Existence and Global Exponential Stability of Almost Automorphic Solution for Clifford-Valued High-Order Hopfield Neural Networks with Leakage Delays

In this paper, we study the existence and global exponential stability of almost automorphic solutions for Clifford-valued high-order Hopfield neural networks by direct method. That is to say, we do not decompose the systems under consideration into real-valued systems, but we directly study Clifford-valued systems. Our methods and results are new. Finally, an example is given to illustrate our main results.

On the one hand, the Clifford algebra was proposed by the British mathematician William K. Clifford [15] in 1878 and is a generalization of the plural, quaternion, and Glassman algebra.Currently, Clifford algebra has been widely used in various fields such as neural computing, computer and robot vision, image and signal processing, and control problems.Studies have shown that Clifford-valued neural networks are superior to commonly used real-valued neural networks [16,17], so they have become an active research field in recent years.However, because the multiplication of Clifford numbers does not satisfy the commutative law, it has brought great difficulties to the research of Cliffordvalued neural networks.Therefore, the current results on the dynamics of Clifford-valued neural networks are still very rare.At present, only a few papers have been published on the dynamics of Clifford-valued neural networks [18][19][20][21][22].It is worth mentioning that the results of these mentioned papers are obtained by decomposing Clifford-valued neural networks into real-valued networks.Therefore, it is meaningful to study the dynamics of Clifford-valued neural networks by direct method.
On the other hand, it is well known that periodic and almost periodic oscillations are important dynamic behaviors of neural networks.Almost automorphy is an extension of almost periodicity and plays an important role in better understanding of almost periodicity.Therefore, almost automorphic oscillation is more complex than almost periodic oscillation.Considering the interaction between neurons in a neural network is very complex, it is meaningful to study the almost automorphic oscillation of neural networks.
In addition, time delays are inevitable and may affect and change the dynamic behavior of dynamic systems [23,24].Therefore, neural networks with various delays have been extensively studied.In particular, since K. Gopalsamy [25] first studied the stability of neural networks with leakage delays, a lot of research has been done on neural networks with leakage delays [6,11,12,26].However, there is no research on the Clifford-valued neural network with leakage delays.
2 Complexity Inspired by the above analysis and discussion, the main purpose of this paper is to study the existence and global exponential stability of almost automorphic solutions for Clifford-valued high-order Hopfield neural networks with leakage delays by direct method; that is, we will study the considered Clifford-valued neural networks directly, instead of converting them to real-valued ones.As far as we know, this is the first paper to study the almost automorphic solutions of Clifford-valued high-order Hopfield neural networks with leakage delays.In addition, this is the first paper to study almost automorphic solutions of Clifford neural networks by direct method.So, our methods and results of this paper are new.Besides, our methods proposed in this paper can be used to study the problem of almost automorphic solutions for other types of Clifford-valued neural networks.
This paper is organized as follows.In Section 2, we introduce some basic definitions and lemmas and give a model description.In Section 3, we study the existence of almost automorphic solutions for Clifford-valued high-order Hopfield neural networks with leakage delays.In Section 4, we investigate the global exponential stability of almost automorphic solutions of the neural networks.In Section 5, an example is given to demonstrate the proposed results.Finally, we draw a brief conclusion in Section 6.
In this paper, we are concerned with the following Clifford-valued high-order Hopfield neural network with delays in the leakage term: where   () ∈ A corresponds to the state of the th unit at time ,   () ∈ R + represents the rate with which the th unit will reset its potential to the resting state when disconnected from the network and external inputs at time ,   () ∈ A denotes the strength of th unit on th unit at time ,   () ∈ A is the second-order synaptic weight of the neural networks,   (),   () ∈ A denote the activation functions,   () ∈ A is the external input on the th at time , and   (),   (), ]  () ∈ R + denote the transmission delays.
We will adopt the following notation: The initial value of system (2) is given by where Denote by (R, R) the set of all uniformly continuous functions from R to R. Let (R, A  ) denote the set of all bounded continuous functions from R to A  .Then ((R, A  ), ‖ ⋅ ‖ 0 ) is a Banach space with the norm ‖‖ 0 = max 1≤≤ sup ∈R {‖  ()‖ A }, where  = ( 1 ,  2 , . . .,   )  ∈ (R, A  ).
Definition .A function  ∈ (R, A  ) is said to be almost automorphic, if for every sequence of real numbers (   ) ∈N there exists a subsequence (  ) ∈N such that is well defined for each  ∈ R, and lim for each  ∈ R.
Since ,  ∈ (R, A), for every sequence of real numbers (   ) ∈N there exists a subsequence (  ) ∈N such that lim for every  ∈ R. Therefore, there exists a positive integer  such that lim for  >  and  ∈ R. Hence, we have Consequently, {(+  −(+  ))} converges to (−()) for each  ∈ R. Similarly, we can obtain that The proof is complete.

The Existence of Almost Automorphic Solutions
In this section, we study the existence of almost automorphic solutions by the contracting mapping principle.Let For any  = ( 1 ,  2 , . . .,   )  ∈ X, we define the norm of  as ‖‖ X = max{‖‖ 0 , ‖  ‖ 0 }, where and then, for every  ∈ X 0 , we have Theorem 6. Assume that ( 1 )-( 3 ) hold.en system ( ) has at least one almost automorphic solution in X 0 .
Thirdly, we will prove that  is a self-mapping from X 0 to X 0 .In fact, for each  ∈ X 0 , we have Fourthly, we will prove that  is a contracting mapping.In fact, for any ,  ∈ X 0 , we have that Hence, by ( 3 ),  is a contracting mapping principle.Therefore, there exists a unique fixed-point  * ∈ X 0 such that  * =  * , which implies that system (2) has an almost automorphic solution in X 0 .The proof is complete.

Global Exponential Stability
In this section, we investigate the global exponential stability of almost automorphic solutions by reduction to absurdity.(

Example
In this section, we give an example to show the feasibility of our results obtained in this paper.

Conclusion
In this paper, we obtained the existence and global exponential stability of almost automorphic solutions for Cliffordvalued high-order Hopfield neural networks by direct method.Our methods and results are new.The methods proposed in this paper can be used to study the problem of almost automorphic solutions of other types of Cliffordvalued neural networks with or without leakage delays such as Clifford-valued BAM networks, Clifford-valued cellular neural networks, and Clifford-valued shunting inhibitory cellular neural networks.Studying the dynamics of the Cliffordvalued neural networks with impulsive effects is our future work.