : are defined, respectively, by be explored further. σ(t) = inf{s ∈ T : s > t}, ρ(t) = sup{s ∈ T : s < t}, In [13-16], Adıvar et al. defined two shift operators, i.e. µ(t) = σ(t) − t. δ+ (forward shift operator) and δ− (backward shift operator), which does not oblige the time scale to be closed under the A point t ∈ T is called left-dense if t > inf T and ρ(t) = t, ± operations t s. The applications of the shift operators δ± left-scattered if ρ(t) < t, right-dense if t < sup T and σ(t) = on time scales; see, for example, [17-19]. Motivated by the t, and right-scattered if σ(t) > t. If T has a left-scattered above works, the main purpose of this paper is to define maximum m, then Tk = T\{m}; otherwise Tk = T. If a new almost periodicity concept with the aid of the shift T has a right-scattered minimum m, then Tk = T\{m}; operators δ±. We first give three definitions, that is, almost otherwise Tk = T. periodicity in shifts δ±, almost periodic function in shifts A function f : T → R is right-dense continuous provided δ± and ∆-almost periodic function in shifts δ±. Under the it is continuous at right-dense point in T and its left-side T Manuscript received January 25, 2021; revised April 26, 2021. This work limits exist at left-dense points in . If f is continuous at was supported in part by the Key Scientific Research Project of Colleges and each right-dense point and each left-dense point, then f is Universities of Henan Province (No.21A110001) and the National Natural said to be a continuous function on T. Science Foundation of China (No.11801011). T → R Meng Hu is an associate professor in the School of Mathematics and A function p : is called regressive provided 1 + Statistics, Anyang Normal University, Anyang, Henan 455000, China (Cor- µ(t)p(t) ≠ 0 for all t ∈ Tk. The set of all regressive and responding author; e-mail: [email protected]). rd-continuous functions p : T → R will be denoted by R = Lili Wang is a lecturer in the School of Mathematics and Statistic- R T R R+ R+ T R { ∈ R s, Anyang Normal University, Anyang, Henan 455000, China (e-mail: ( , ). We define the set = ( , ) = p : ay [email protected]). 1 + µ(t)p(t) > 0, ∀ t ∈ T}.

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l(ε) If r is a regressive function, then the generalized expo- that in any interval of length δ± (·), there exists at least a nential function er is defined by p ∈ E{ε, f} such that { ∫ } t p ∗ |f(δ±(t)) − f(t)| < ε, ∀t ∈ T , er(t, s) = exp ξµ(τ)(r(τ))∆τ s p where δ± := δ±(p, t), p is called the ε-translation constant for all s, t ∈ T, with the cylinder transformation of f, l(ε) is called the inclusion length of E{ε, f}. { Log(1+hz) ̸ ∈ ∞ ∗ h , if h = 0, Remark 2. In Definition 2, if there exists a ω [ωT, + )T ξh(z) = ω ∗ z, if h = 0. and ω ∈ E{ε, f} such that f(δ±(t)) = f(t), ∀t ∈ T , the smallest constant ω is called the period of f. Let p, q : T → R are two regressive functions, define p p Remark 3. If T = R or T = Z, then the operator δ±(t)) = p ⊕ q = p + q + µp q, ⊖p = − , p ⊖ q = p ⊕ (⊖q). ± 1 + µp t p associated with the initial point t0 = 0, then Definition 2 is converted to the traditional definition of almost periodic T → R Lemma 1. ([7]) Assume that p, q : are two regressive function on R or Z. functions, then (i) e0(t, s) ≡ 1 and ep(t, t) ≡ 1; Now, we give two examples to illustrate the Definition 2 (ii) ep(σ(t), s) = (1 + µ(t)p(t))ep(t, s); is more generality. 1 T R (iii) ep(t, s) = = e⊖p(s, t); Example 1. Let = and t0 = 1. The operators ep(s,t) { (iv) ep(t, s)ep(s, r) = ep(t, r); t , if t ≥ 0, ∆ ⊖ δ−(p, t) = p for p ∈ [1, +∞) (v) (∫e⊖p(t, s)) = ( p)(t)e⊖p(t, s); b pt, if t < 0, (vi) p(t)ep(c, σ(t))∆t = ep(c, a) − ep(c, b). a and { δ± A comprehensive review on the shift operators on time pt, if t ≥ 0, δ (p, t) = for p ∈ [1, +∞) scales can be found in [13-16]. + t , if t < 0, Let T∗ is a non-empty subset of the time scale T and p ∗ ∗ t0 ∈ T is a fixed constant, define operators δ± :[t0, +∞)× are backward and forward shift operators (on the set R = ∗ ∗ ∗ T → T . The operators δ+ and δ− associated with t0 ∈ T R − {0}) associated with the initial point t0 = 1. (called the initial point) are said to be forward and backward Consider the function ∗ ( ) shift operators on the set T , respectively. The variable s ∈ ln |t| f(t) = cos π , t ∈ R∗. [t0, +∞)T in δ±(s, t) is called the shift size. The values 1 ∗ ln( 2 ) δ+(s, t) and δ−(s, t) in T indicate s units translation of the ∈ T∗ By direct computation, term t to the right and the left, respectively. The sets { ±1 ≥ D { ∈ ∞ × T∗ ∈ T∗} p f(tp ), if t 0, ± := (s, t) [t0, + )T : δ±(s, t) f(δ±(t)) = t f( ±1 ), if t < 0, ( p ) are the domains of the shift operator δ±, respectively. Here- | | ± 1 ∗ ln t ln( ) after, T is the largest subset of the time scale T such that = cos p π ∞ × T∗ → T∗ ln( 1 ) the shift operators δ± :[t0, + ) exist. ( 2 ) Next, we give three definitions with the aid of the shift ln |t| ln( 1 ) = cos π ± p π . operators δ±, that is, almost periodicity in shifts δ±, almost 1 1 ln( 2 ) ln( 2 ) periodic function in shifts δ± and ∆-almost periodic function 2n ∈ Z in shifts δ±. Let l = p = 2 , n , for any given ε > 0, then | p − | ∀ ∈ R∗ Definition 1. (Almost periodicity in shifts δ±) Let T is a f(δ±(t)) f(t) = 0 < ε, t . ( ) time scale with the shift operators δ± associated with the ln |t| ∗ Therefore, cos ln( 1 ) π is an almost periodic function in initial point t0 ∈ T . The time scale T is said to be almost 2 ( ) ln |t| ∈ ∞ ∗ ± periodic in shifts δ± if there exists p (t0, + )T such that shifts δ . Moreover, cos ln( 1 ) π is a 4-periodic function ∗ 2 (p, t) ∈ D± for all t ∈ T , that is, in shifts δ±. ( ) ∗ ln |t| {p ∈ (t0, +∞)T∗ :(p, t) ∈ D±, ∀t ∈ T } ̸= ∅. Remark 4. The function f(t) = cos 1 π is not almost ln( 2 ) periodic in the sense of the definition of almost periodic Remark 1. In Definition 1, if function which has been defined in [9], but is almost periodic ∗ ωT := inf{p ∈ (t0, +∞)T∗ :(p, t) ∈ D±, ∀t ∈ T } ̸= t0, in shifts δ±. See, Figure1, T Z Z { n ∈ Z } ∪ { } then ωT is called the period of the time scale T. Example 2. Let = q , q = q : n , q > 1 0 , and t0 = 1. The operators Definition 2. (Almost periodic function in shifts δ±) Let T t is an almost periodic time scale in shifts δ±. A real-valued δ−(p, t) = , for p ∈ [1, +∞) ∗ p function f defined on T is almost periodic in shifts δ± if the ε-translation set of f and p ∗ E{ε, f} = {(p, t) ∈ D± : |f(δ±(t)) − f(t)| < ε, ∀t ∈ T } δ+(p, t) = pt, for p ∈ [1, +∞) ∗ is a relatively dense set in T∗ for all ε > 0; that is, for are backward and forward shift operators (on the set qZ = Z any given ε > 0, there exists a constant l(ε) > t0, such q ) associated with the initial point t0 = 1.

