Stability and Hopf bifurcation analysis on a simplified BAM neural network with delays

Physica D 200 (2005) 185–204

Stability and Hopf bifurcation analysis on a simplified BAM neural network with delays Yongli Songa,∗ , Maoan Hana , Junjie Weib a

b

Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, PR China Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, PR China Received 5 April 2004; received in revised form 17 October 2004; accepted 22 October 2004 Communicated by M. Ding

Abstract A delay-differential equation modelling a bidirectional associative memory (BAM) neural network with three neurons is investigated. Its dynamics are studied in terms of local analysis and Hopf bifurcation analysis. By analyzing the associated characteristic equation, its linear stability is investigated and Hopf bifurcations are demonstrated. The stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold reduction. Numerical simulation results are given to support the theoretical predictions. © 2004 Elsevier B.V. All rights reserved. Keywords: Neural network; Time delay; Hopf bifurcations; Periodic solutions; Stability

1. Introduction Recently, there has been increasing interest in investigating the dynamics of neural networks since Hopfied [15] constructed a simplified neural network model. Based on the Hopfied neural network model, Marcus and Westervelt [21] argued that time delays always occur in the signal transmission and proposed a neural network model with delay. Afterward, a variety of artificial models has been established to describe neural networks with delays (see, for example, [1,2,11,12,17–20,38,39]). In [12,18,19], a class of two-layer heteroassociative networks, called bidirectional associative memory (BAM) neural networks with or without axonal signal transmission delays, has been proposed and applied in many fields such as pattern recognition and automatic control. The bidirectional associated memory neural networks with or without delays has been widely studied (see [6,7,12,20,22,31], and the ∗

Corresponding author. Tel.: +86 21 54747534. E-mail address: [email protected] (Y. Song).

0167-2780/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physd.2004.10.010

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Y. Song et al. / Physica D 200 (2005) 185–204

references therein). However, most work focused on establishing the local and global stability. It is well known that studies on neural dynamical systems not only involve a discussion of stability properties, but also involve many dynamic behaviors such as periodic phenomenon, bifurcation and chaos. In many applications, the properties of periodic solutions are of great interest. In delay differential equations, periodic solutions can arise through the Hopf bifurcation. Thus, there are extensive literatures on bifurcation analysis of some special neural networks (see [3,5,8– 11,13,16,24,25,32,35–37]). Among them, either only a single delay or only networks of two neurons is considered. In this paper, we study the bifurcation properties of a simplified bidirectional associative memory (BAM) neural networks with three neurons and multiple delays. The delayed bidirectional associative memory neural network is described by the following system
 x˙ i (t) = −µi xi (t) + m j=1 cji fi (yj (t − τji )) + Ii , (1) n y˙ j (t) = −νj yj (t) + i=1 dij gj (xi (t − νij )) + Ij , where cji , cij (i = 1, 2, . . . , n, j = 1, 2, . . . , m) are the connection weights through the neurons in two layers: the I-layer and the J-layer; µi and µj describe the stability of internal neuron processes on the I-layer and the J-layer, respectively. On the I-layer, the neurons whose states are denoted by xi (t) receive the inputs Ii and the inputs outputted by those neurons in the J-layer via activation functions fi , while on the J-layer, the neurons whose associated states denoted by yj (t) receive the inputs Ij and the inputs outputted by those neurons in the I-layer via activation functions gj . Although system (1) can be mathematically regarded as a Hopfied-type neural network with dimension n + m, it is really produces many nice properties due to the special structure of connection weights and has practical applications in storing paired patterns or memories. For further details, we refer to [12,18,19,22]. For the sake of implicity, we assume that the time delay from the I-layer to another J-layer is τ1 while the time delay from the J-layer back to the I-layer is τ2 , and there are only one neuron in the I-layer and two neurons in the J-layer. The architecture of this special case of system (1) is illustrated in Fig. 1. This simplified BAM neural network model can be described by the following system    x˙ 1 (t) = −µ1 x1 (t) + c21 f1 (y1 (t − τ2 )) + c31 f1 (y2 (t − τ2 )), y˙ 1 (t) = −µ2 y1 (t) + c12 f2 (x1 (t − τ1 )), (2)   y˙ 2 (t) = −µ3 y2 (t) + c13 f3 (x1 (t − τ1 )). where µi > 0(i = 1, 2, 3), cj1 (j = 2, 3) and c1i (i = 2, 3) are real constants. It is necessary to point out that Wang and Zou [31] studied the stability and Hopf bifurcations of system (1) with n = m, µi = νj = 1. Unfortunately, the method in [31] cannot be applied to system (2). Our aim in this paper is to study the stability of the zero solution of system (2) and Hopf bifurcations. Taking the sum of the delays, τ = τ1 + τ2 , as a parameter, we shall show that when the delay τ passes through a critical value, the zero solution loses its stability and a Hopf bifurcation occurs. The remainder of this paper is organized as follows. In the next section, we shall consider the stability and the local Hopf bifurcation. In Section 3, based on the normal form method and the center manifold reduction introduced by Hassard et al. [14], we derive the formulae determining the direction, stability and the period of the bifurcating

Fig. 1. Architecture of the model.

