Anti-Periodic Dynamics of Quaternion-Valued Fuzzy Cellular Neural Networks with Time-Varying Delays on Time Scales

A class of quaternion-valued fuzzy cellular neural networkswith time-varying delays on time scales is proposed. Based on inequality analysis techniques on time scales, a fixed point theorem and the theory of calculus on time scales, the existence, and global exponential stability of anti-periodic solutions for this class of neural networks are established.The obtained results are completely new and supplement to the known results. Finally, a numerical example is given to illustrate the feasibility of our results.


Introduction
Since Yang and Yang [1] first introduced fuzzy cellular neural networks (FCNNs) combining fuzzy operations (fuzzy AND and fuzzy OR) with cellular neural networks, FCNNs have been successfully applied in many fields such as physics, chemistry, biology, economics, sociology, medicine, and meteorology [2].Because all of their applications heavily rely on their dynamics, in recent years, a lot of meaningful results regarding the dynamics of them are obtained by many researchers (see [3][4][5][6][7][8][9][10][11] and reference therein).For instance, authors in [8] studied the global stability of equilibria of FCNNs, authors in [9] obtained some sufficient conditions for the existence and stability of a unique periodic solution of FCNNs, authors in [10] investigated the existence and global exponential stability of anti-periodic solutions for neutral type FCNNs with time-varying delays and  operator on time scales, the author in [11] studied the almost periodicity of FCNNs with multi-proportional delays, and in other accounts authors investigated other behaviors of FCNNs with time delays.
On the one hand, because quaternion-valued neural networks (QVNNs) as an extension of the real-valued neural networks and complex-valued neural networks can be extensively applied to the fields of robotics, attitude control of satellites, computer graphics, ensemble control, color night vision, and image compression ( [12][13][14]) and one of the benefits by using quaternion is the three-dimensional geometrical affine transformation that can be represented efficiently and compactly, the study of dynamical behaviors for QVNNs has received much attention of many scholars and some good results have been obtained for the stability [15][16][17][18][19], dissipativity [20], periodicity [21], pseduo almost periodicity [22], and synchronization of QVNNs [23,24].
On the other hand, as we know, a special case of the quasi-periodicity of functions is the anti-periodicity and -anti-periodic functions are 2-periodic functions, but not all periodic functions are anti-periodic ones.Since the signal transmission process of neural networks can often be described as an anti-periodic process, the problem of antiperiodic solutions for various types of neural networks has been investigated by many authors ( [10,[25][26][27][28][29][30][31][32][33][34]).
Moreover, continuous time and discrete time systems are very important in implementation and applications.In addition, in a realistic system, the interaction among agents can happen at any time, and maybe some continuous time intervals accompany some discrete moments.So it is necessary and significant to consider both continuous time and discrete time cases at the same time in networked systems.Fortunately, the time scale theory, which was introduced by Hilger [35], can unify the study of continuous and discrete analysis, and the study of dynamic equations on time scales can contain, link, and extend the classical theory of differential and difference equations [36].Recently, the theory of time scale calculus has been applied in real-valued neural networks [37][38][39][40][41][42][43] and complex-valued networks [44].However, to the best of our knowledge, the existence and global stability of anti-periodic solutions of quaternion-valued fuzzy cellular neural networks (QVFCNNs) on time scales have not been considered yet.Besides, it is well known that time delays are unavoidable in real neural network systems and they may cause the changes of the dynamical behaviors of neural networks [11,25,40,41].
Motivated by the above discussion, our main aim of this paper is to study the existence and global stability of antiperiodic solutions of QVFCNNs with time-varying delays.The innovation points of this paper are summarized as follows: (1) We propose a class of QVFCNNs with time-varying delays on time scales which can unify the continuous time and discrete time cases of QVFCNNs and, what is more, which can contain the QVFCNNs that their time argument may vary in some continuous time intervals accompanying some discrete moments.
(2) The QVFCNNs proposed in this paper contain realvalued FCNNs and complex-valued FCNNs as their special cases.
(4) Our results are completely new and supplement to the known results, and our results show that if the coefficients of leakage terms of QVFCNNs with timevarying delays on time scales are positive regressive, then both the continuous time and discrete time QVFCNNs with time-varying delays have the same dynamics for the anti-periodicity.
This paper is organized as follows.In Section 2, we give the model description and introduce some definitions and preliminary lemmas and transform the quaternionvalued system (10) into an equivalent real-valued system.In Section 3, we establish the existence of anti-periodic solutions of the considered network based on a fixed point theorem.In Section 4, by using some inequality techniques, we derive some sufficient conditions for the global exponential stability of anti-periodic solutions of the considered network.In Section 5, we give an example to show the feasibility and effectiveness of our main results.This paper ends with a brief conclusion in Section 6.