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T∗ ε > 0 1.5 is a relatively dense set in for all ; that is, for any given ε > 0, there exists a constant l(ε) > t0, such l(ε) · 1 that in any interval of length δ± ( ), there exists at least a p ∈ E∆{ε, f} such that p ∆p ∗ 0.5 |f(δ±(t))δ± (t) − f(t)| < ε, ∀t ∈ T , p where δ± := δ±(p, t), p is called the ε-translation constant 0 of f, l(ε) is called the inclusion length of E∆{ε, f}.

−0.5 Lemma 2. Let f ∈ C(T, R) is an almost periodic function in shifts δ±, then f(t) is bounded on T.

−1 Proof: We only consider the forward shift operator δ+, the other case is similar. For given ε ≤ 1, there exists a −1.5 l · −100 −50 0 50 100 constant l, such that in{ any} interval of length δ+( ), there p ( ) exists at least a p ∈ E ε, f , such that |f(δ (t)) − f(t)| < ln |t| + Fig. 1. Graph of f(t) = cos 1 π . ε, ∀t ∈ T. In addition, f ∈ C(T, R), then in the limited ln( 2 ) l interval [t0, δ+(t0)]T, there exists a constant M > 0, such that{ |f(}t)∩| < M. For any given t ∈ T, we can take p ∈ p ∈ Consider the function E ε, f [δ−(t, t0), δ−(t, t0 + l)]T, then we have δ+(t) l p [t , δ (t )]T. Hence, we can obtain |f(δ (t))| < M and ln t Z 0 + 0 + f(t) = (−1) ln q , t ∈ q . | p − | ∈ T | | f(δ+(t)) f(t) < 1. So for all t , we have f(t) < Let l = p = q2n, n ∈ Z, then M + 1. This completes the proof. ∈ Rn × p ln t ±2n ln t Deﬁnition 4. ([9]) Let x , and A(t) is an n n rd- − ln q − ln q f(δ±(t)) = ( 1) = ( 1) = f(t), continuous matrix function on T, the linear system and for any given ε > 0, x∆(t) = A(t)x(t), t ∈ T, (2) p Z |f(δ±(t)) − f(t)| = 0 < ε, ∀t ∈ q . is said to admit an exponential dichotomy on T if there exist