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periodic solution at the critical value of τ. To verify the theoretic analysis, numerical simulations are given in this section. Finally, a conclusion is drawn in Section 4.

2. Stability and local Hopf bifurcations Letting u1 (t) = x1 (t − τ1 ), u2 (t) = y1 (t), u3 (t) = y2 (t) and τ = τ1 + τ2 , we can rewrite (2) as the following equivalent system    u˙ 1 (t) = −µ1 u1 (t) + c21 f1 (u2 (t − τ)) + c31 f1 (u3 (t − τ)), u˙ 2 (t) = −µ2 u2 (t) + c12 f2 (u1 (t)), (3)   u˙ 3 (t) = −µ3 u3 (t) + c13 f3 (u1 (t)). We make the following assumption on functions fi : (H1 )fi ∈ C1 , fi (0) = 0, for i = 1, 2, 3. Under the hypothesis (H1 ), the linearization of system (3) at (0, 0, 0) is    u˙ 1 (t) = −µ1 u1 (t) + α21 (u2 (t − τ)) + α31 (u3 (t − τ)), u˙ 2 (t) = −µ2 u2 (t) + α12 (u1 (t)),   u˙ 3 (t) = −µ3 u3 (t) + α13 (u1 (t)), where αij = cij fj (0). The associated characteristic equation of (4) is   λ + µ1 −α21 e−λτ −α31 e−λτ   det  −α12 λ + µ2 0  = 0. −α13 0 λ + µ3

(4)

(5)

That is λ3 + (µ1 + µ2 + µ3 )λ2 + (µ1 µ2 + µ1 µ3 + µ2 µ3 )λ + µ1 µ2 µ3 − [(α21 α12 + α31 α13 )λ + (α31 α13 µ2 + α21 α12 µ3 )]e−λτ = 0.

(6)

The equilibrium (0, 0, 0) is stable if all roots of (6) have negative real parts. Thus, we need to investigate the distribution of roots of Eq. (6). For the simplicity of notations, we first investigate the distribution of roots of the following third degree exponential polynomial equation λ3 + a2 λ2 + a1 λ + a0 + (b1 λ + b0 )e−λτ = 0,  where ai , bj ∈ R(i = 0, 1, 2; j = 0, 1) and 1j=0 bj2 = 0. Obviously, iω(ω > 0) is a root of Eq. (7) if and only if ω satisfies

(7)

−ω3 i − a2 ω2 + a1 ωi + a0 + (b1 ωi + b0 )(cos ωτ − i sin ωτ) = 0. Separating the real and imaginary parts, we have
a2 ω2 − a0 = b0 cos ωτ + b1 ω sin ωτ, −ω3 + a1 ω = b0 sin ωτ − b1 ω cos ωτ,

(8)

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which implies ω6 + (a22 − 2a1 )ω4 + [a12 − 2a0 a2 − b12 ]ω2 + a02 − b02 = 0.

(9)

Let z = ω2 and denote p = a22 − 2a1 ,

q = a12 − 2a0 a2 − b12 ,

r = a02 − b02 .

(10)

Then, Eq. (9) becomes z3 + pz2 + qz + r = 0.

(11)

Denote h(z) = z3 + pz2 + qz + r.

(12)

Since limz→+∞ h(z) = +∞, we conclude that if r < 0, then Eq. (12) has at least one positive root. By (12), we have dh(z) = 3z2 + 2pz + q. dz Clearly, if ∆ = p2 − 3q ≤ 0, then the function h(z) is monotonically increasing in z ∈ [0, ∞). Thus, for r ≥ 0 and ∆ ≤ 0, Eq. (12) has no positive roots for z ∈ [0, ∞). On the other hand, when r ≥ 0 and ∆ > 0, the following equation 3z2 + 2pz + q = 0. has two real roots z∗1

−p + = 3

√

∆

and

(13)

z∗2

−p − = 3

√

∆

.

(14)

√ √ Obviously, h (z∗1 ) = 2 ∆ > 0 and h (z∗2 ) = −2 ∆ < 0. It follows that z∗1 and z∗2 are the local minimum and the local maximum of h(z), respectively. Hence, we have the following. Lemma 2.1. Suppose that r ≥ 0 and ∆ > 0. Then, Eq. (11) has positive roots if and only if z∗1 > 0, h(z∗1 ) ≤ 0. Proof. Noticing that r ≥ 0, z∗1 is the local minimum of h(z) and limz→+∞ h(z) = +∞, we immediately know that the sufficiency is true. In what follows, we need to prove the necessity. Suppose the contrary that either z∗1 ≤ 0 or z∗1 > 0 and h(z∗1 ) > 0. Since h(z) is increasing for z ≥ z∗1 and h(0) = r ≥ 0, we know that h(z) has no positive real roots for z∗1 ≤ 0. If z∗1 > 0 and h(z∗1 ) > 0, since z∗2 is the local maximum value, it follows that h(z∗1 ) < h(z∗2 ). It follows from h(0) = r ≥ 0 that when z∗1 > 0 and h(z∗1 ) > 0, h(z) has no positive real roots, too. This completes the proof.
Summarizing the above discussions, we obtain the following: Lemma 2.2. For the polynomial Eq. (11), we have the following results. (i) If r < 0, then Eq. (11) has at least one positive root. (ii) If r ≥ 0 and ∆ = p2 − 3q ≤ 0, then Eq. (11) has no positive roots. √ (iii) If r ≥ 0 and ∆ = p2 − 3q > 0, then Eq. (11) has positive roots if and only if z∗1 = 13 (−p + ∆) > 0 and h(z∗1 ) ≤ 0.

