Anti-periodic dynamics on high-order inertial Hopfield neural networks involving time-varying delays

Abstract: Taking into accounting time-varying delays and anti-periodic environments, this paper deals with the global convergence dynamics on a class of anti-periodic high-order inertial Hopfield neural networks. First of all, with the help of Lyapunov function method, we prove that the global solutions are exponentially attractive to each other. Secondly, by using analytical techniques in uniform convergence functions sequence, the existence of the anti-periodic solution and its global exponential stability are established. Finally, two examples are arranged to illustrate the effectiveness and feasibility of the obtained results.


Introduction
Due to the engineering backgrounds and strong biological significance, Babcock and Westervelt [1,2] introduced an inertial term into the traditional multidirectional associative memory neural networks, and established a class of second order delay differential equations, which was called as the famous delayed inertial neural networks model. Arising from problems in different applied sciences such as mathematical physics, control theory, biology in different situations, nonlinear vibration, mechanics, electromagnetic theory and other related fields, the periodic oscillation is an important qualitative property of nonlinear differential equations [3][4][5][6][7][8][9]. Consequently, assuming that the activation functions are bounded and employing reduced-order variable substitution which convert the inertial systems into the first order differential equations, the authors in [10,11] and [12] have respectively gained the existence and stability of anti-periodic solution and periodic solution for addressed inertial neural networks models. Manifestly, the above transformation will raise the dimension in the inertial neural networks system, then some new parameters need to be introduced. This will increase huge amount of computation and be attained hard in practice [13,14]. For the above reasons, most recently, avoiding the reduced order method, the authors in [15] and [16] respectively developed some non-reduced order methods to establish the existence and stability of periodic solutions for inertial neural networks with time-varying delays.
It has been recognized that, in neural networks dynamics touching the communication, economics, biology or ecology areas, the relevant state variables are often considered as proteins and molecules, light intensity levels or electric charge, and they are naturally anti-periodic [17][18][19]. Such neural networks systems are often regarded as anti-periodic systems. Therefore, the convergence analysis and stability on the anti-periodic solutions in various neural networks systems with delays have attracted the interest of many researchers and some excellent results are reported in [20][21][22][23][24][25][26][27]. In particular, the antiperiodicity on inertial quaternion-valued high-order Hopfield neural networks with state-dependent delays has been established in [28] by employing reduced-order variable substitution. However, few researchers have utilized the non-reduced order methods to explore such topics on the following highorder inertial Hopfield neural networks involving time-varying delays: associating with initial value conditions: respectively the first-order term and the second-order term of the neural network, A j , B j and Q j are the nonlinear activation functions, and q i j , η i jl , ξ i jl : R → R + are bounded and continuous functions, , and i, j, l ∈ D := {1, 2, · · · , n}. Motivated by the above arguments, in this paper, without adopting the reduced order method, we propose a novel approach involving differential inequality techniques coupled with Lyapunov function method to demonstrate the existence and global exponential stability of anti-periodic solutions for system (1.1). Particularly, our results are new and supplement some corresponding ones of the existing literature [19][20][21][22][23][24][25][26][27][28]. In a nutshell, the contributions of this paper can be summarized as follows. 1) A class of anti-periodic high-order inertial Hopfield neural networks involving time-varying delays are proposed; 2) Under some appropriate anti-periodic assumptions, all solutions and their derivatives in the proposed neural networks model are guaranteed to converge to the anti-periodic solution and its derivative, respectively; 3) Numerical results including comparisons are presented to verify the obtained theoretical results.
The remaining parts of this paper are organized as follows. In Section 2, we make some preparations. In Section 3, the existence and the global exponential stability of the anti-periodic solution are stated and demonstrated. Section 4 shows numerical examples. Conclusions are drawn in Section 5.

