Anti-periodicity on high-order inertial Hopfield neural networks involving mixed delays

This paper deals with a class of high-order inertial Hopfield neural networks involving mixed delays. Utilizing differential inequality techniques and the Lyapunov function method, we obtain a sufficient assertion to ensure the existence and global exponential stability of anti-periodic solutions of the proposed networks. Moreover, an example with a numerical simulation is furnished to illustrate the effectiveness and feasibility of the theoretical results.


Introduction
The inertial neural networks model, which was first proposed by Babcock and Westerwelt [1,2], is one of the other popular artificial neural network models used in a variety of application areas. This type of neural networks has received much attention by many researchers. In particular, numerous works have been devoted to study the dynamic behaviors on inertial neural networks with time-varying delays and some excellent results are reported, for example, stability [3][4][5], Hopf bifurcation [6][7][8][9][10][11], and synchronization [12][13][14]. To the best of our knowledge, the dynamics analysis on inertial neural networks is usually to convert them into a first-order differential system by reducing order variable substitution under the assumption that the activation functions are bounded [15][16][17]. However, the authors in [12,[18][19][20][21] pointed out that the above method not only raises the dimension in the inertial neural networks system, but also increases huge amount of computation which makes it difficult to realize in practice. For the above reasons, the authors of [19,20] and [21], respectively, developed some non-reduced order techniques to investigate the stability and synchronization of inertial neural networks with different types of time delays.
On the other hand, in neural networks dynamics involving fields such as communication, economics, biology or ecology, the relevant state variables are usually considered as proteins and molecules, light intensity levels or electric charges, which are naturally antiperiodic [22][23][24]. Considering this factor, many recurrent neural network models involv-ing time-varying delays and anti-periodic environments have been widely investigated in [16,17,[24][25][26]. It is worth noting that the high-order Hopfield neural networks have the advantages of faster convergence speed, larger storage capacity and stronger fault tolerance than lower-order neural networks [27][28][29]. Consequently, Yao [30] studied the existence and global exponential stability of anti-periodic solutions for a class of proportional delayed high-order inertial Hopfield neural networks with time-varying delays.
In recent years, the authors in [21] have mentioned that many parallel routes with a series of different axon sizes and lengths appear in neural networks, and it is desired to explain the dynamics behaviors of neural networks by involving continuously distributed delays. Furthermore, the dynamic behaviors of many recurrent neural networks with continuously distributed delays have been revealed in [27,[31][32][33][34][35]. However, few articles have considered the anti-periodic problem for the following high-order inertial Hopfield neural networks (HIHNNs) involving time-varying delays and continuously distributed delays: (1.1) and the initial value conditions: where BC((-∞, 0], R) is the set of all continuous and bounded functions from (- Motivated by the previous discussions, in this paper, without adopting the reduced order method, we shall install new results concerning the anti-periodic dynamics for HIHNNs with time-varying delays and continuously distributed delays. Some sufficient conditions ensuring the existence and global exponential stability on the anti-periodic solution of system (1.1) are established by using differential inequalities and the Lyapunov function method, which improve and complement some earlier publications [16,17,[36][37][38][39][40].
We organize the paper as follows. In Sect. 2, some assumptions and an important lemmas are listed. Section 3 presents the main results and their detailed proof. Section 4 gives a numerical example to demonstrate the feasibility of the main results. Conclusions are drawn in Sect. 5.

Preliminary results
In this section, some assumptions and a key lemma are provided.
Remark 2.1 According to (G 1 ) and the basic theory on functional differential equation with infinite delay in [41], one can show that all solutions of (1.1) and (1.2) exist in [0, +∞).
According to (G 2 ) and the boundedness of (1.1), one can select a constant λ > 0 such that Set A straightforward computation yields It follows from (G 1 ) and PQ ≤ 1 2 (P 2 + Q 2 ) (P, Q ∈ R) that which, together with (2.4) and (2.5), entails This indicates that W (t) ≤ W (0) for all t ∈ [0, +∞), and Manifestly, Combining with the Cauchy-Schwarz inequality, one can pick a constant M > 0 such that which proves Lemma 2.1.
Remark 2.2 More precisely, according to Lemma 2.1, we know that, if y(t) is an equilibrium point or a T-anti-periodic solution of (1.1), then all solutions of the system (1.1) and their derivatives are exponentially convergent to y(t) and y (t), respectively. Referring to the definition of stability adopted in [5, 18-21, 40, 42-45], this indicates that y(t) is globally exponentially stable.

Anti-periodicity of HIHNNs (1.1)
Now, we set out to present the main result of this paper as follows. Proof Let v(t) = (v 1 (t), v 2 (t), . . . , v n (t)) be a solution of system (1.1) with initial conditions: Clearly, for any nonnegative integer m, for all i ∈ D, t + (m + 1)T ≥ 0. It is easy to see that (-1) m+1 v(t + (m + 1)T) is a solution of (1.1), and u(t) = -v(t + T) satisfies system (1.1) involving initial values: According to Lemma 2.1, we can take a constant Thus, Thus, together with the facts that and (-1) m+1 v i t + (m + 1)T then, we can show that there exists a continuous differentiable function κ(t) = (κ 1 (t), κ 2 (t), . . . , κ n (t)) such that {(-1) m v(t + mT)} m≥1 and {((-1) m v(t + mT)) } m≥1 are uniformly convergent to κ(t) and κ (t) on any compact set of R, respectively. Moreover, involves that κ(t) is T-anti-periodic on R. It follows from (G 1 )-(G 4 ) and the continuity on (3.2) that {v (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 ob- which involves the fact that κ(t) is a T-anti-periodic solution of (

A numerical example
In this section, we give an example with a simulation to demonstrate the feasibility and the validity of our theoretical results.

Conclusions
In this paper, abandoning the traditional reduced order method, we explore the global convergence dynamics on a class of anti-periodic high-order inertial Hopfield neural networks with bounded time-varying delays and unbounded continuously distributed delays. Some sufficient conditions have been obtained to guarantee that every solution and its derivative of the addressed model is exponentially convergent to a anti-periodic solution and its derivative by combining differential inequality techniques with the Lyapunov function method. It should be mentioned that the results obtained in this manuscript are novel, and the method adopted provides a possible effective approach for studying other types high-order inertial neural networks with mixed delays.