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Suppose that Eq. (11) has positive roots. Without loss of generality, we assume that it has three positive roots, defined by z1 , z2 and z3 , respectively. Then, Eq. (9) has three positive roots ω1 =

√ z1 ,

ω2 =

√ z2

and ω3 =

√ z3 .

By (8), we have cos ωk τ =

b1 ωk4 + (a2 b0 − a1 b1 )ωk2 − a0 b0 b02 + b12 ωk2

.

Thus, if we denote


  b1 ωk4 + (a2 b0 − a1 b1 )ωk2 − a0 b0 1 (j) arccos + 2jπ , τk = ωk b02 + b12 ωk2

(15) (j)

where k = 1, 2, 3; j = 0, 1, . . ., then ±iωk is a pair of purely imaginary roots of Eq. (7) with τk . Define (0)

τ0 = τk0 =

(0)

min {τk }, ω0 = ωk0 .

(16)

k∈{1,2,3}

Note that when τ = 0, Eq. (7) becomes λ3 + a2 λ2 + (a1 + b1 )λ + a0 + b0 = 0.

(17)

Till now, we can employ a result from Ruan and Wei [26] to analyze (7), which is, for the convenience of the reader, stated as follows. Lemma 2.3. Consider the exponential polynomial n−1 −λτ1 + · · · + pn−1 λ + p(1) P(λ, e−λτ1 , . . . , e−λτm ) = λn + p1 λn−1 + · · · + pn−1 λ + p(0) n + [p1 λ n ]e (0)

(0)

(1)

(1)

−λτm + · · · + [p1 λn−1 + · · · + pn−1 λ + p(m) , n ]e (m)

(m)

(i)

where τi ≥ 0 (i = 1, 2, . . . , m) and pj (i = 0, 1, . . . , m; j = 1, 2, . . . , n) are constants. As (τ1 , τ2 , . . . , τm ) vary, the sum of the order of the zeros of P(λ, e−λτ1 , . . . , e−λτm ) on the open right half plane can change only if a zero appears on or crosses the imaginary axis. Using Lemmas 2.2 and 2.3, we can easily obtain the following results on the distribution of roots of the transcendental Eq. (7). Lemma 2.4. For the third degree exponential polynomial Eq. (7), we have (i) if r ≥ 0 and ∆ = p2 − 3q ≤ 0, then all roots with positive real parts of Eq. (7) has the same sum as those of the polynomial Eq. (17) for all τ ≥ 0; (ii) if either r < 0 or r ≥ 0, ∆ = p2 − 3q > 0, z∗1 > 0 and h(z∗1 ) ≤ 0, then all roots with positive real parts of Eq. (7) has the same sum as those of the polynomial Eq. (17) for τ ∈ [0, τ0 ). Remark 2.5. Lemma 2.4 is a generalization of the result on the zeros of a third degree exponential polynomial in Ruan and Wei [25], where they consider only Eq. (7) with b1 = 0. Let λ(τ) = α(τ) + iω(τ)

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be the root of Eq. (7) near τ = τk satisfying (j)

α(τk ) = 0,

(j)

ω(τk ) = ωk .

Then, the following transversality condition holds. Lemma 2.6. Suppose that zk = ωk2 and h (zk ) = 0, where h(z) is defined by (12). Then, (j)

Re{λ(τk )} = 0, dτ and the sign of Re{λ(τk )}/dτ is consistent with that of h (zk ). (j)

Proof. Substituting λ(τ) into Eq. (7) and differentiating the resulting equation in τ, we obtain {3λ2 + 2a2 λ + a1 + [b1 − τ(b1 λ + b0 )]e−λτ }

dλ = λ(b1 λ + b0 )e−λτ . dτ

Thus, 

dλ dτ

−1

=

(3λ2 + 2a2 λ + a1 )eλτ b1 τ + − . λ(b1 λ + b0 ) λ(b1 λ + b0 ) λ

(18)

It follows from (8) that [λ(b1 λ + b0 )]τ=τ (j) = −b1 ωk2 + ib0 ωk ,

(19)

k

and [(3λ2 + 2a2 λ + a1 )eλτ ]τ=τ (j) = [(a1 − 3ωk2 ) cos ωk τk − 2a2 ωk sin ωk τk ] (j)

(j)

k

(j)

(j)

+ [2a2 ωk cos ωk τk + (a1 − 3ωk2 ) sin ωk τk ]i.