Preliminaries
To study the existence and uniqueness of anti-periodic solutions to system (1.1), we first require the following assumptions and some key lemmas: Assumptions: for all u, v ∈ R.
x n (t)) and y(t) = (y 1 (t), y 2 (t), · · · , y n (t)) as two solutions of system (1.1) satisfying Then, there are two positive constants λ and Proof. Denote x(t) = (x 1 (t), x 2 (t), · · · , x n (t)) and y(t) = (y 1 (t), y 2 (t), · · · , y n (t)) as two solutions of (1.1) and (1. According to (F 2 ) and the periodicity in (1.1), one can select a constant λ > 0 such that Define the Lyapunov function by setting Straightforward computation yields that It follows from (F 1 ) and PQ ≤ 1 which, together with (2.4) and (2.5), entails that This indicates that K(t) ≤ K(0) for all t ∈ [0, +∞), and Note that (α i w i (t)e λt + γ i w i (t)e λt ) 2 = (α i w i (t) + γ i w i (t)) 2 e 2λt and α i |w i (t)|e λt ≤ |α i w i (t)e λt + γ i w i (t)e λt | + |γ i w i (t)e λt |, one can find a constant M > 0 such that which proves Lemma 2.1. Remark 2.2. Under the assumptions adopted in Lemma 2.1, if y(t) is an equilibrium point or a periodic solution of (1.1), one can see y(t) is globally exponentially stable. Moreover, the definition of global exponential stability can be also seen in [13,16].
3. Anti-periodicity of system (1.1) Now, we set out the main result of this paper as follows. Proof. Denote κ(t) = (κ 1 (t), κ 2 (t), · · · , κ n (t)) be a solution of system (1.1) satisfying: With the aid of (F 1 ), one can see that where t ∈ R and i, j, l ∈ D.
Consequently, for any nonnegative integer m, Clearly, (−1) m+1 κ(t + (m + 1)T ) (t + (m + 1)T ≥ 0) satisfies (1.1), and v(t) = −κ(t + T ) is a solution of system (1.1) involving initial values: Thus, with the aid of Lemma 2.1, we can pick a constant M = M(ϕ κ , ψ κ , ϕ v , ψ v ) satisfying Hence, Consequently, and Therefore, (3.3) suggests that there exists a continuous differentiable function y(t) = (y 1 (t), y 2 (t), · · · , y n (t)) such that {(−1) m κ(t + mT )} m≥1 and {((−1) m κ(t + mT )) } m≥1 are uniformly convergent to y(t) and y (t) on any compact set of R, respectively. Moreover, involves that y(t) is T −anti-periodic on R. It follows from (F 1 )-(F 3 ) and the continuity on (3.2) that {(κ (t + (m + 1)T )} m≥1 uniformly converges to a continuous function on any compact set of R. Furthermore, for any compact set of R, setting m −→ +∞, we obtain which involves that y(t) is a T −anti-periodic solution of (1.1). Again from Lemma 2.1, we gain that y(t) is globally exponentially stable. This finishes the proof of Theorem 3.1. Remark 3.1. For inertial neural networks without high-order terms respectively, suppose and the authors gained the existence and stability on periodic solutions in [10,11] and anti-periodic solutions in [12]. Moreover, the reduced-order method was crucial in [10][11][12] when anti-periodicity and periodicity of second-order inertial neural networks were considered. However, (3.4) and (3.5) have been abandoned in Theorem 3.1 and the reduced-order method has been substituted in this paper. Therefore, our results on anti-periodicity of high-order inertial Hopfield neural networks are new and supplemental in nature.

Conclusion
In this paper, abandoning the reduced order method, we apply inequality techniques and Lyapunov function method to establish the existence and global exponential stability of anti-periodic solutions for a class of high-order inertial Hopfield neural networks involving time-varying delays and anti-periodic environments. The obtained results are essentially new and complement some recently published results. The method proposed in this article furnishes a possible approach for studying anti-periodic on other types high-order inertial neural networks such as shunting inhibitory cellular neural networks, BAM neural networks, Cohen-Grossberg neural networks and so on.