Synchronization of Caputo fractional neural networks with bounded time variable delays

: One of the main problems connected with neural networks is synchronization. We examine a model of a neural network with time - varying delay and also the case when the connection weights ( the in ﬂ u - ential strength of the j th neuron to the i th neuron ) are variable in time and unbounded. The rate of change of the dynamics of all neurons is described by the Caputo fractional derivative. We apply Lyapunov functions and the Razumikhin method to obtain some su ﬃ cient conditions to ensure synchronization in the model. These su ﬃ cient conditions are explicitly expressed in terms of the parameters of the system, and hence, they are easily veri ﬁ able. We illustrate our theory with a particular nonlinear neural network.


Introduction
Nowadays, the synchronization of fractional-order delayed neural networks has attracted more and more attention. Some synchronization results have been obtained, for instance, in [1][2][3], where the authors studied the synchronization of fractional-order memristor-based neural networks with delay. The problem of synchronization of the fractional neural network with delay is studied in [4]. In [5,6], the synchronization of fractional-order neural networks with multiple time delays is investigated. Note that in most of the known literature models, the connection weights are constants and the controllers are proportional to the error with a constant.
Motivated by the above discussions, the main goal of this paper is to study synchronization of neural networks with delay and with the Caputo fractional derivative. Note that the studies in this paper are more general than the existing ones in the literature. We study the general case of time-varying self-regulating parameters of all units and also time-varying functions of the connection between two neurons in the network. For example, in [4] the model is considered in the case all the connection weights are constants, the self-inhibition rate is described by a constant, and for a very special output depending on the Lipschitz constants of the activation functions. Also, we are applying Mittag-Leffler function with one parameter, which is deeply connected with the applied fractional derivative, and on the other side is a generalization of the exponential function applied in [1,5,7]. The study in this paper is based on the application of Lyapunov functions. To deal with the presence of the delay, we apply the Razumikhin method. By constructing an appropriate Lyapunov function, applying a special type of its derivative (as introduced in [8]), and a comparison principle with time delay, we obtain some sufficient conditions to ensure synchronization in the model. The theoretical results are illustrated in an example.
The main contributions of this paper could be summarized as follows: • We include in the model the Mittag-Leffler function in both the self-inhibition rate and the connection weights, which is more appropriate when we are modeling with a Caputo fractional derivative. • We consider the variable coefficients in both the self-inhibition rate and the connection weights.
• We study the bounded variable in time delay.
• We consider output coupling controller depending on the delay and with variable in time input matrices.
• We apply the Razumikhin method and Dini fractional derivatives of Lyapunov functions, which are connected with both the presence of the delay and Caputo fractional derivative in the system.

System description
We will consider the model based on an analog circuit, consisting of capacitors, resistors, and amplifiers. The input-output characteristics of the amplifiers in the circuits are modeled by some functions known as activation functions. In the literature, these functions are assumed to satisfy a wide variety of assumptions, mainly they are Lipschitz functions. Initially, we will give some basic definitions from fractional calculus. We fix a fractional order ∈ ( ) q 0, 1 everywhere in the paper, as it is usually done in many applications in science and engineering (for some physical interpretation of the fractional order see, for example, [9]).
We recall that the Mittag-Leffler function with one parameter is defined as In our further investigations, we will use the following notations: We set up the following initial condition to model (2): . Note that model (2) is more general than the existing ones in the literature. For example, the synchronization of model (2) is studied in [4] in the case all the connection weights are constants, the self-inhibition rate is described by a constant, and for a very special output depending on the Lipschitz constants of the activation functions.
A similar model is studied in [13], but the connection weights are bounded, which is not satisfied in this case because of the presence of Mittag-Leffler function.
System (2) is considered as the driven system, and the corresponding response system (slave system) is described by the following fractional-order differential equations: with initial condition denotes the state variable of the response system (4) and = ( u u u , , 1 2 … ) u , n indicates the synchronous controller to be designed. Definition 2. The master delay Caputo fractional system (2) and the slave delay Caputo fractional system (4) are globally asymptotically synchronized if, for any initial functions are solutions of the master delay Caputo fractional system (2) with initial condition (3) and the slave delay Caputo fractional system (4) with initial condition (5), respectively.
The main goal of the paper is to implement appropriate controllers ( ) u t i , = … i n 1, 2, , for the response system, such that the controlled response system (2) and (3) could be synchronized with the drive system (4).
One approach to study synchronization problems is based on applying Lyapunov-like functions. Thus, we start by defining what is a Lyapunov function and then its derivative among the fractional equation.

Definition 3. [8]
Let ⊂ + J be a given interval. We will say that the function n and it is locally Lipschitzian with respect to its second argument.
In connection with the Caputo fractional derivative, it is necessary to define in an appropriate way the derivative of the Lyapunov functions among a nonlinear Caputo fractional differential equation The studied model is a partial case of the nonlinear system (6). Some existence and uniqueness results of global solutions of Caputo fractional differential equations are given in [12].
In this paper, we will use Dini fractional derivative of the Lyapunov function ∈ ( We will illustrate the application of the above defined Dini fractional derivative among the fractional system (6) to a particular Lyapunov function.  (1).

Example 1. Consider the particular Lyapunov function
. Then, the Dini fractional derivative of the Lyapunov function V among the fractional system (6) is Using the formula In the sequel, we will use the following comparison result: , n is such that, for any point Then, max Θ, Θ , for 0. x , (⋅) y are solutions of the initial value problems for the driven system (2), (3) and for the response system (4), respectively, with controller defined by (11). Then, the synchronization of system (4) is equivalent to asymptotic stability of the zero solution of the system for the error functions given by

( )) F e t F y t x t f y t f x t G e t G y t x t g y t g x t and .
j j j j j j j j jj j j j j j j We assume the following:  Theorem 1. Let Assumptions A1-A3 be satisfied. Then, the master delay Caputo fractional system (2), (3) and the slave delay Caputo fractional system (4) are globally asymptotically synchronized.
Proof. Consider the Lyapunov function 0 be a solution of the initial value problem (12) and let ≥ t 0 be a point such that where we get for the Dini fractional derivative of the Lyapunov function V among the fractional system (12) at the point t and the function ψ defined above: Applying the formula If the initial functions ; 0 t and, therefore, the master delay Caputo fractional system (2) and the slave delay Caputo fractional system (4) are globally asymptotically synchronized. □

An application
As an example, we will study a partial case of the driven system (2) and the corresponding slave system (4) with the controller with time delay taken as output coupling given by (11).
Consider the following master delay Caputo fractional system with three agents, i.e., = n 3: x s ϕ s s  . Therefore, according to Theorem 1, the master delay Caputo fractional system (20) and the slave delay Caputo fractional system (21) are globally asymptotically synchronized. The solutions of both systems are graphed in Figures 1-3. Also, in Figure 4, the differences between components of both solutions are graphed. As it can be observed, these differences approach 0 very fast. Remark 3. The synchronization of Caputo delay neural networks was also studied in [14]. But, in that case, the connection weights (the influential strength of the jth neuron to the ith neuron) at the current time as well as at the delayed time are bounded from above. Now, because of the Mittag-Leffler function, this condition is not satisfied and the results from [14] could not be applied to the master delay Caputo fractional system (20) and the slave delay Caputo fractional system (21).