(20)

From (18)–(20) and (10), we can easily obtain  2      (3λ + 2a2 λ + a1 )eλτ 2b2 λ + b1 d(Reλ(τ)) −1 = Re + Re (j) (j) dτ λ(b1 λ + b0 ) λ(b1 λ + b0 ) τ=τ (j) τ=τ τ=τ k

k

=

k

1 (j) (j) {−b1 ωk2 [(a1 − 3ωk2 ) cos ωk τk − 2a2 ωk sin ωk τk ] Λ (j)

(j)

+ ωk b0 [2a2 ωk cos ωk τk + (a1 − 3ωk2 ) sin ωk τk ] − b12 ωk2 } =

1 (j) (j) {(a1 − 3ωk2 )ωk [b0 sin ωk τk − b1 ωk cos ωk τk ] Λ (j)

(j)

+ 2a2 ωk2 [b0 cos ωk τk + b1 ωk sin ωk τk ] − b12 ωk2 } 1 1 {3ωk6 + 2(a22 − 2a1 )ωk4 + [a12 − 2a0 a2 − b12 ]ωk2 } = (3ωk6 + 2pωk4 + qωk2 ) Λ Λ z 1 k = {zk (3z2k + 2pzk + q)} = h (zk ), Λ Λ

=

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where Λ = (b1 ωk2 )2 + (b0 ωk )2 > 0. Thus, we have  sign

d(Reλ(τ)) dτ



 (j)

τ=τk

= sign

d(Reλ(τ)) dτ

−1 (j)

τ=τk

= sign

z

 h (zk ) =  0.

k 

Λ

Notice that Λ, zk > 0. Therefore, we conclude that the sign of (d(Reλ(τ))/dτ)τ=τ (j) is determined by that of h (zk ). k

This proves the lemma.




Now we can apply the results obtained above to study (6), and then obtain the stability and Hopf bifurcation of system (3). Comparing (6) with (7), we get the following: a2 = µ1 + µ2 + µ3 > 0, b1 = −α21 α12 − α31 α13 ,

a1 = µ1 µ2 + µ1 µ3 + µ2 µ3 > 0,

a0 = µ1 µ2 µ3 > 0,

b0 = −α31 α13 µ2 − α21 α12 µ3 .

(21)

This, together with (10), means that p = a22 − 2a1 = µ21 + µ22 + µ23 > 0, q = a12 − 2a0 a2 − b12 = µ21 µ22 + µ21 µ23 + µ22 µ23 − (α21 α12 + α31 α13 )2 , r=

a02

− b02

=

µ21 µ22 µ23

− (α31 α13 µ2 + α21 α12 µ3

(22)

)2 .

Further, we can compute h(z) = z3 + pz2 + qz + r,

∆ = p2 − 3q,

z∗1 =

−p + 3

√

∆

.

In addition, Routh-Hurwitz criterion implies that if (H2 )

a0 + b0 > 0,

a2 (a1 + b1 ) > a0 + b0 ,

then all roots of Eq. (6) with τ = 0 have negative real parts. Thus, from Lemmas 2.4 and 2.6, we have the following: Theorem 2.7. Let ai , bj (i = 0, 1, 2; j = 0, 1); p, q, r and τj are defined by (21), (22), (15), respectively. Suppose that (H1 ) and (H2 ) hold. Then, the following results hold. (i) When r ≥ 0 and ∆ = p2 − 3q ≤ 0, all roots of Eq. (6) have negative real parts for all τ ≥ 0, and the zero solution of system (2) is asymptotically stable for all τ ≥ 0. (ii) If either r < 0 or r ≥ 0, ∆ = p2 − 3q > 0, z∗1 > 0 and h(z∗1 ) ≤ 0 hold, then h(z) has at least positive root zk , (0) and all roots of Eq. (6) have negative real parts for τ ∈ [0, τk ), and system (2) is asymptotically stable for (0) τ ∈ [0, τk ). (iii) If all the conditions as stated in (ii) and h (zk ) = 0 hold, then system (2) undergoes a Hopf bifurcation at the (j) zero solution when τ = τk (j = 0, 1, 2, . . .). 3. Direction and stability of the Hopf bifurcation In the previous section, we obtained some conditions under which system (2) or system (3) undergoes Hopf bifurcation at τ = τ (j) (j = 0, 1, 2, . . .). In this section, we assume that system (3) undergoes a Hopf bifurcation

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at the zero equilibrium when τ = τ (j) (j = 0, 1, 2, . . .), i.e., a family of periodic solutions bifurcate from the zero equilibrium. In the sequel, using the normal form theory and center manifold reduction due to Hassard et al. [14], we are able to determine the Hopf bifurcation direction, i.e., make clear whether the bifurcating branch of periodic solution exists locally for τ > τ (j) or τ < τ (j) , and determine the properties of these bifurcating periodic solutions, for example, stability on the center manifold and period. In what follows, it is necessary to assume that fi ∈ C3 , i = 1, 2, 3. For convenience, let xi (t) = ui (τt) and τ = τ (j) + µ, where τ (j) is defined by (15) and µ ∈ R, then the system (3) can be written as an FDE in C = C([−1, 0], R3 ) as x˙ (t) = Lµ (xt ) + F (µ, xt ),

(23)

where xt (θ) = x(t + θ) ∈ C, and Lµ : C −→ R, F : R × C −→ R are given, respectively, by       −µ1 0 0 φ1 (0) 0 α21 α31 φ1 (−1)       Lµ (φ) = (τ (j) + µ)  α12 −µ2 0   φ2 (0)  + (τ (j) + µ)  0 0 0   φ2 (−1)  , α13 φ3 (0) 0 −µ3 0 0 0 φ3 (−1) and



l1 φ22 (−1) + l2 φ23 (−1) + l3 φ32 (−1) + l4 φ33 (−1) + h.o.t.