ln t positive constant k, α, projection P and the fundamental Therefore, (−1) ln q is an almost periodic function in shifts solution matrix X(t) of (2),satisfying ln t 2 δ±. Moreover, (−1) ln q is a q -periodic function in shifts δ±. −1 |X(t)PX (σ(s))|0 ≤ ke⊖α(t, σ(s)), ln t Remark 5. The function f(t) = (−1) ln q is not almost s, t ∈ T, t ≥ σ(s), periodic in the sense of the definition of almost periodic −1 |X(t)(I − P )X (σ(s))|0 ≤ ke⊖α(σ(s), t), function which has been defined in [9], since there is not ∈ T ≤ any positive constant p such that f(t + p) = f(t) holds. But s, t , t σ(s), f(t) is almost periodic in shifts δ±. See, Figure2, where | · |0 is the Euclidean norm. ∗ ∗ ∗ ··· ∗ T Definition 5. The solution x = (x1, x2, , xn) of a 1.5 system is said to be exponentially stable, if there exists a positive α such that for any ξ ∈ [δ−(τ0, t0), t0]T, τ0 > t0, 1 there exists N = N(ξ) ≥ 1 such that for any solution T x = (x1, x2, ··· , xn) satisfying

0.5 ∗ ∗ ∥x − x ∥ ≤ N∥φ − x ∥e⊖α(t, ξ), t ∈ [t0, +∞)T,

0 where φ is the initial condition, and ∥ − ∗∥ | − ∗ | 2n, n=1,2,... φ x = max sup φi(ξ) xi (ξ) . 1≤i≤n ∈ −0.5 ξ [δ−(τ0,t0),t0]T Lemma 3. ([20]) If the following conditions ∑n ∑n −1 + ∆ ≤ ∈ ∞ (1) D xi (t) aijxj(t)+ bijx¯j(t), t [t0, + )T, j=1 j=1 −1.5 i, j = 1, 2, ··· , n, 0 50 100 150 200 250 300 ∑n where a ≥ 0(i ≠ j), b ≥ 0, x¯ (t ) > 0, x¯ (t) = ln t ij ij i 0 i Fig. 2. Graph of f(t) = (−1) ln q , q = 2. i=1 sup xi(s), and τ0 > 0 is a constant; s∈[δ−(τ0,t),t]T f − Definition 3. (∆-almost periodic function in shifts δ±) Let (2) M := (aij + bij)n×n is an M-matrix; T is an almost periodic time scale in shifts δ±. A real-valued hold, then there exists a constant γi > 0, a > 0, such that ∗ function f defined on T is ∆-almost periodic in shifts δ± the solutions of inequality (1) satisfies ( ) if the ε-translation set of f ∑n xi(t) ≤ γi x¯j(t0) e⊖a(t, t0), ∆{ } E ε, f = j=1 p ∆p ∗ {(p, t) ∈ D± : |f(δ±(t))δ± (t) − f(t)| < ε, ∀t ∈ T } ∀ t ∈ (t0, +∞)T, i = 1, 2, ··· , n.

