Stability analysis of almost periodic solutions for discontinuous bidirectional associative memory (BAM) neural networks with discrete and distributed delays

: This paper aims to discuss a class of discontinuous bidirectional associative memory (BAM) neural networks with discrete and distributed delays. By using the set-valued map, differential inclusions theory and fundamental solution matrix, the existence of almost-periodic solutions for the addressed neural network model is firstly discussed under some new conditions. Subsequently, based on the non-smooth analysis theory with Lyapunov-like strategy, the global exponential stability result of the almost-periodic solution for the proposed neural network system is also established without using any additional conditions. The results achieved in the paper extend some previous works on BAM neural networks to the discontinuous case and it is worth mentioning that it is the first time to investigate the almost-periodic dynamic behavior for the BAM neural networks like the form in this paper. Finally, in order to demonstrate the effectiveness of the theoretical schemes, simulation results of two topical numerical examples are delineated.


Introduction 1.Previous works
Bidirectional associative memory (BAM) neural networks introduced by Kosko in [1,2] have been widely investigated due to their extensive applications in signal processing, image processing, pattern recognition, and so on.During the past sever decades, many studies have investigated the dynamical behaviors of delayed BAM neural networks.See [3][4][5][6][7][8][9][10][11][12][13][14][15][16] and the related references therein.For example, on the basis of the M-matrix approach, the homeomorphism property and by constructing a novel Lyapunov functional, Guo et al. [6] considered the existence, uniqueness, and exponential stability of complex-valued memristor-based BAM neural networks with time delays; by using the Banach's fixed point theorems and differential inequality techniques, Aouiti et al. [4] established the existence and global exponential stability results of pseudo almost periodic solution for neutral delay BAM neural networks with time-varying delay in leakage terms.
From the above references, we can see that the activation functions are continuous, Lipschitz continuous or even smooth.But, discontinuous activation can always exist since the influences from the external environment.In recent years, neural networks with discontinuous right-hand sides have been widely investigated owe to their practical application in mechanics, automatic control, and other various science and engineering fields.Due to the existence of the discontinuous right-hand sides, instability of neural networks often happens.

Motivations of this paper
However, compared with the BAM neural networks and discontinuous systems, little attention has been devoted to the dynamic behavior of BAM neural networks with discontinuous activations so far, see [27,28].In addition, though the almost periodic solutions for the BAM neural networks with continuous activations with has been considered [29], to the best of our knowledge, to date, the almost-periodic dynamical behaviors for the BAM neural networks with discontinuous activations, discrete and distributed delays have not been touched yet.
In order to solve the deficiencies mentioned above partially, in this paper, we consider a class of discontinuous BAM neural networks with discrete and distributed delays.

Highlights of this paper
The main contributions of this paper are summarized as follows.
(1) Most of the existing results on the stability analysis of the BAM neural networks ignore the influence of the discontinuous activations, discrete and distributed time-delays on the system.And, a lot of previous results on the BAM neural networks assumed that the activation functions are continuous, Lipschitz continuous or even smooth.In this paper, we focus on the study of discontinuous BAM neural networks with discrete and distributed delays.Hence, a lot of previous neural network models can be included by our system, such as [4,5,9,[27][28][29].(2) We study the almost periodic dynamical behaviors of discontinuous BAM neural networks with discrete and distributed delays for the first time.By using the set-valued map and differential inclusions theory, we prove the existence result of almost-periodic solution under new certain conditions.(3) According to the definition of the globally exponentially stability, the globally exponentially stability of the proposed discontinuous BAM neural network are given without using any additional conditions.(4) Two simulation examples are shown to illustrate the correctness of the established results.
The remainder part of this paper is organized as follows.In Section 2, some basic definitions and preliminary lemmas are introduced.In Section 3, some new criteria are given to establish the existence result of almost periodic solutions.In Section 4, global exponential stability analysis of the almost periodic solution is provided.In Section 5, we provide two numerical examples to demonstrate the theoretical results.Finally, some conclusions are stated in Section 6.
2 System description and preliminaries

System description
Consider the following discontinuous BAM neural networks with discrete and distributed delays: where i = 1, 2, … , n, j = 1, 2, … , m, u i (t), v j (t) denote the potential of the cell i and j at time t; a i , b j are positive constants, denoting the rate which the cell i and j rest their potential to the resting state when isolated from the other cells and inputs; c ij (t), d ij (t), e ij (t), p ji (t), q ji (t), h ji (t) denote the first and second-order connection weights of the neural networks; f i (⋅) and g j (⋅) denote the activation functions of the ith neurons and the jth neurons, respectively; I i (t), J j (t) are the ith and the jth components of an external inputs source introduced from outside the networks to the cell i and j; τ ij (t), σ ji (t) and δ ij (t), κ ij (t) denote the discrete time-varying delay and distribute time-varying delay, respectively and τ ij (t), σ ji (t) satisfy where τ and σ are positive constants.
Remark 1.According to the ordinary differential equation theory, periodic solution should be defined up to ∞.But, given the long-term dynamical behaviors, the periodic parameters of dynamical system are often influenced by the uncertain perturbations.That means, parameters can be regarded as periodic up to a small error.In this situation, almost periodic behavior is more practical than the periodic behavior for the dynamical systems.Moreover, it must be pointed out that we will prove that the maximal existing interval of the almostperiodic solutions for the proposed discontinuous BAM neural network system (2.1) can be extended to +∞ in this paper.
Throughout the paper, we guarantee that the following conditions are always true.
(H1) f j and g i , i = 1, … , n, j = 1, … , m are piecewise continuous, i.e., f j and g i are continuous in R except a countable set of jump discontinuous points, and in every compact set of R, have only a finite number of jump discontinuous points.