 F (µ, φ) = (τ (j) + µ)  m1 φ12 (0) + m2 φ13 (0) + h.o.t.

(24)

  ,

(25)

n1 φ12 (0) + n2 φ13 (0) + h.o.t. where φ(θ) = (φ1 (θ), φ2 (θ), φ3 (θ))T ∈ C, l1 = c21 f1 (0)/2!, l2 = c21 f1 (0)/3!, l3 = c31 f1 (0)/2!, l4 = c31 f1 (0)/ 3!, m1 = c12 f2 (0)/2!, m2 = c12 f2 (0)/3!, n1 = c13 f3 (0)/2!, n2 = c13 f3 (0)/3!. From the discussions in Section 2, we know that if µ = 0, then system (23) undergoes a Hopf bifurcation at the zero equilibrium and the associated characteristic equation of system (23) has a pair of simple imaginary roots ±iτ (j) ω0 . By the Riesz representation theorem, there exists a function η(θ, µ) of bounded variation for θ ∈ [−1, 0], such that  0 dη(θ, 0)φ(θ) for φ ∈ C. (26) Lµ φ = −1

In fact, we can choose



−µ1

0

 η(θ, µ) = (τ (j) + µ)  α12 −µ2 α13 0
0, θ = 0, where δ is defined by δ(θ) = 1, θ = 0. For φ∈C1 ([−1, 0], R3 ), define

0



0

  0  δ(θ) − (τ (j) + µ)  0 0 −µ3

 dφ(θ)   θ∈[−1, 0),   dθ , A(µ)φ =  0    dη(µ, s)φ(s), θ = 0,  −1



α21 0 0

 α31  0  δ(θ + 1), 0

(27)

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and
R(µ)φ =

0, θ∈[−1, 0), F (µ, φ), θ = 0.

Then, system (23) is equivalent to x˙ t = A(µ)xt + R(µ)xt ,

(28)

where xt (θ) = x(t + θ) for θ ∈ [−1, 0]. For ψ∈C1 ([0, 1], (R3 )∗ ), define
∗

A ψ(s) =

− dψ(s) s ∈ (0, 1], ds , 0 −1 ψ(−t)dη(t, 0), s = 0,

and a bilinear inner product  ¯ ψ(s), φ(θ) = ψ(0)φ(0) −

0



θ

−1 ξ=0

¯ − θ)dη(θ)φ(ξ)dξ, ψ(ξ

(29)

where η(θ) = η(θ, 0). Then, A(0) and A∗ are adjoint operators. By Theorem 2.7, we know that ±iτ (j) ω0 are eigenvalues of A(0). Thus, they are also eigenvalues of A∗ . (j) Suppose that q(θ) = (1, α, β)T eiθω0 τ is the eigenvector of A(0) corresponding to iτ (j) ω0 . Then, A(0)q(θ) = iτ (j) ω0 q(θ). It follows from the definition of A(0) and (27) that    (j) (j)  0 iω0 + µ1 −α21 e−iω0 τ −α31 e−iω0 τ     τ (j)  −α12 q(0) = iω0 + µ2 0  0, 0 −α13 0 iω0 + µ3 which yields  q(0) = (1, α, β) = 1, T

α12 α13 , µ2 + iω0 µ3 + iω0

T .

Similarly, it can be verified that q∗ (s) = D(1, α∗ , β∗ )eisω0 τ where α21 eiω0 τ , µ2 − iω0 (j)

α∗ =

(j)

is the eigenvector of A∗ corresponding to −iω0 τ (j) ,

α31 eiω0 τ . µ3 − iω0 (j)

β∗ =



By (29), we get ∗

¯ q (s), q(θ) = D(1, α∗ , β∗ )(1, α, β)T
¯ 1 + αα∗ + ββ∗ − =D

− 

0 −1

0



θ

−1 ξ=0

(j) (j) ¯ α∗ , β∗ )e−i(ξ−θ)ω0 τ dη(θ)(1, α, β)T eiξω0 τ dξ D(1,

(1, α∗ , β∗ )θe

 iθω0 τ (j)

dη(θ)(1, α, β)

¯ + αα∗ + ββ∗ + (αα21 + βα31 )τ (j) e−iω0 τ (j) }. = D{1

T

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Y. Song et al. / Physica D 200 (2005) 185–204

Thus, we can choose D=

¯ ∗ 1 + αα

1 ∗ ¯ ¯ 31 )τ (j) eiω0 τ (j) ¯ 21 + βα + ββ + (αα

such that q∗ (s), q(θ) = 1. In the following, we follow the ideas in Hassard et al. [14] and by using the same notations as there to compute the coordinates describing the center manifold C0 at µ = 0. Let xt be the solution of Eq. (23) when µ = 0. Define z(t) = q∗ , xt ,

W(t, θ) = xt (θ) − 2Re{z(t)q(θ)}.

