Dynamical Behaviors of the Stochastic Hopfield Neural Networks with Mixed Time Delays

and Applied Analysis 3 Lemma 4 (see [43]). Suppose that system (5) satisfies the local Lipschitz condition and the following assumptions hold. (A2) There are two functionsV ∈ C2,1(Rn×[−τ,∞); R+) and U ∈ C(R n ×[−τ,∞); R + ) and two probability measures μ(⋅) and μ(⋅) on [−τ, 0] such that lim ‖x‖→∞ inf 0≤t≤∞ V (x, t) = ∞, (9) while for all (φ, t) ∈ C([−τ, 0]; Rn) × R, LV (φ, t) ≤ α 1 − α 2 V (φ (0) , t)


Introduction
The well-known Hopfield neural networks were firstly introduced by Hopfield [1,2] in early 1980s.Since then, both the mathematical analysis and practical applications of Hopfield neural networks have gained considerable research attention.The Hopfield neural networks have already been successfully applied in many different areas such as combinatorial optimization, knowledge acquisition, and pattern recognition, see, for example, [3][4][5].In both the biological and artificial neural networks, the interactions between neurons are generally asynchronous, which give rise to the inevitable signal transmission delays.Also, in electronic implementation of analog neural networks, time delay is usually time-varying due to the finite switching speed of amplifiers.Note that continuously distributed delays have gained particular attention, since a neural network usually has a spatial nature due to the presence of an amount of parallel pathways of a variety of axon sizes and lengths.
Recently, it has been well recognized that stochastic disturbances are ubiquitous and inevitable in various systems, ranging from electronic implementations to biochemical systems, which are mainly caused by thermal noise and environmental fluctuations as well as different orders of ongoing events in the overall systems [6,7].Therefore, considerable attentions have been paid to investigate the dynamics of stochastic neural networks, and many results on stochastic neural networks with delays have been reported in the literature, see, for example,  and references therein.Among which, some sufficient criteria on the stability of uncertain stochastic neural networks were derived in [8][9][10].Almost sure exponential stability of stochastic neural networks was discussed in [11][12][13][14][15].In [16][17][18][19][20][21][22], mean square exponential stability and pth moment exponential stability of stochastic neural networks were investigated; Some sufficient criteria on the exponential stability for impulsive stochastic neural networks were established in [23][24][25][26].In [27], the stability of discrete-time stochastic neural networks was analyzed, while exponential stability of stochastic neural networks with Markovian jump parameters is investigated in [28][29][30].These references mainly considered the stability of equilibrium point of stochastic neural networks.What do we study when the equilibrium point does not exist?
Except for stability property, boundedness and attractor are also foundational concepts of dynamical systems.They play an important role in investigating the uniqueness of equilibrium, global asymptotic stability, global exponential stability, the existence of periodic solution, its control and synchronization [31,32], and so on.Recently, ultimate boundedness and attractor of several classes of neural networks 2 Abstract and Applied Analysis with time delays have been reported.Some sufficient criteria were derived in [33,34], but these results hold only under constant delays.Following, in [35], the globally robust ultimate boundedness of integrodifferential neural networks with uncertainties and varying delays was studied.After that, some sufficient criteria on the ultimate boundedness of neural networks with both varying and unbounded delays were derived in [36], but the concerned systems are deterministic ones.In [37,38], a series of criteria on the boundedness, global exponential stability, and the existence of periodic solution for nonautonomous recurrent neural networks were established.In [39][40][41], the ultimate boundedness and attractor of the stochastic Hopfield neural networks with timevarying delays were discussed.To the best of our knowledge, for stochastic neural networks with mixed time delays, there are few published results on the ultimate boundedness and weak attractor.Therefore, the arising questions about the ultimate boundedness, weak attractor, and asymptotic stability of the stochastic Hopfield neural networks with mixed time delays are important yet meaningful.
The left of the paper is organized as follows and some preliminaries are in Section 2, Section 3 presents our main results, a numerical example and conclusions will be in Sections 4 and 5, respectively.
Remark 2. We note that assumption (A1) is less conservative than that in [8,9,39], since the constants  −  ,  +  ,  −  and  +  are allowed to be positive, negative numbers, or zeros.
The following lemmas will be used in establishing our main results.Lemma 3 (see [44]).For any positive definite matrix  > 0, scalar  > 0, vector function  : [0, ] →   such that the integrations concerned are well defined, and the following inequality holds: Lemma 4 (see [43]).Suppose that system (5) satisfies the local Lipschitz condition and the following assumptions hold.
Theorem 5 shows that there exists  0 > 0 such that for any  ≥  0 , {‖()‖ ≤ } ≥ 1 − .Let   denote by Clearly,   is closed, bounded, and invariant.Moreover, lim sup with no less than probability 1 − , which means that   attracts the solutions infinitely many times with no less than probability 1 − , so we may say that   is a weak attractor for the solutions.
Theorem 6. Suppose that all conditions of Theorem 5 hold, then there exists a weak attractor   for the solutions of system (1).

Remark 7.
Compared with [39][40][41], assumption (A1) is less conservative than that in [39] and system (1) includes mixed time delays, which is more complex than that in [39][40][41].In addition, Lemma 4 is firstly used to investigate the dynamical behaviors of stochastic neural networks with mixed time delays.The bound for L may be in a much weaker form.Our results do not only deal with the asymptotic moment estimation but also the path wise (almost sure) estimation.

Numerical Example
In this section, a numerical example is presented to demonstrate the validity and effectiveness of our theoretical results.

Conclusion
In this paper, by using the theory of stochastic functional differential equations and linear matrix inequality, new results and sufficient criteria on the asymptotic stability, ultimate boundedness, and attractor of stochastic Hopfield neural networks with mixed time delays are established.A numerical example is also presented to demonstrate the correctness of our theoretical results.

Figure 1 :
Figure 1: Time trajectories (a) as well as the set   and several typical phase portraits (b) for the system in Example 8 (color online).Where initial values for  < 0 are chosen as () = (50, 80).For (b), only phase portraits for  ≥ 0 are shown.