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∫ +∞ Consider the following almost periodic system in shifts δ± p −1 p = X(δ±(t))(I − p)X (σ(δ±(s))) ∆ t x (t) = A(t)x(t) + f(t), t ∈ T, (3) p ∆p ×f(δ±(s))δ± (s)∆s, where A(t) is an almost periodic matrix function in shifts and f(t) is a ∆-almost periodic function in shifts δ±, then δ±, f(t) is a ∆-almost periodic vector function in shifts δ±, ∆ we can check that x(t) is an almost periodic solution in shifts and δ± (·, t) are bounded. ( ) ( ) δ± of system (3). Let A(t) = a (t) ,Au = sup(a (t)) , 1 ≤ ij n×n ij n×n Next, we show that the solution x(t) is globally exponen- i, j ≤ n, t ∈ T . tially stable and uniqueness. ∗ ∗ ∗ ··· ∗ T Theorem 1. If the linear system (2) admits exponential Let x = (x1, x2, , xn) is a solution of (3), and u T dichotomy, and −A is an M-matrix, then system (3) has a x = (x1, x2, ··· , xn) is an arbitrary solution of (3). From unique almost periodic solution in shifts δ±, which is globally system (3), we have exponentially stable. And the solution − ∗ ∆ − ∗ ∫ (x(t) x (t)) = A(t)x(t) A(t)x (t). (5) t −1 x(t) = eA(t)P eA (σ(s))f(s)∆s Assume that the initial condition of (3) is −∞ ∫ T +∞ ϕ(s) = (ϕ1(s), ··· , ϕn(s)) , s ∈ [δ−(τ0, t0), t0]T, τ0 > 0, − − −1 eA(t)(I P )eA (σ(s))f(s)∆s, (4) t then the initial condition of (5) is ˆ ∗ where eA(t) is the fundamental solution matrix of (2). ϕ(s) = ϕ(s) − x , s ∈ [δ−(τ0, t0), t0]T. Proof: First, we prove that x(t) is a bounded almost Let V (t) = |x(t) − x∗(t)|, the upper right derivative periodic solution in shifts δ± of system (3). In fact, D+V ∆(t) along the solutions of system (5) is as follows x∆(t) − A(t)x(t) + ∆ − ∗ − ∗ ∆ ∫ D V (t) = sign(x(t) x (t))(x(t) x (t)) t u ∆ −1 ≤ A V (t) + OV (t), = eA (t) P eA (σ(s))f(s)∆s −∞ where O is an n × n-matrix with all its elements are zeros, +e (σ(t))P e−1(σ(t))f(t) A ∫ A “≤” denotes the relationship between the components of +∞ − ∆ − −1 vectors of the two sides, respectively. eA (t) (I P )eA (σ(s))f(s)∆s t Since −(A + O) = −A is an M-matrix, according to −1 +e (σ(t))(I − P )e (σ(t))f(t) Lemma 3, then there exist constants α > 0, γ0 > 0, for any A ∫ A t ξ ∈ [δ−(τ0, t0), t0]T, − −1 A(t)eA(t) P eA (σ(s))f(s)∆s ∗ −∞ |x (t) − x (t)| ∫ i[ i ] +∞ −1 ∗ − ≤ | − | ⊖ +A(t)eA(t) (I P )eA (σ(s))f(s)∆s γ0 sup ϕi(ξ) xi (ξ) e α(t, t0) ∈ t ξ [δ−(τ[0,t0),t0]T ] = e (σ(t))(P + I − P )e−1(σ(t))f(t) A A ≤ γ0 | − ∗ | sup ϕi(ξ) xi (ξ) e⊖α(t, ξ), = f(t). e⊖α(t0, ξ) ξ∈[δ−(τ0,t0),t0]T ∈ ∞ By Lemma 1, Lemma 2 and Deﬁnition 4, t [t0, + )T. ∫ e⊖ (t ,ξ) t Then, there exists a positive constant η > α 0 , such | | −1 γ0 x 0 = eA(t)P eA (σ(s))f(s)∆s that −∞ ∫ ∞ ∗ ∗ + ∥x − x ∥ ≤ N∥ϕ − x ∥e⊖ (t, ξ), t ∈ [t , +∞)T, −1 α 0 (6) − eA(t)(I − P )e (σ(s))f(s)∆s A ηγ t 0 N = N(ξ) = 0 > 1, ∥x∥ = ( ∫ where e⊖ (t ,ξ) and t α 0 max sup |xi(t)|. ≤ e⊖α(t, σ(s))∆s 1≤i≤n t∈[t0,+∞)T ∫−∞ ) ∗ ∗ ∗ ··· ∗ T +∞ By Deﬁnition 5, the solution x = (x1, x2, , xn) is + e⊖α(σ(s), t)∆s k|f|0 globally exponentially stable, that is, the solution of system ( t ) (3) is globally exponentially stable. 1 1 → ∞ → ≤ − k|f| , In inequality (6), let t + , then e⊖α(t, ξ) 0, so we α ⊖α 0 can get x = x∗. Hence, system (3) has a unique solution. This completes the proof. that is, x(t) is a bounded solution of system (3). Furthermore, Lemma 4. ([9]) Let ci(t) is an almost periodic function in ∫ p T − ∈ R+ ∀ ∈ T δ±(t) shifts δ± on , where ci(t) > 0, ci(t) , t and p −1 { } X(δ±(t))PX (σ(s))f(s)∆s −∞ min inf ci(t) = me > 0, ∫ 1≤i≤n t∈T t p −1 p p ∆p = X(δ±(t))PX (σ(δ±(s)))f(δ±(s))δ± (s)∆s; then the linear system −∞ ( ) ∫ ∞ ∆ + x (t) = diag − c1(t), −c2(t), ··· , −cn(t) x(t) (7) p −1 X(δ±(t))(I − P )X (σ(s))f(s)∆s p T δ±(t) admits an exponential dichotomy on .

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{ } By Lemma 4 and Theorem 1, we can obtain the following Since min inf ai(t) > 0, according to Corollary 1, the i∈N result. uniqueness solution of system (8) can be expressed as the Corollary 1. In system (3), if following form

yz(t) = A(t) = diag(−a1(t), −a2(t), ··· , −an(t)), { ∫ [ { } t ∑n and min inf ai(t) =a ˆ > 0, then the system (3) has a e−a1 (t, σ(s)) d1j(s)fj(ψj(δ−(τ1j, s))) 1≤i≤n t∈T −∞ j=1 unique almost periodic solution in shifts δ±, which is defined ] ∑n in (4). + c1j(s)gj(ψj(s)) + I1(s) ∆s, ··· , j=1 ∫ [ III.AN APPLICATION t ∑n − − In this section, by using the definitions and the results e an (t, σ(s)) dnj(s)fj(ψj(δ (τnj, s))) −∞ developed in Section 2, we shall study the existence and j=1 ] } ∑n exponential stability of almost periodic solution in shifts δ± + c (s)g (ψ (s)) + I (s) ∆s . (9) of system (1). nj j j n Firstly, we make the following assumptions j=1 Define a mapping Φ: B → B, and (H1) ai(t), cij(t), dij(t),Ii(t) are almost periodic functions in shifts δ± defined on T, i, j ∈ N. Φ(z)(t) = yz(t), ∀ z ∈ B. ∈ R R (H2) fj, gj C( , ) and satisfy fj(0) = 0, gj(0) = 0, Set { } respectively. Moreover, there exists positive constants B∗ | ∈ B ∥ − ∥ ≤ w1w2 f g | − | ≤ f | − | | − = z z , z z0 , Lj ,Lj such that fj(x) fj(y) Lj x y , gj(x) 1 − w1 | ≤ g| − | ∈ N gj(y) {Lj x y ,}j . then B∗ is a closed convex subset of B. According to the − ∀ ∈ (H3) λ = min inf ai(t) > 0, and 1 µ(t)ai(t) > 0, t definition of the norm on B, we have i∈N t∈T { ∫ } T, i ∈ N. t ∆ · T ∥ ∥ (H4) δ± ( , t) are bounded functions on . z0 = max sup e−ai (t, σ(s))Ii(s)∆s i∈N t∈T −∞ By Lemma 2, we know that all almost periodic functions { } Ii in shifts δ± are bounded. For convenience, we denote a = ≤ max = w2. ∈N sup |a(t)|, a = inf |a(t)| for any a(t) ∈ AP S(T), where i ai ∈T t∈T t Therefore, AP S(T) is a set of all almost periodic functions in shifts w2 δ± on the time scale T. ∥z∥ ≤ ∥z − z0∥ + ∥z0∥ = . 1 − w1 Theorem 2. Assume that (H1) − (H4) hold, and First, we show that the mapping Φ is a self-mapping from { [ ]} ∗ ∗ ∗ ∑n ∑n B to B . In fact, for any z ∈ B , we have 1 ¯ f g w1 = max dijL + c¯ijL < 1, ∈N j j ∥Φ(z) − z0∥ i ai { ∫ j=1 j=1 t