Preliminaries
Let f (t) be a continuous ω-antiperiodic function defined on R. We define For any vector v = (v 1 , v 2 , …, v n ) ⊤ and matrix D = (d ij ) n×n , we define the norm as Consider the dynamic system defined by the following differential equation where t 1 ∈ R + or +∞ and the set-valued function K[f (t, x(t))] is defined as follows where co(S) denotes the convex closure of set S, B (x, δ) is the open ball with the center at x ∈ R and the radius δ ∈ R, μ(N) represents the Lebesgue measure of the set N. By Definition 2.1, u i (t)(i = 1, 2, …, n) and v j (t)(j = 1, 2, …, m) are the solutions of initial value problem (2.1) on [0, b) and b ∈ (0, +∞], if u i (t)(i = 1, 2, …, n) and v j (t)(j = 1, 2, …, m) are absolutely continuous on any compact subinterval of [0, b) and satisfy the following inclusion: Then, for i = 1, 2, … , n, j = 1, 2, … , m, the set-valued maps: have nonempty compact convex values.Thus, they are upper semi-continuous and measurable.According to the measurable selection theorem, if u i (t) and v j (t) are the solution of system (2.1), there exists a measurable function for almost all for a.e.t ∈ [ −ς, b) and Definition 2.2.[28] The solution z * (t) = (u * (t), v * (t)) ⊤ of system (2.1) is said to be globally exponentially stable, if, for any z(t) = (u(t), v(t)) ⊤ of system (2.1), there exist constants α > 0 and λ > 0 such that Suppose that x(t) : [0, +∞) → R n is absolutely continuous on any compact interval of [0, +∞).We give a chain rule for computing the time derivative of the composed function V(x(t)) : [0, +∞) → R as follows.
3 Existence of almost periodic solution Theorem 3.1.Suppose that the assumptions (H1) and (H2) hold, then for any solution z(t) = (u(t), v(t)) ⊤ of discontinuous BAM neural network system (2.1), there exist two positive constants and M such that Proof: Define the set-valued maps By (H2), one can easily see that the above two set-valued maps are upper semi-continuous with nonempty compact convex values.Besides, according to the theorems in [22,30], the local existence of a solution z(t) = (u(t), v(t)) ⊤ of (2.1) is clear.That implies, the IVP of (2.1) has at least one solution for some b ∈ [0, +∞] and the derivative of u i (t) and v j (t) are measurable selections from and Next, we show that lim That is to say that the maximal existing interval of u (t) can be extended to +∞.From Definition 2.1, there exists Then system (2.1) can be rewritten as where Solving (3.1), we can obtain From Lemma 2, (H2) and (H3), it follows that Then, we can obtain Thus, lim t→b − ||z(t)|| < +∞, which implies that b = +∞.Therefore, by (3.2), we have Furthermore, by (H1), we can see that f j has a finite number of discontinuous points on any compact interval of R. Particularly, we can choose that f j has a finite number of discontinuous points on compact interval [−M, M].Without loss of generality, let f j discontinuous at points {ρ Let us consider a series of continuous functions: and Let Note that, γ j (t) ∈ co[f j (v j (t))] for a.e.t ∈ [ −ς, +∞) and j = 1, 2, … , m, we have In a similar way, let Then, from (3.4) and (3.5), it follows that (H4).The delays τ ij (t) and σ ij (t) are continuously differentiable function and satisfy τ' where Then any solution z(t) = (u(t), v(t)) ⊤ of the discontinuous BAM neural network system (2.1) associated with an output κ(t) = (η(t), γ(t)) ⊤ is asymptotically almost periodic, i.e., for any ε > 0, there exist T > 0, l = l(ε) and ω = ω(ε) in any interval with the length of l(ε), such that Proof.From (H3), it follows that, for any ε > 0, there exists l = l(ε) such that for any α ∈ R there exists ω ∈ [ω, ω + l] satisfying the following inequalities: where Furthermore, in view of (H1) and Let and Then, by (3.6) and (3.7), we can have , and ϑ i (t) can be arbitrarily choosen in [−1, 1] if u i (t + ω) = u i (t).In particular, let ϑ i (t) be as follows Then, we have Similarly, define , and θ j (t) can be arbitrarily choosen in In particular, let θ j (t) be as follows Then, we have Let