(30)

On the center manifold C0 , we have W(t, θ) = W(z(t), z¯ (t), θ), where W(z, z¯ , θ) = W20 (θ)

z2 z¯ 2 z3 + W11 (θ)z¯z + W02 (θ) + W30 (θ) + · · · , 2 2 6

z and z¯ are local coordinates for center manifold C0 in the direction of q∗ and q¯ ∗ . Note that W is real if xt is real. We consider only real solutions. For the solution xt ∈C0 of (26), since µ = 0, we have z˙ = iτ (j) ω0 z + q∗ (θ), F (0, W(z, z¯ , θ) + 2Re{zq(θ)}) = iτ (j) ω0 z + q¯ ∗ (0)F (0, W(z, z¯ , 0) + 2Re{zq(0)})

def

= iτ (j) ω0 z + q¯ ∗ (0)F0 (z, z¯ ).

We rewrite this equation as z˙ (t) = iτ (j) ω0 z(t) + g(z, z¯ ) with g(z, z¯ ) = q¯ ∗ (0)F0 (z, z¯ ) = g20

z2 z¯ 2 z2 z¯ + g11 z¯z + g02 + g21 + ···. 2 2 2

(31)

By (30), we have xt (θ) = (x1t (θ), x2t (θ), x3t (θ)) = W(t, θ) + zq(θ) + z¯ q¯ (θ) and q(θ) = (1, α, β)T eiθω0 τ , and then (j)

z2 z¯ 2 (1) (1) + W11 (0)z¯z + W02 (0) + O(|(z, z¯ )|3 ), 2 2 2 z z¯ 2 (j) (j) (2) (2) (2) ¯ iω0 τ + W20 (−1) + W11 (−1)z¯z + W02 (−1) + O(|(z, z¯ )|3 ), x2t (−1) = zαe−iω0 τ + z¯ αe 2 2 2 2 z z ¯ (j) (j) (3) (3) (3) ¯ iω0 τ + W (−1) + W (−1)z¯z + W (−1) + O(|(z, z¯ )|3 ). x3t (−1) = zβe−iω0 τ + z¯ βe 20 11 02 2 2 (1)

x1t (0) = z + z¯ + W20 (0)

It follows together with (25) that g(z, z¯ ) = q¯ ∗ (0)F0 (z, z¯ )



2 (−1) + l x3 (−1) + l x2 (−1) + l x3 (−1) + h.o.t. l1 x2t 2 2t 3 3t 4 3t

2 (0) + m x3 (0) + h.o.t. ¯ (j) (1, α∗ , β∗ )  = Dτ  m1 x1t 2 1t 2 (0) + n x3 (0) + h.o.t. n1 x1t 2 1t

  

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¯ (j) {(l1 α2 e−2iω0 τ (j) + l3 β2 e−2iω0 τ (j) + α∗ m1 + β∗ n1 )z2 + 2(l1 |α|2 + l3 |β|2 + m1 α∗ + n1 β∗ )z¯z = Dτ + (l1 α¯ 2 e2iω0 τ

(j)

(j) (2) (3) + l3 β¯ 2 e2iω0 τ + α∗ m1 + β∗ n1 )¯z2 + [(2l1 αW11 (−1) + 2l3 βW11 (−1) + 3l2 α2 α¯

¯ −iω0 τ + 3l4 β2 β)e

(j)

iω0 τ ¯ ¯ 20 (−1) + l3 βW + (l1 αW 20 (−1))e (2)

(3)

(j)

(1)

+ 2(m1 α∗ + n1 β∗ )W11 (0)

(1)

+ (m1 α∗ + n1 β∗ )W20 (0) + 3(m2 α∗ + n2 β∗ )]z2 z¯ + · · ·}. Comparing the coefficients with (31), we have   ¯ (j) l1 α2 e−2iω0 τ (j) + l3 β2 e−2iω0 τ (j) + α∗ m1 + β∗ n1 ; g20 = 2Dτ ¯ (j) (l1 |α|2 + l3 |β|2 + m1 α∗ + n1 β∗ ); g11 = 2Dτ ¯ (j) (l1 α¯ 2 e2iω0 τ (j) + l3 β¯ 2 e2iω0 τ (j) + α∗ m1 + β∗ n1 ); g02 = 2Dτ ¯ −iω0 τ (j) ¯ (j) [(2l1 αW (2) (−1) + 2l3 βW (3) (−1) + 3l2 α2 α¯ + 3l4 β2 β)e g21 = 2Dτ 11 11     (2) ¯ (3) (−1) eiω0 τ (j) + 2 m1 α∗ + n1 β∗ W (1) (0) ¯ 20 (−1) + l3 βW + l1 αW 20 11  (1)  ∗ ∗ ∗ ∗ + m1 α + n1 β W20 (0) + 3(m2 α + n2 β )].