− then system (1) has a unique almost periodic solution in = max sup e ai (t, σ(s)) i∈N t∈T −∞ w1w2 shifts δ± in ∥z − z0∥ ≤ , where [ 1−w1 ∑n ∫ { t × dij(s)fj(ψj(δ−(τij, s))) z = e− (t, σ(s))I (s)∆s, ··· , j=1 0 a1 1 ] } ∫ −∞ ∑n t } + cij(s)gj(ψj(s)) ∆s e−an (t, σ(s))In(s)∆s , −∞ j=1 { ∫ { } t I ≤ max sup e−a (t, σ(s)) i ∈N i and w2 = max a . i t∈T −∞ i∈N i [ { ( ) } ∑n T × ¯ f | | Proof: Let B = z|z = ψ1, ψ2, ··· , ψn , where z dijLj ψj(δ−(τij, s)) is a continuous almost periodic function in shifts δ± on time j=1 ] } scale T with the norm ∑n g| | { } + c¯ijLj ψj(s) ∆s ∥z∥ = max sup |ψi(t)| , j=1 i∈N { ∫ t∈T t ≤ then B is a Banach space. max sup e−ai (t, σ(s)) i∈N ∈T ∈ B [ t −∞ ] } For any z , consider the solution yz(t) of the nonlinear ∑n ∑n almost periodic dynamic system in shifts δ± × ¯ f g ∥ ∥ dijLj + c¯ijLj ∆s z ∑n j=1 j=1 ∆ − { [ n n ]} yi (t) = ai(t)yi(t) + dij(t)fj(ψj(δ−(τij, t))) ∑ ∑ ≤ 1 ¯ f g ∥ ∥ j=1 max dijLj + c¯ijLj z i∈N a ∑n i j=1 j=1 + cij(t)gj(ψj(t)) + Ii(t), i ∈ N. (8) w1w2 = w1∥z∥ ≤ , j=1 1 − w1

