A STABILIZATION APPROACH FOR A NOVEL CHAOTIC FRACTIONAL-ORDER DISCRETE NEURAL NETWORK

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. This paper aims to make a contribution to the topic of the stabilization of chaotic dynamics in fractionalorder discrete systems by presenting a novel three-dimensional fractional-order discrete neural network formulated by the h-fractional difference operator. A novel theorem is illustrated with the aim of stabilizing the chaotic trajectories of the three-dimensional fractional-order discrete neural network at zero through proposing a quite simple linear control laws. Finally, certain simulation results are carried out to highlight the effectiveness of the stabilization approach proposed herein.


INTRODUCTION
For a long period of time, the consideration and expansion of fractional calculus were only limited to its case formulated by continuous-time [1]. This formulation plays a major role in many sciences and fields. During the early modern era, many researchers give appropriate Diaz and Osier are the first researchers introduced the discrete fractional calculus and that is in the year 1974. Also, they carried out many major research works concerning to this new field [2]. Lately, many researchers focus their studies on the FoNN in its discrete-time form not only on its continuous-time form [17,18,19]. Afterwards, many categories of applications related to the Fractional-order Discrete Neural Networks (FoDNNs), like bidirectional associative memory and others are studied in [20]. Besides, certain synchronization between specific types of FoDNNs with/without time delays is discussed in [13]. However, here is how this article is arranged. Some preliminaries and notations associated with the fractional h-difference operators are presented in the second section. In the third section, A the FoDNN model as well as its equivalent form are formulated in view of the fractional h-difference operator. Section four illustrates the proposed stabilization approach, while the fifth section contains the main conclusions of this study.

PRELIMINARIES
Due to the fact that the discrete fractional calculus is considered a relatively new topic that has not yet settled, some preliminaries and notations associated with this topic are presented here for completeness. Definition 1. [21] Let X : (hN) a → R 0n which a is a starting point. Then, the ν th -order h-sum is given by: where ν > 0 and the h-falling factorial function is defined as: Definition 2.
[10] For the function x (t) defined on (hN) a , the Caputo-like difference operator is defined by: where ∆X (t) = X(t+h)−X(t) h and n = ν + 1 in which ν > 0 and ν / ∈ N.
Next, we briefly illustrate a significant result reported in [22] with the aim of identifying some convenient stability conditions of the zero equilibrium point for the following nonlinear fractional-order difference system: Theorem 1. Let x = 0 be an equilibrium point of the nonlinear fractional-order discrete system (3). If there exists a positive definite and decrescent scalar function V (t, X (t)), such that C h ∆ ν a V (t, X (t)) ≤ 0 , then the equilibrium point is asymptotically stable.
In the following content, we state an extremely useful technical inequality associated with identifying a proper Lyapunov's function which is considered the heart of Lyapunov's direct method that will be used later on.

THE CHAOTIC FRACTIONAL-ORDER DISCRETE NEURAL NETWORK
In this part, a novel version of the fractional-order discrete neural network, which has been recently proposed in [8], is established by using the fractional-order h-difference operator. In particular, the authors in [8] have operated the ν-Caputo operator to formulate a three-dimensional continuous fractional-order Hopfield neural network model which was formerly proposed by NOVEL CHAOTIC FRACTIONAL-ORDER DISCRETE NEURAL NETWORK 5517 Zhang et al. in [23]. Herein, in view of the Caputo h-difference operator, we can formulate the following three-dimensional fractional-order discrete neural network model: Based on a key result reported in [24], system (5) can be converted into an equivalent form consisting of the following three discrete integral equation: The Predictor-corrector method is adopted here to transfer the above three formulas given in system (6)

STABILIZATION APPROACH
In this section, a novel stability result through using the Lyapunov direct method is stated and derived with the aim of achieving a proper control for the chaos which occurred to the  fractional-order discrete neural network (5). In other words, we will endeavor to propose some proper linear control laws that would stabilize all the three states of system (5) at the origin.
Afterwards, we will verify whether this task is successfully carried out by performing several NOVEL CHAOTIC FRACTIONAL-ORDER DISCRETE NEURAL NETWORK 5519 numerical simulations. For this purpose, let us begins with the following theorem that represents the main result of our work.

CONCLUSION
In this paper, a novel three-dimensional fractional-order discrete neural network has been formulated using the h-fractional difference operator. For the purpose of stabilizing the chaotic trajectories of such proposed model at zero, a novel theorem, which proposes certain simple linear control laws, has been stated and derived. Several numerical simulations have been carried out to verify the theoretical analysis.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.