W. Xie et al.: Discontinuous BAMNNs
Consider the following suitable Lyapunov function: (3.12) Obviously, V (t) is regular.Meanwhile, the solutions (u i (t), v j (t)) ⊤ , (u i (t + ω), v j (t + ω)) ⊤ of the discontinuous system (2.1) are all absolutely continuous.Then, V (t) is differential for a.e.t ≥ 0.Then, by applying Lemma 1, for a.e.t ≥ 0, we can get that in view of (3.6)-(3.8),we further obtain which together with (H4) and (3.8) gives Thus, by (3.12), we have which leads to Moreover, from (3.12), W (0) is a constant.Then, we can choose a sufficiently large T > 0 such that which together with (3.13) and the arbitrariness of ε gives Therefore, the proof is complete.□ 4 Global exponential stability Theorem 4.1.Suppose that the assumptions (H1)-(H4) are satisfied, then the almost periodic solution of the discontinuous BAMNN system (2.1) is globally exponentially stable.
Proof.Suppose that (u ⊤ is a solution of BAMNN system (2.1) with initial conditions ]. Consider the following Lyapunov function: Obviously, U (t) is regular.Meanwhile, the ω-periodic solution (u * (t), v * (t)) ⊤ and any solution (u(t), v(t)) ⊤ of the discontinuous system (2.1) are all absolutely continuous.Then, In particular, let θi (t) be as follows Then, we have , and θj (t) can be arbitrarily choosen in In particular, let θj (t) be as follows Then, we have Now, by applying Lemma 1, calculate the time derivative of U (t) along the solution trajectories of system (2.1) in the sense of (2.1), then we can get for a.e.t ≥ 0 that Note that, by (2.1), we have which together with (H4) yields dU(t) dt < 0, for a.e.t ≥ 0 .
Furthermore, from (4.1), it follows that where ξ m : = min 1≤i≤n+m {ξ i }.Moreover, since U (0) is a constant.Thus, by Definition 2.2, we can see that the proof is complete.□
Proof.The proof is similar to that of Theorem 4.1, we omit it here.
Then, for i, j = 1, we can have ς = max{τ, δ} = 0.2.Moreover, let It is easy to see that the activation function f i (x) and g j (x) are discontinuous, bounded, monotonically nonincreasing.This fact can be seen in Figure 5.
As a result, the coefficients of system (6.2) satisfy all the conditions in Theorem 5.1-5.3,thus we can conclude that system (6.2) possesses a unique almost periodic solution which is globally exponentially stable.This fact can be shown by the the following Figures 6-8.(a) Time-domain behavior of the state variables u 1 (t), u 2 (t), v 1 (t) and v 2 (t) for system (6.2) with random initial conditions; (b) Phase plane behavior of the state variables (u 1 (t), u 2 (t)) and (v 1 (t), v 2 (t)) for system (6.2). (t), u 2 (t) and v 1 (t) for system (6.2);(b) Three-dimensional trajectory of state variables u 1 (t), u 2 (t) and v 2 (t) for system (6.2). (t), v 2 (t) and u 1 (t) for system (6.2);(b) Three-dimensional trajectory of state variables v 1 (t), v 2 (t) and u 2 (t) for system (6.2).

W. Xie et al.: Discontinuous BAMNNs
Remark 2. From Example 6.1 and Example 6.2, one can see the activations are discontinuous, unbounded and non-monotonic, that means the activations are not continuous, Lipschitz continuous or smooth, which are different from the related references in the literature, such as [3, 4, 6-8, 10, 14-16].The results established in the present paper extend the previous work about BAM neural networks to the discontinuous cases.Remark 3. Since the existence and globally exponential stability of the almost-periodic solutions of the discontinuous BAM neural networks with discrete and distributed delays has not been studied before, it is clearly to see that all results obtained in references cited therein are invalid for Example 6.1 and 6.2.Here a novel proof is put to use to establish some criteria which guarantee the existence and globally exponential stability of the almost-periodic solutions of the discontinuous BAM neural networks with discrete and distributed delays.

Conclusion
This paper presents a class of discontinuous BAM neural networks with discrete and distributed delays.Under the framework of the Filippov solution, by applying differential inclusions theory, fundamental solution matrix of coefficients, inequality technique and the non-smooth analysis theory with Lyapunov-like strategy, the existence and global exponential stability of the almost-periodic solutions for the addressed BAM neural networks have been proved.In addition, two typical numerical examples and the corresponding simulations have been presented at the end of this paper to illustrate the effectiveness and feasibility of the proposed criterion.It should be pointed out that it is the first time to investigate the almost-periodic dynamic behavior for the discontinuous BAM neural networks with discrete and distributed delays.Consequently, some related works known can be extended.
Moreover, little attention has been devoted to the study of the almost-periodic dynamic behavior of the BAM neural networks with discontinuous activations so far.This method affords a possible method to analyse the existence and globally exponential stability of the almost-periodic solutions problem of other delayed BAM neural networks with discontinuous activations, such as CG(Cohen-Grossberg)BAM neural networks with discontinuous activations, neutral type BAM neural networks with discontinuous activations, interval general BAM neural network with discontinuous activations and so on.These issues will be the topic of our future research.