Model Description and Preliminaries
The quaternion was invented in 1843 by Hamilton [45].The skew field of quaternion is denoted by where   ,   ,   ,   are real numbers and the elements , , and  obey the Hamilton's multiplication rules: 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 the backward jump operators ,  : T → T are defined by Assume that  : T → R is a function and let  ∈ T  .Then we define  Δ () to be the number (provided it exists) with the property that given any  > 0, there is a neighborhood U of t (i.e.,  = ( − ,  + ) ∩ T for some  > 0) such that       ( ()) −  () −  Δ () ( () − )      ≤  | () − | for all  ∈ .We call  Δ () the delta derivative of  at .
Definition 3 (see [46]).We say that a time scale T is periodic if there exists  > 0 such that if  ∈ T, then  ±  ∈ T. For T ̸ = R, the smallest positive  is called the period of the time scale.
Definition 4 (see [46]).Let T ̸ = R be a periodic time scale with period .We say that the function  : T → R is periodic with period  if there exists a natural number  such that  = , ( + ) = () for all  ∈ T and  is the smallest positive number such that ( + ) = ().
Remark 5. From [47], we know that if T is -periodic, then the graininess function  is -periodic.
Lemma 6 (see [47]).Let T be -periodic and suppose  : T × T → R satisfies the assumptions of Lemma  T → R is said to be -anti-periodic on T if ( + ) = −() for all  ∈ T,  > 0 is a constant.Definition 8. Let T be an -periodic time scale.A function  =   +    +    +    : T → Q  is called an -antiperiodic, where   : T → R  ,  ∈ {, , , } fl  if for every  ∈ ,   is -anti-periodic.
In this paper, we consider the following QVFCNN with time-varying delays on time scales: where  ∈ {1, 2, . . ., } fl Λ,  ∈ T, T is a periodic time scale;  is the number of neurons in layers;   () ∈ Q and  ℎ () ∈ Q are the state of the th neuron at time  and the deviations of the ℎth neuron at time , respectively;   () > 0 represents the rate with which the th neuron will reset its potential to the resting state in isolation when they are disconnected from the network and the external inputs at time The initial conditions of system (10) are where Lemma 9 (see [8]).Suppose  and  are two states of system (15).Then we have In order to overcome the inconvenience of the noncommutativity of quaternion multiplication, in the following, we will first decompose system (10) into the vector form of the four real-valued systems.To do so, for we assume that the activation functions  ℎ ( ℎ ) and  ℎ ( ℎ ) of ( 10) can be expressed as where Remark 10.In system (10), if  ℎ =   ℎ +   ℎ  ∈ C, where   ℎ ,   ℎ ∈ R and the activation functions  ℎ ,  ℎ : C → C are complex variable functions, that is, where   ℎ ,   ℎ : R 2 → R, ℎ ∈ Λ,  = , , and all the quaternion-valued coefficients of (10) are transformed to the complex-valued coefficients, then system (10) degenerates to a complex-valued system; if all of the activation functions and coefficients of (10) are real variable functions, then system (10) degenerates to a real-valued system.

Global Exponential Stability of Anti-Periodic Solutions
In this section, we will study the global exponential stability of anti-periodic solutions of (15).
Remark 16.In view of Remark 11, we have that, under conditions of Theorems 14 and 15, system (10) has a unique anti-periodic solution that is globally exponentially stable.
which implies that conditions ( 3 ) and ( 4 ) are also satisfied.Therefore, according to Theorems 14 and 15, (10) has a unique -anti-periodic solution, which is globally exponentially stable (Figures 1 and 2).

Conclusion
In this paper, we have proposed a class of QVFCNNs with time-varying delays on time scales which can unify the continuous time and discrete time cases of QVFCNNs and contain real-valued FCNNs and complex-valued FCNNs as their special cases.Based on a novel method that is different from those used in [10,[25][26][27][28][29][30][31][32][33][34], we established the existence and global exponential stability of anti-periodic solutions of the QVFCNNs.To the best of our knowledge, this is the first paper to study the existence of anti-periodic solutions for QVFCNNs with time-varying delays on time scales.The method of this paper may be applied to study other types of the QVNNs on times scales such as quaternionvalued recurrent neural networks, quaternion-valued BAM neural networks, quaternion-valued shunting inhibitory cellular neural networks, quaternion-valued Cohen-Grossberg neural networks, and so on.

2 Figure 1 : 2 Figure 2 :
Figure 1: Responses of the four parts of  with continuous time t.