Exponential Synchronization of the Hopfield Neural Networks with New Chaotic Strange Attractor

This paper studies the problems of exponential synchronization for the Hopfield neural networks with impulsive effects and Gui chaotic strange attractor. By employing the Lyapunov functional method of impulsive functional differential equations, some criteria for synchronization between two impulsive neural networks are derived. An illustrative example is provided to show the effectiveness and feasibility of the proposed method and results.


Introduction
Recently, the issue of controlled synchronization in complex dynamical networks has become a rather significant topic in both theoretical research and practical applications [1][2][3].It is known that in theory and practice, impulsive control has been widely used to stabilize and synchronize chaotic systems.For example, Yang and Chua derived some sufficient conditions for the stabilization and synchronization of Chua's oscillators via impulsive control [4]; Xie, Wen and Li obtained sufficient conditions for the stabilization and synchronization of the Lorenz system via impulsive control with varying impulsive intervals [5].The Hopfield neural networks with impulse effect are studied, where the criteria on the existence, uniqueness and global stability of periodic solution are obtained.Further, Gui chaos strange attractor was also found [6][7][8][9][10].
Motivated by the above discussions, the aim of this paper is to study the synchronization of Hopfield neural networks with a Gui chaotic strange attractor.By employing the Lyapunov-like stability theory of impulsive functional differential equations, some criteria for synchronization of Hopfield neural networks are derived.
The remainder of the paper is organized as follows: Section 2 describes the issue of synchronization of coupled impulsive systems with a Gui chaotic strange attractor.In Section 3, some sufficient conditions for the synchronization are derived by constructing suitable Lyapunov-like function.In Section 4, an illustrative example is given to show the effectiveness of the proposed method.Conclusions are given in Section 5.

Preliminaries and Problem Formulation
In this paper, we consider the following nonautonomous Hopfield neural networks model with impulses to the first equality of (1) there exist the limits ( ) According to the above convention, we assume ( ) ( ) . Throughout this paper, we assume that: (H1) Functions ( ) j f u satisfy the Lipschitz condition, i.e., there are constants 0 , ( , ) .u u R f f (H2) There exists a positive integer p , such that ( ) , , As is known to all that (1) can exhibit chaotic phenomenons [6-10] .In order to show it clearly, we give the following example.
Example 1.Consider a two-dimensional neural network with impulsive effects, which can be described by the following impulsive differential equations: Obviously, ( ) j f x satisfy (H1).Now we investigate the influence of the period T of impulsive effect on the system (2).Set , then (H2) isn't satisfied.Periodic oscillation of system (2) will be destroyed by impulses effect.Numeric results show that system (2) still has a global attractor which can be a Gui chaotic strange attractor.Every solutions of system (2) will finally tend to the new chaotic strange attractor which is useful in exponential synchronization (see Fig. 1-3).Ƒ For the purpose of synchronization, we introduce the response system that is driven by (1) via a set of signals , e t y t x t be the synchronization error, ( ) i x t and ( ) i y t are the state variables of drive system (1) and response system (3).The error system of the impulsive synchronization is given by ITA 2017 where ( ( )) ( ( )) ( ( )) j j j j j j g e t f y t f x t .Note that the origin is the equilibrium point of system (4).If ( ) i e t tends exponentially to origin in evolution, exponential synchronization between two systems would be realized.Our aim is to find some criteria on the impulsive gains ik J such that drive system (1) and respond system (3) are exponentially synchronized for any initial condition.

Main results
In this section, we investigate the exponential synchronization of system ( 1) and (3) by using Lyapunov like functional method.Theorem 1.Under the assumption (H1), (H2) and (H3), the system (1) and ( 3) are exponentially synchronized, if there exist positive constants 1, 0 and By inequality (5), choose a small 0 Consider the following Lyapunov function: Calculating the upper right dini-derivative D V of V along the solution of system (4) at the continuous points , 0, By Lemma 1, we have the following inequalities: 1   t .According to Definition 1, we conclude that the drive system (1) and the response system (3) are exponentially synchronized.This completes the proof.Ƒ

A Simulation Example
In this section, we give an example to illustrate the effectiveness of the results obtained in the previous sections.Consider a two-dimensional neural networks with impulsive effects.Taking (2) as the drive system in Example 1.The response system is constructed as follows: Then the error system of drive system (2) and respond system (9) is constructed as follows:  Numeric results show that system (9) still has a Gui chaotic strange attractor [6-10] .The phase plot of Gui chaotic strange attractor of response system is shown in Fig. 4. It is easy to check the conditions in Theorem 1 are satisfied.Therefore, systems (2) and ( 9) exhibit exponential synchronization.Define ( ) ( ) ( ) e t y t x t and the errors (10) between systems (2) and ( 9) are depicted in Fig. 5.

Conclusions
In this paper, the conditions for the exponential synchronization of a class of neural networks with impulsive effects are derived by utilizing Lyapunov functional method.A numerical simulation is given to show the effectiveness and feasibility of the proposed method.As far as we know, there is no paper to deal with such a problem.