Global Robust Exponential Stability and Periodic Solutions for Interval Cohen-Grossberg Neural Networks with Mixed Delays

A class of interval Cohen-Grossberg neural networks with time-varying delays and infinite distributed delays is investigated. By employing H-matrix and M-matrix theory, homeomorphism techniques, Lyapunov functional method, and linear matrix inequality approach, sufficient conditions are established for the existence, uniqueness, and global robust exponential stability of the equilibrium point and the periodic solution to the neural networks. Our results improve some previously published ones. Finally, numerical examples are given to illustrate the feasibility of the theoretical results and further to exhibit that there is a characteristic sequence of bifurcations leading to a chaotic dynamics, which implies that the system admits rich and complex dynamics.


Introduction
In the past two decades, neural networks have received a great deal of attention due to the extensive applications in many areas such as signal processing, associative memory, pattern recognition, and parallel computation and optimization.It should be pointed out that the successful applications heavily rely on the dynamic behaviors of neural networks.Stability, as one of the most important properties of neural networks, is crucially required when designing neural networks.
In electronic implementation of neural networks, there exist inevitably some uncertainties caused by the existence of modeling errors, external disturbance, and parameter fluctuation, which would lead to complex dynamic behaviors.Thus, it is important to investigate the robustness of neural networks against such uncertainties and deviations (see [1][2][3][4][5][6][7][8] and references therein).In [4][5][6], employing homeomorphism techniques, Lyapunov method, -matrix and matrix theory, and linear matrix inequality (LMI) approach, Shao et where   () is time-varying delay which is variable with time due to the finite switching speed of amplifiers.Recently, the stability of neural networks with time-varying delays has been extensively investigated, and various sufficient conditions have been established for the global asymptotic and exponential stability in [9][10][11][12][13].Generally, neural networks usually have a spatial extent due to the presence of a multitude of parallel pathways with a variety of axon sizes and lengths.It is desired to model them by introducing continuously distributed delays over a certain duration of time such that the distant past has less influence compared to the recent behavior of the state (see [14][15][16]).However, the distributed delays were not taken into account in system (1).
As an important neural networks, Cohen-Grossberg neural networks (CGNNs) include Hopfield neural networks, cellular neural networks, and other neural networks.CGNNs have aroused a tremendous surge of investigation in these years.Whereas, for the interval CGNNs, fewer robust stability results have been reported in contrast to the results on Hopfield neural networks [17][18][19].On the other hand, the research of neural networks involves not only the dynamic analysis of equilibrium point but also that of the periodic oscillatory solution, which is very important in learning theory due to the fact that learning usually requires repetition [20,21].Some important results for periodic solutions of neural networks have been obtained in [7,[22][23][24][25][26][27] and references therein.Motivated by the works of [4][5][6] and the discussions above, the objective of this paper is to investigate the global robust exponential stability and periodic solutions of the following CGNNs with time-varying and distributed delays: or equivalently where where   () denotes the state of the th neuron at time , α (  ()) denotes a positive, continuous, and bounded amplification function; that is, 0 <   ≤ α (  ()) ≤   < +∞, β (  ()) denotes an appropriate behaved function,   (  ()) denotes the activation function,   () denotes the timevarying delay associated with the th neuron, satisfying 0 ≤   () ≤  and 0 ≤ τ  () ≤  < 1,   () > 0 represents the delay kernel function, which is a real-valued continuous function,  is the connection weight matrix,  is the timevarying delayed connection weight matrix,  is the infinite distributed delayed connection weight matrix,   () is the external input bias.The coefficients   ,   , and   can be intervalised as follows: (H2) For the activation functions   (⋅) ( = 1, 2, . . ., ), there exist constants   > 0 such that (H3) The delay kernels   (⋅) ( = 1, 2, . . ., ) satisfy for some positive constant .
The organization of this paper is as follows.In Section 2, some preliminaries are given.In Section 3, sufficient conditions are presented for the existence, uniqueness, and global robust exponential stability of the equilibrium point for system (2) with the external constant input bias (i.e.,   () ≡   ,   is a constant).In Section 4, sufficient conditions are given which guarantee the uniqueness and global exponential stability of periodic solutions for system (2) when the timevarying delay   () and the external input bias   () are continuously periodic functions.Numerical examples are provided to illustrate the effectiveness of the obtained results in Section 5. A concluding remark is given in Section 6 to end this work.
Definition 2 (see [30]).Let . ., }, where   (R) denotes the set of all  ×  matrices with entries from R. Then a matrix  is called an -matrix if  ∈   and all successive principal minors of  are positive.

Global Robust Exponential Stability of the Equilibrium Point
In this section, in system (2), we assume that the external input bias   () ≡   ,   is a constant ( = 1, 2, . . ., ), and we will give a new sufficient condition for the existence and uniqueness of the equilibrium point for system (2) and analyze the global robust exponential stability of the equilibrium point.
The proof of Theorem 8 is similar to that in [32], therefore we omit it here.
On the other hand, where Combining Theorem 8, we get that system (2) is globally robustly exponentially stable.The proof is complete.
Remark 10.Letting  =  be a positive scalar matrix in Theorem 9, we can get a robust exponential stability criterion based on LMI.
It can be seen that the main results in [4] is a special case of Theorem 9. Therefore, the obtained results in this paper improve the results in [4][5][6].Also, our results generalize some previous ones in [33][34][35] as mentioned in [4].In addition, in [19], the authors dealt with the robust exponential stability of CGNNs with time-varying delays.However, the distributed delays were not taken into account.Therefore, our results in this paper are more general than those reported in [19].
Considering that  * is a nonnegative matrix, we develop a new approach based on H-matrix theory.The obtained robust stability criterion is in terms of the matrices  * and   * , which can reduce the conservativeness of the robust results to some extent.
Remark 14.The periodic oscillatory behavior of the neural networks is of great interest in many applications.For instance, this phenomena of periodic solutions for neural networks coincide with the fact that learning usually requires repetition and periodic sequences of neural impulse are also of fundament significance for the control of dynamic functions of the body such as heart beat and respiration which occur with great regularity.
Remark 15.In [23], the authors studied the existence and attractivity of periodic solutions for two class of CGNNs with discrete time delays or finite distributed time delays, respectively.In this paper, we incorporated time-varying delays and infinite distributed delays into CGNNs and derived the uniqueness and global exponential stability of periodic solutions.In [24,27], two classes of CGNNs with distributed delays were investigated, and sufficient conditions were established to guarantee the uniqueness and global exponential stability of periodic solutions of such networks by using Lyapunov functional and the properties of -matrix, whereas, the time-varying delays were ignored in the models.Thus, our results effectually improve or complement the results in [23,24,27].

Conclusion
In this paper, we discussed a class of interval CGNNs with time-varying delays and infinite distributed delays.By employing -matrix and -matrix theory, Lyapunov functional method, and LMI approach, sufficient conditions were established for the existence, uniqueness, and global robust exponential stability of the equilibrium point and the periodic solution to the neural networks.It was shown that the obtained results improve or complement the previously published results.Numerical simulations demonstrated the main results and further showed that chaotic phenomena may occur for the system, which coincide with the fact of recognition character of human beings.On the other hand, it is well known that chaotic synchronization has been successfully applied to secure communication; chaotic behaviors of neural networks imply that they may be used to create secure communication systems.

Figure 1 :
Figure 1: Time responses of the state variables () with different initial values in Example 1.