(32)

In order to determine g21 , in the sequel, we need to compute W20 (θ) and W11 (θ). From (28) and (30), we have
AW − 2Re{¯q∗ (0)F0 q(θ)}, θ∈[−1, 0), def ˙ ˙ W = x˙ t − z˙ q − z¯ q¯ = = AW + H(z, z¯ , θ), (33) ∗ AW − 2Re{¯q (0)F0 q(0)} + F0 , θ = 0, where H(z, z¯ , θ) = H20 (θ)

z2 z¯ 2 + H11 (θ)z¯z + H02 (θ) + · · · . 2 2

(34)

On the other hand, note that on the center manifold C0 near to the origin, ˙ = Wz z˙ + Wz¯ z¯˙ . W This, together with (33)–(34), reads to (A − 2iτ (j) ω0 )W20 (θ) = −H20 (θ), AW11 (θ) = −H11 (θ), . . . .

(35)

By (33), we know that for θ∈[−1, 0), H(z, z¯ , θ) = −¯q∗ (0)F0 q(θ) − q∗ (0)F¯ 0 q¯ (θ) = −gq(θ) − g¯ q¯ (θ).

(36)

Comparing the coefficients with (34) gives that H20 (θ) = −g20 q(θ) − g¯ 02 q¯ (θ),

(37)

H11 (θ) = −g11 q(θ) − g¯ 11 q¯ (θ).

(38)

and

From (35), (37) and the definition of A, we can obtain ˙ 20 (θ) = 2iτ (j) ω0 W20 (θ) + g20 q(θ) + g¯ 02 q¯ (θ). W

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Noticing that q(θ) = (1, α, β)T eiθω0 τ , we have (j)

W20 (θ) =

ig20 i¯g02 (j) (j) (j) q(0)eiτ ω0 θ + (j) q¯ (0)e−iτ ω0 θ + E1 e2iτ ω0 θ , τ (j) ω0 3τ ω0

(1)

(2)

(39)

(3)

where E1 = (E1 , E1 , E1 ) ∈ R3 is a constant vector. Similarly, from (35) and (38), we can obtain W11 (θ) = −

ig11 i¯g11 (j) (j) q(0)eiτ ω0 θ + (j) q¯ (0)e−iτ ω0 θ + E2 , τ (j) ω0 τ ω0

(1)

(2)

(40)

(3)

where E2 = (E1 , E2 , E2 ) ∈ R3 is also a constant vector. In what follows, we shall seek appropriate E1 and E2 . From the definition of A and (35), we obtain  0 dη(θ)W20 (θ) = 2iτ (j) ω0 W20 (0) − H20 (0), −1

and



0

−1

dη(θ)W11 (θ) = −H11 (0),

(41)

(42)

where η(θ) = η(0, θ). From (33) and (34), we have  (j) (j)  l1 α2 e−2iω0 τ + l3 β2 e−2iω0 τ   H20 (0) = −g20 q(0) − g¯ 02 q¯ (0) + 2τ (j)  m1 ,

(43)

n1 and



 l1 |α|2 + l3 |β|2   H11 (0) = −g11 q(0) − g¯ 11 q¯ (0) + 2τ (j)  m1 .

(44)

n1 Substituting (39) and (43) into (41) and noticing that

 iτ ω0 I −



0

(j)

−1

e

iθω0 τ (j)

dη(θ) q(0) = 0

and

 −iτ ω0 I − (j)



0 −1

e

−iθω0 τ (j)

dη(θ) q¯ (0) = 0,

we obtain

 2iτ (j) ω0 I −

0 −1





(j)  e2iθω0 τ dη(θ) E1 = 2τ (j) 

l1 α2 e−2iω0 τ

(j)

(j)  + l3 β2 e−2iω0 τ  m1 , n1

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which leads to   (j) (j)  (j) (j)  l1 α2 e−2iω0 τ + l3 β2 e−2iω0 τ 2iω0 + µ1 −α21 e−2iω0 τ −α31 e−2iω0 τ     2iω0 + µ2 0 m1  E1 = 2  .  −α12 −α13 0 2iω0 + µ3 n1 It follows that

(1)

  l1 α2 e−2iω0 τ (j) + l3 β2 e−2iω0 τ (j) 2  =  m1 A  n1

(2)

  2iω0 + µ1 2  =  −α12 A  −α13

E1

E1

(3)

E1

  2iω0 + µ1 2  =  −α12 A  −α13

l1 α2 e−2iω0 τ

(j)

−α21 e−2iω0 τ 2iω0 + µ2 0

+ l3 β2 e−2iω0 τ m1

(j)

n1 −α21 e−2iω0 τ

(j)

l1 α2 e−2iω0 τ

(j)

2iω0 + µ2 0

(j)

