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1 Introduction

Over the past several tens of years, more and more scholars devoted many efforts to study the artificial neural networks (NNs) dynamical behaviors since they have appli- cation value in many different fields, such as image processing, associative memories, and classification of patterns [9, 16, 18, 22, 23, 27, 42]. NNs fall into several categories including Hopfield NNs, cellular NNs, BAM NNs (BAMNNs), and Cohen–Grossberg NNs. Among them, Kosko firstly introduce the BAMNNs in 1988 [18]. BAMNNs are constructed of neurons ordered in double layers. Generally speaking, the neurons in one layer are completely incorporated to neurons in another layer. BAMNNs have been applied successfully to pattern recognition due to its generalization of the single-layer auto associative Hebbian correlator to a two-layer pattern-matched hetero associative circuit.

BAMNNs are deemed as one of the most important NNs. Compared with the general NNs, BAMNNs are consisted of neurons distributed in two layers. The neurons distributed in one layer are fully interconnected with the neurons distributed in the other layer, while there is no interconnection between the neurons distributed in the same layer. In implementation of NNs, time delays appeared because of the finite switching speed of amplifiers and neurons, which leads to network instability or oscillatory behavior. For these reasons, more and more researches pay more attention to the stability for NNs with time delays [3–6, 15, 38, 41].

Synchronization can be divided into few types including projective synchronization (PS), lag synchronization, phase synchronization, impulsive synchronization, and gener- alized synchronization [1, 8, 10, 13, 17, 20, 21, 30, 36, 37]. Lag synchronization indicates that the two systems exist a coincidence of shifted-in-time states like .(.) .(t σ), . (where σ > 0 is propagation delay). Generalized synchronization means that the drive and response systems have some functional relation like .(.) .(.(.)) (where .). Complete synchronization means the state variables were equal, while evolving in time like .(.) .(.) (where .). Impulsive synchronization means that the system behaviors abrupt changes at certain moments. PS means that the two systems can be synchronized by a scaling factor P. like .(.) P.v(.) (where .). Different from other forms of synchronization, PS is more suitable to clearly represent the nonlinear systems fragile nature, and there is a typical advantage in PS due to unpredictability of proportional constant will additionally enhance the communication security [1]. There- fore, PS of chaotic nonlinear systems received hot research attention [1, 8, 10, 36, 37].

In [8], the writers introduced the analysis for PS of fractional-order memristor-based NNs based on the fractional-order differential inequality and Caputo’s fractional deriva- tion. In [10], based on the matrix measure and Halanay inequality, the authors achieved the weak PS of coupled NNs. In [36], the authors studied the PS of fractional-order NNs via adaptive control. In [37], by introducing the sliding mode control, the authors studied the PS os NNs. In [1], authors derived sufficient conditions achieving the GDPS of BAMNNs by applying Lyapunov functional approach and differential inclusion theory. However, there are potential space to discuss the PS of BAMNNs with time-varying delay via adaptive controller.

Based on the above discussion, we study the PS of BAMNNs with time-delays. This paper has the following contributions. First, the definition of PS for BAMNNs is intro- duced. What is more, based on the adaptive controller and Lyapunov theory [11, 39, 40], some novel and useful conditions are given to ensure the PS of considered NNs. In par- ticular, the synchronization between drive-response systems develops into the complete synchronization as P. = 1. The synchronization between drive-response systems devel- ops into the anti-synchronization as P. = 1. The synchronization problem develops into the chaos problem as P. = 0. Finally, an example is given to prove the adaptability of the theoretical results. Complete synchronization and anti-synchronization are the special case of PS.

The structure of this paper is as follows. Definition, assumptions, lemma, and the sys- tem description are given in Section 2. By designing a novel type of adaptive controller, we derive some sufficient criteria of PS in Section 3. In Section 4, an example is given to prove the adaptability of results. Lastly, the conclusion about this paper is given in Section 5.

2 Model description and preliminaries

In this paper, we consider the following BAMNNs system:

where xi(t) and yj(t) indicate the state variable of the neurons at time t, respectively; fj(.) and gi(.) are activation functions; τ (.) and σ(.) denote the time-varying delays, which satisfy 0 ≤ τ (.) ≤ τ and 0 ≤ σ(.) ≤ σ; ci, dj denote the self connection of the ith, the jth neurons; aji, bji, pij, and hij are connection weights; i= 1, 2, . . . , m and j= 1, 2, . . . , n} ; m ≥ 2 and n ≥ 2 are the number of units in NNs; Ii and Jj are the input of the ith and jth neurons.

Assumption 1

τ (t) and σ(t) are differential functions with 0 ≤ τ˙(t) ≤ ε < 1, 0 ≤ σ˙ (t) ≤ µ < 1.

Remark 1

From Assumption 1 we obtained the following inequality:

Assumption 2

Solutions xi(t, x0), yj(t, y0), i ∈ I, j ∈ J , t ≥ 0, of system (1) are

bounded with x0, y0 ∈ R being the initial values. That is, there exist positive constants Mf and Mg such that

Assumption 3

For all j ∈ J , u ∈ R, there exist positive constants Lf and Lg such that

Remark 2

If the activation functions satisfy Assumption 3 [2], then from Lagrange mean value theorem, it is not difficult to check that fj(u) and gj(u) satisfy the globally Lipschitz condition. That is,

Where

Consider system (1) as the drive system, then the response system is written by

where qi(t) and rj(t) are the controllers to be designed later.