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which implies that Φ(z)(t) ∈ B∗. Therefore, the mapping Φ Let u(t) = x(t) − x∗(t), then for i ∈ N, the system (1) is a self-mapping from B∗ to B∗. can be written as Next, we show that the mapping Φ is a contraction ∑n ∗ ∆ − mapping of B . In fact, in view of (H1) − (H4), for any ui (t) = ai(t)ui(t) + cij(t)pj(uj(t)) z, z˜ ∈ B, j=1 ( ) ( ) ∑n T ˜ ˜ ˜ T z = ψ1, ψ2, ··· , ψn , z˜ = ψ1, ψ2, ··· , ψn , + dij(t)qj(uj(δ−(τij, t))), (10) j=1 we have where ∥Φ(z) − Φ(˜z)∥ { ∫ − ∗ t pj(uj(t)) = gj(xj(t)) gj(xj (t)), qj(uj(δ−(τij, t))) ∗ − = max sup e ai (t, σ(s)) = fj(xj(δ−(τij, t))) − fj(x (δ−(τij, t))). i∈N ∈T j [ t −∞ ∑n The initial condition of system (10) is Ψ(s) = ψ(s)−x∗, s ∈ × d (s)[f (ψ (δ−(τ , s))) ij j j ij [δ−(ˆτ, t0), t0]T. j=1 From (H2), we have −f (ψ˜ (δ−(τ , s)))] j j ij ] } |p (u )| ≤ Lg|u |, |q (u )| ≤ Lf |u |, j ∈ N. ∑n j j j j j j j j + c (s)[g (ψ (s)) − g (ψ˜ (s))] ∆s ij j j j j Let Vi(t) = |ui(t)|, calculating the upper right derivative + ∆ j=1 { ∫ D V (t) along the solutions of system (10), t ≤ + ∆ max sup e−ai (t, σ(s)) D Vi (t) i∈N ∈T −∞ [ t ∆ ∑n = sign(ui(t))ui (t) n n × |d (s)||f (ψ (δ−(τ , s))) ∑ ∑ ij j j ij ≤ − | | g| | ¯ f | | j=1 ai ui(t) + c¯ijLj uj(t) + dijLj u¯j(t) j=1 j=1 −f (ψ˜ (δ−(τ , s)))| j j ij ] } ∑n ∑n ∑n ≤ −a V (t) + c¯ LgV (t) + d¯ Lf V (t), + |c (s)||g (ψ (s)) − g (ψ˜ (s))| ∆s i i ij j j ij j j ij j j j j j=1 j=1 j=1 { ∫ t that is ≤ max sup e−ai (t, σ(s)) + ∆ g f i∈N ∈T D V (t) ≤ (−A + CL )V (t) + DL V (t), t ∈ T, [ t −∞ ∑n f where “≤” denotes the relationship between the components × d¯ L |ψ (δ−(τ , s)) − ψ˜ (δ−(τ , s))| ij j j ij j ij of vectors of the two sides, respectively. j=1 ] } − g f ∑n Since A (CL + DL ) is an M-matrix, according to g| − ˜ | Lemma 3, there exist constants µ > 0, r > 0, such that + c¯ijLj ψj(s) ψj(s) ∆s j=1 { ∫ Vi(t) = |ui(t)| t ∗ ≤ ≤ r sup |ψ (ζ) − x (ζ)|e⊖ (t, t ), i ∈ N, max sup e−ai (t, σ(s)) i µ 0 i∈N ∈T ζ∈[δ−(ˆτ,t ),t ]T [ t −∞ ] } 0 0 ∑n ∑n × ¯ f g ∥ − ∥ that is dijLj + c¯ijLj ∆s z z˜ ∗ j=1 j=1 |x (t) − x (t)| { [ ]} i i ∑n ∑n ≤ | − ∗ | 1 ¯ f g r sup ψi(ζ) x (ζ) e⊖µ(t, t0) ≤ max dijL + c¯ijL ∥z − z˜∥ ∈ ∈N j j ζ [δ−(ˆτ,t0),t0]T i ai j=1 j=1 ≤ r ∥ − ∗∥ ∈ N ∥ − ∥ ψ x e⊖µ(t, ζ), i . = w1 z z˜ . e⊖µ(t0, ζ) N = N(ζ) = r This implies that the mapping Φ is a contraction mapping. Let e⊖ (t ,ζ) , then ∗ ∗ µ 0 Hence, Φ has exactly one ﬁxed point z in B such that ∗ ∗ ∥x − x ∥ ≤ N∥ψ − x ∥e⊖ (t, ζ), t ∈ T. Φ(z∗) = z∗, that is, system (1) has a unique almost periodic µ B∗ ∗ ∗ ∗ ··· ∗ T solution in shifts δ± in . This completes the proof. From Deﬁnition 5, the solution x = (x1, x2, , xn) of system (1) is globally exponentially stable. This completes Theorem 3. Assume that (H ) − (H ) and the conditions 1 4 the proof. of Theorem 2 hold. Furthermore, if A − (CLg + DLf ) ·· · is an M-matrix, where A = diag(a1, a2, , an), C = ¯ g g g ··· g f IV. NUMERICAL SIMULATIONS (¯cij)n×n,D = (dij)n×n,L = diag(L1,L2, ,Ln),L = diag(Lf ,Lf , ··· ,Lf ), then the almost periodic solution in Consider the following system 1 2 n  shifts δ± of system (1) is globally exponentially stable. ∑2  ∆ −  yi (t) = ai(t)yi(t) + cij(t)gj(yj(t)) Proof: According to Theorem 2, system (1) has  j=1 ∑2 an almost periodic solution in shifts δ±, denote it as (11)  + d (t)f (y (δ−(τ , t))) + I (t), ∗ ∗ ∗ ∗ T  ij j j ij i x (t) = (x1(t), x2(t), . . . , xn(t)) . Assume that x(t) =  j=1 T (x1(t), x2(t), . . . , xn(t)) is an arbitrary solution of (1). yi(s) = ϕi(s), s ∈ [δ−(ˆτ, t0), t0]T, i = 1, 2.

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The ﬁrst two cases. Let T = R and t0 = 1, the operators p 3 δ±(p, t) = δ±(t), which have been deﬁned in Example 1. Case I. Almost periodic solution in shifts δ±. Let 2

diag(a (t), a (t)) 1 [ 1 2 ] ln |t| 2 + sin( 1 π) 0 ln( 2 ) 0 = ln |t| ,

− y1 0 2 cos( ln( 1 ) π) 2 −1 (cij(t))2×2 [ √ | | ] ln t −2 0 0.5 sin 2( 1 π) ln( 2 ) = ln |t| , 0.3 sin( 1 π) 0 −3 ln( 2 ) (dij(t))2×2 [ ] −4 ln |t| 0 50 100 150 200 0.4 sin 2( 1 π) 0 t ln( 2 ) √ = ln |t| , 0 0.5 sin 2( ln( 1 ) π) [ √ ] 2 3 ln |t| 3 sin 5( 1 π) 2.5 ln( 2 ) (Ii(t))2×1 = ln |t| , 3 cos 2( 1 π) 2 ln( 2 ) fj(yj(δ−(τij, t))) = tanh(yi(t)), 1.5 1 1 g (y (t)) = (|y (t) + 1| − |y (t) − 1|). j j 2 j j