Let ei(t) = ui(t) Pixi(t) and zj(t) = vj(t) P˜j yj(t), then the error system can be represented as

where ϕj(zj(·)) = fj(vj(·)) − P˜j fj(yj(·)), ψi(ei(·)) = gi(ui(·)) − Pigi(xi(·)), Pi, and P j are scaling constants

Definition 1

(See [31].) The drive-response system (1) and (2) are PS if there exist bounded continuously differentiable scaling constants Pi and P˜j satisfying

where ǁ·ǁ stands for the Euclidean vector norm

Lemma 1

(See [29].) If function F(u) : [0;1) ! R is uniformly continuous and limu!1F(v) dv exists and is bounded, then F(u) ! 0 as u ! +1.

3 Main results

In this section, we will get some effective conditions to achieve the PS between the drive- response systems. Now introduce the controllers qi(t), rj(t) as follows:

where

ki, k˜j , di, and d˜j are arbitrary positive constants, which to be given later. According to controller (3), we can derive the following theorem.

Theorem 1.

Suppose that Assumptions 1, 2, and 3 hold. The drive-response systems (1) and (2) are PS if the response system (2) is controlled under the adaptive controller (3).

Proof. First, giving the definitions of ψi(ei(t)) and ϕj(zj(t)), one has

Based on the Lagrange mean value theorem, one has

where

Using Assumption 2, we get

which leads to

Where

Here

where αi, βi, ξj, and γj are positive constants.

Calculating the V (t) derivation, one has

where the inequality 2xy ≤ x2 + y2 for all x, y E R is used. Take the αi, βi, ξj, and γj large enough such that

where ϵi > 0, νj > 0 are arbitrarily chosen constants.

Let ϵ = mini∈I{ϵi} > 0, ν = minj∈J {νj} > 0, then we get

Therefore,

Hence, V (t) ≤ V (0) for all t ∈ [0, +∞), which drives that e(t), z(t), e˙(t), and z˙(t) are bounded for all t ∈ [0, +∞). Consequently, the derivative of e(t)Te(t) and z(t)Tz(t)

are bounded. By integrating of (4) we get

therefore,

Based on the Lemma 1, we get

Based on the Definition 1, the drive-response systems can achieve PS under the adap- tive control law (3). The proof is thus completed.

Remark 3.

In [8, 10, 36, 37], the authors considered the PS of NNs based on some types of stability techniques, for example, Halanay inequality, Lyapunov–Krasovskii, and linear inequality. Very recently, the GDPS (general decay projective synchronization) of NNs was investigated in [1]. However, the Lagrange mean value theorem is introduced in this work. From this point, the results in this paper are quite distinct from the previous works.

Corollary 1.

Let τ = τ (t), σ = σ(t), then system (1) and (2) become as

and

If Assumptions 1, 2, and 3 hold, the above drive-response NNs are PS under the above adaptive controller.

Assumption 4.

If gj(v) and hj(v) are bounded, then there exist positive constants Gj and Wj such that

Corollary 2.

If Assumption 4 holds, then the NNs (1) and (2) are PS according to the adaptive controller (3).

Remark 4.

It is worth to point out that in most of the existing works [14, 31, 32], the authors achieved the PS by introducing very specific but complex controllers, which are sometimes too difficult to implement physically. For instance, in order to offset the unmatched terms caused by scaling factor Pi when computing the derivative of error system, most of the above-mentioned works constructed very complex controllers, which were consisted by linear terms ei(t), zj(t) , ei(t σ(t)), and zj(t τ (t)) relevant to the activation functions fj(ej(t)), gi(zi(t)), fj(ej(t σ(t))), and gi(zi(t τ (t))). However, in some special cases, for example, when the solutions of drive a system are bounded, we can optimize the controller by removing the terms relevant to the terms fj(ej(t)), gi(zi(t)), fj(ej(t σ(t))), and gi(zi(t τ (t))). Thus, it is interesting to develop a simple and easy implementing controller for realizing PS between derive-response chaotic systems.

4 Numerical simulations

In this section, a numerical example is shown to describe the adaptability of the derived results in the paper.

Example. When n = 2, consider the following BAMNNs system:

The chaotic attractor of the drive system (5) with the initial values x1(s) = 0.5,

x2(s) = 0.1, y1(s) = 0.5, and y2(s) = 0.1 (s [ 1, 0]) are shown in Fig. 1.

The response system is given as

where ci, di, aji, bji, pij, hij, fj, gj, σ(t), τ (t), Ii, and Jj have the same value as in those system (5). The controllers qi(t) and rj(t) are written as follows:

qi(t) = αi(t) sign ei(t) βi(t)ei(t),

rj(t) = −γj(t) sign zj(t) − ξj(t)zj(t),

where ei(t) = ui(t) − Pixi(t), zj(t) = vj(t) − P˜j yj(t).

It can be easily checked that Assumptions 1 and 3 hold with Lf = Lf = 1, Lg = L2 = 1, ε = µ = 1. Moreover, from system (5) in Fig. 1 we can easy find that the solu- tions of system (5) are bounded and Assumption 3 holds. Consequently, based on Theo- rem 1, systems (5) and (6) are PS. When Pi = P˜j = 1, the estimation of synchronization errors are given in Fig. 2, and the state trajectories of drive-response systems are given in Fig. 3. In Fig. 4, the time evolution of the controllers gains βi, αi, γi, ξi is given. Figures 5 and 6 show the time estimations of synchronization curves and errors for p˜j = −1.

The time evolution of the controllers gains as Pi = P˜j = −1 are shown in Fig. 7.

5 Conclusion

In this paper, the PS problem of BAMNNs with time-varying delay is studied by adapting a novel adaptive controller. Some sufficient conditions are given by using inequality technique and Lyapunov theory. Finally, an example is given to prove the effectiveness of the obtained results. The results given in this paper can be seen as the extension and improvement of some existing works on the PS of BAMNNs with or without time-varying delays.

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