y2 0.5

By direct computation, we can obtain 0 f g ∈ N −0.5 Lj = Lj = 1, i, j , λ = 1 > 0,Q = 0.9 < 1, −1

and −1.5 [ ] 0.6 −0.5 −2 A − (CLg + DLf ) = 0 50 100 150 200 −0.3 0.5 t

is an M-matrix. 5 (1,5) According to Theorem 2 and Theorem 3, we can conclude 4 that (11) has an almost periodic solution in shifts δ±, and the solution is exponential stability. See Figure 3. 3 (3,3) Case II. Periodic solution in shifts δ±. Let 2 diag(a (t), a (t)) [ 1 2 ] y2 ln |t| 1 (5,1) 2 + sin( ln( 1 ) π) 0 = 2 , − ln |t| 0 2 cos( ln( 1 ) π) 0 [ 2 ] ln |t| 0 0.5 sin( 1 π) −1 ln( 2 ) (cij(t))2×2 = ln |t| , 0.3 sin( ln( 1 ) π) 0 [ 2 ] −2 ln |t| −4 −3 −2 −1 0 1 2 3 4 5 y1 0.4 sin( 1 π) 0 ln( 2 ) (dij(t))2×2 = ln |t| , Fig. 3. Dynamic behaviors of system (11), the parameters are given in 0 0.5 sin( ln( 1 ) π) [ ] 2 Case I. ln |t| 3 sin( ln( 1 ) π) (I (t)) × = 2 , i 2 1 ln |t| Case III. Almost periodic solution. Let 3 cos( 1 π) ln( 2 ) [ ] 2 + sin t 0 fj(yj(δ−(τij, t))) = tanh(yi(t)), diag(a (t), a (t)) = , 1 2 0 2 − cos t 1 | | − | − | [ √ ] gj(yj(t)) = ( yj(t) + 1 yj(t) 1 ). 0 0.5 sin 2t 2 (c (t)) × = , ij 2 2 0.3 sin t 0 Then system (11) is a 4-periodic system in shifts δ±. Simi- [ ] 0.4 sin 2t 0 √ larly to the computation of Case I, according to Theorem 2 (d (t)) × = , ij 2 2 0 0.5 sin 2t and Theorem 3, we can conclude that (11) has a 4-periodic [ √ ] solution in shifts δ±, and the solution is exponential stability. 3 sin 5t (I (t)) × = , See Figure 4. i 2 1 3 cos 2t T R The following two cases. Let = and t0 = 0, the fj(yj(δ−(τij, t))) = tanh(yi(t)), operators δ±(p, t) = t − p, then system (11) is deduced into 1 g (y (t)) = (|y (t) + 1| − |y (t) − 1|). the traditional almost periodic system. j j 2 j j

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3 3

2.5 2 2

1.5 1 1

y1 0 y1 0.5

0 −1 −0.5

−1 −2 −1.5

−3 −2 0 50 100 150 200 0 10 20 30 40 50 t t

3.5 3.5

3 3

2.5 2.5

2 2

1.5 1.5

y2 1 y2 1

0.5 0.5

0 0

−0.5 −0.5

−1 −1

−1.5 −1.5 0 50 100 150 200 0 10 20 30 40 50 t t

5 (1,5) 5 (1,5)

4 4

3 (3,3) 3 (3,3)

2 2 y2 y2 1 (5,1) 1 (5,1)

0 0

−1 −1

−2 −2 −3 −2 −1 0 1 2 3 4 5 −2 −1 0 1 2 3 4 5 y1 y1 Fig. 4. Dynamic behaviors of system (11), the parameters are given in Fig. 5. Dynamic behaviors of system (11), the parameters are given in Case II. Case III. [ ] 3 sin t (I (t)) × = , Similarly to the computation of Case I, according to Theorem i 2 1 3 cos t 2 and Theorem 3, we can conclude that (11) has an almost fj(yj(δ−(τij, t))) = tanh(yi(t)), periodic solution, and the solution is exponential stability. 1 See Figure 5. gj(yj(t)) = (|yj(t) + 1| − |yj(t) − 1|). 2 Case IV. Periodic solution. Let Similarly to the computation of Case I, according to Theorem [ ] 2 and Theorem 3, we can conclude that (11) has a periodic 2 + sin t 0 diag(a (t), a (t)) = , solution, and the solution is exponential stability. See Figure 1 2 0 2 − cos t [ ] 6. 0 0.5 sin t (c (t)) × = , ij 2 2 0.3 sin t 0 V. CONCLUSION [ ] 0.4 sin t 0 From the examples and numerical simulations in Section (d (t)) × = , ij 2 2 0 0.5 sin t IV, the coefﬁcients of the ﬁrst two cases are almost periodic

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3 see [21-24]. We leave this for future work.

2.5 REFERENCES 2 [1] C. Niu, X. Chen, “Almost periodic sequence solutions of a discrete 1.5 Lotka-Volterra competitive system with feedback control,” Nonlinear Anal. RWA., vol. 10, pp3152-3161, 2009. 1 [2] Y. Xia, J. Cao, M. Lin, “New results on the existence and uniqueness of

y1 0.5 almost periodic solution for BAM neural networks with continuously distributed delays,” Chaos Soliton. Fract., vol. 31, pp928-936, 2007. 0 [3] Y. Hamaya, “Existence of an almost periodic solution in a difference

−0.5 equation with infinite delay,” J. Difference Equ. Appli., vol. 9, pp227- 237, 2003. −1 [4] N. Boukli-Hacene, K. Ezzinbi, “Weighted pseudo almost periodic solutions for some partial functional differential equations,” Nonlinear −1.5 Anal., vol. 71, pp3612-3621, 2009. −2 [5] E.H.A. Dads, P. Cieutat, K. Ezzinbi, “The existence of pseudo-almost 0 10 20 30 40 50 periodic solutions for some nonlinear differential equations in Banach t spaces,” Nonlinear Anal., vol. 69, pp1325-1342, 2008. [6] Y. Li, X. Fan, “Existence and globally exponential stability of almost 3.5 periodic solution for Cohen-Grossberg BAM neural networks with variable coefficients,” Appl. Math. Model., vol. 33, pp2114-2120, 2009. 3 [7] M. Bohner, A. Peterson, Dynamic equations on time scales: An Introduction with Applications, Birkhauser, Boston, 2001. 2.5 [8] S. Hilger, “Analysis on measure chains-a unified approach to continuous and discrete calculus,” Result. Math., vol. 18, pp18-56, 1990. 2 [9] Y. Li, C. Wang, “Almost periodic functions on time scales and 1.5 applications,” Discrete Dyn. Nat. Soc., vol. 2011, Article ID 727068. [10] M. Hu, H. Lv, “Almost periodic solutions of a single-species system y2 1 with feedback control on time scales,” Adv. Difference Equ., vol. 2013, 2013:196. 0.5 [11] Y. Zhi, Z. Ding, Y. Li, “Permanence and almost periodic solution for an enterprise cluster model based on ecology theory with feedback 0 controls on time scales,” Discrete Dyn. Nat. Soc., vol. 2013, Article ID 639138. −0.5 [12] H, Zhang, Y. Li, “Almost periodic solutions to dynamic equations on −1 time scales,” J. Egypt. Math. Soc., vol. 21, pp3-10, 2013. 0 10 20 30 40 50 [13] M. Adıvar, “Function bounds for solutions of Volterra integro dynamic t equations on the time scales,” Electron. J. Qual. Theo., vol. 7, pp1-22, 2010. 5 (1,5) [14] M. Adıvar, Y.N. Raffoul, “Existence of resolvent for Volterra integral equations on time scales,” B. Aust. Math. Soc., vol. 82, no.1, pp139- 155, 2010. 4 [15] M. Adıvar, Y.N. Raffoul, “Shift operators and stability in delayed dynamic equations,” Rend. Sem. Mat. Univ. Politec. Torino, vol. 68, no.4, pp369-396, 2010. 3 (3,3) [16] M. Adıvar, “A new periodicity concept for time scales,” Math. Slovaca, vol. 63, no.4, pp817-828, 2013. [17] E. Çetin, “Positive periodic solutions in shifts δ for a nonlinear first-

y2 2 order functional dynamic equation on time scales,” Adv. Difference Equ., vol. 2014, 2014:76. 1 (5,1) [18] E. C¸etin, F.S. Topal, “Periodic solutions in shifts δ for a nonlinear dynamic equation on time scales,” Abstr. Appl. Anal., vol. 2012, Article ID 707319. 0 [19] M. Hu, L. Wang, Z. Wang, “Positive periodic solutions in shifts δ for a class of higher-dimensional functional dynamic equations with impulses on time scales,” Abstr. Appl. Anal., vol. 2014, Article ID −1 −2 −1 0 1 2 3 4 5 509052. y1 [20] M. Hu, L. Wang, “Unique existence theorem of solution of almost Fig. 6. Dynamic behaviors of system (11), the parameters are given in periodic differential equations on time scales,” Discrete Dyn. Nat. Soc., Case IV. vol. 2012, Article ID 240735. [21] L. Xu, Y. Liao, “On an Almost Periodic Gilpin-Ayala Competition Model of Phytoplankton Allelopathy,” IAENG International Journal of Applied Mathematics, vol. 47, no. 2, pp182-190, 2017. [22] L. Wang, “Dynamics of an Almost Periodic Single-Species System functions in shifts δ±, but not almost periodic functions; and with Harvesting Rate and Feedback Control on Time Scales,” IAENG if we choose suitable shift operators, almost periodic function International Journal of Computer Science, vol. 46, no.2, pp237-242, in shifts δ± will be deduced into almost periodic function, 2019. [23] Y. Shi, C. Li, “Almost Periodic Solution, Local Asymptotical Stability see cases III and IV; that is, almost periodic function which and Uniform Persistence for a Harvesting Model of Plankton Al- has been deﬁned in [9] is a special case of almost periodic lelopathy with Time Delays,” IAENG International Journal of Applied function in shifts δ±. Moreover, we can study the existence Mathematics, vol. 49, no. 3, pp307-312, 2019. [24] M. Hu, S. Zhang, X. Ding, “Extinction in a Nonautonomous Compet- of almost periodic solution in shifts δ± on more general time itive Lotka-Volterra System with Impulses and Delays,” Engineering scales, even the time scale is not satisfy additivity or the time Letters, vol. 28, no.2, pp529-534, 2020. scale is bounded. Therefore, the obtained results in this paper improve and supplement that of the previous studies. We would like to point out here that the obtained results in this paper can be used to study many other dynamic systems;

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