−α31 e−2iω0 τ 0

2iω0 + µ3  (j) −α31 e−2iω0 τ   ; 0  2iω0 + µ3   (j) + l3 β2 e−2iω0 τ   ; m1   n1

where

   2iω0 + µ1 −α21 e−2iω0 τ (j) −α31 e−2iω0 τ (j)      A =  −α12 . 2iω0 + µ2 0     −α13 0 2iω0 + µ3

Similarly, substituting (40) and (44) into (42), we can get 

µ1

−α21 −α31

  −α12 µ2 −α13 0

0 µ3





 l1 |α|2 + l3 |β|2    m1  E2 = 2  , n1

and hence, (1)

E2 = (2)

E2 = (3)

E2 = where

2µ2 µ3 (l1 |α|2 + l3 |β|2 ) + 2n1 α31 µ2 + 2m1 α21 µ3 ; B 2m1 µ1 µ3 + 2n1 α12 α31 − 2m1 α13 α31 + α12 µ3 (l1 |α|2 + l3 |β|2 ) ; B 2n1 µ1 µ2 + 2m1 α21 α13 + 2α13 µ2 (l1 |α|2 + l3 |β|2 ) − n1 α12 α21 ; B

   µ1 −α21 −α31      B =  −α12 µ2 0 .    −α13 0 µ3 

(j)

    ;  

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Therefore, all gij in (35) have been expressed in terms of the parameters and the delay given in (26). Further, we can compute the following values:   i |g02 |2 g21 2 c1 (0) = (j) g11 g20 − 2|g11 | − , + 2 3 2τ ω0 κ2 = −

Re{c1 (0)} , Re{λ (τ (j) )}

β2 = 2Re{c1 (0)}, T2 = −

Im{c1 (0)} + κ2 Im{λ (τ (j) )} , τ (j) ω0

which determine the quantities of bifurcating periodic solutions on the center manifold at the critical value τ (j) , i.e., κ2 determines the directions of the Hopf bifurcation: if κ2 > 0 (κ2 < 0), then the Hopf bifurcation is supercritical(subcritical) and the bifurcating periodic solutions exist for τ > τ (j) (τ < τ (j) ); β2 determines the stability of the bifurcating periodic solutions: the bifurcating periodic solutions are stable (unstable) if β2 < 0 (β2 > 0); and T2 determines the period of the bifurcating periodic solutions: the period increase (decrease) if T2 > 0 (T2 < 0). In particular, we consider the neuron network model    x˙ 1 (t) = −νx1 (t) + c21 f (y1 (t − τ2 )) + c31 f (y2 (t − τ2 )), y˙ 1 (t) = −νy1 (t) + c12 f (x1 (t − τ1 )), (45)   y˙ 2 (t) = −νy2 (t) + c13 f (x1 (t − τ1 )), with ν > 0, ci1 (i = 2, 3) > 0 and c1j (j = 2, 3) < 0. This BAM neuron network model means that the output from the neurons y1 and y2 in the J-layer excites the neuron x1 in 1-layer, while the output from the neurons x1 in the I-layer inhibits the neurons y1 and y2 in 2-layer. We make a further assumption on the function f as follows: (H3 ) f ∈ C3 , uf (u) > 0 for u = 0, f  (0) = 0, f  (0) = 0 and f  (0) = 0. A specific function which has been widely used in the neuron network models as a transfer function given by f (u) = tanh(u) does satisfy the above conditions. For system (45), we have α12 = c12 f  (0),

α13 = c13 f  (0), α21 = c21 f  (0),

α31 = c31 f  (0).

(46)

It follows from (21) and (22) that a2 = 3ν > 0,

a1 = 3ν2 > 0,

b1 = −(c12 c21 + c13 c31 )f 2 (0) > 0,

a0 = ν3 > 0, b0 = −ν(c12 c21 + c13 c31 )f 2 (0) > 0.

(47)

It follows that the condition (H2 ) holds, and p = 3ν2 > 0,

q = 3ν4 − (c12 c21 + c13 c31 )2 f 4 (0),

Thus, if ν4 < (c12 c21 + c13 c31 )2 f 4 (0) < 3ν4 , then we have q > 0,

r < 0.

r = (a0 + b0 )(a0 − b0 ).

(48)

Y. Song et al. / Physica D 200 (2005) 185–204

So, h(z) = z3 + pz2 + qz + r has only one positive real root denoted by z0 , and h (z0 ) > 0. Let ω0 = τ1 + τ2 . It follows from (15) and (43) that


  ν2 − ω02 1 (j) τ = arccos + 2jπ . ω0 (c12 c21 + c13 c31 )f 2 (0)

199

√ z0 , τ =

(49)

By Lemmas 2.4 and 2.6, we have the following: Theorem 3.1. Assume that f (0) = 0, f  (0) = 0 and ν4 < (c12 c21 + c13 c31 )2 f 4 (0) < 3ν4 hold, and τ (j) is defined by (49). (i) When τ ∈ [0, τ (0) ), all roots of the corresponding characteristic equation of (45) have negative real parts. Meanwhile, when τ = τ (0) , the corresponding characteristic equation of (45) has only a pair of simple purely imaginary roots ±iω0 , and all other roots have negative real parts. However, when τ > τ (0) , the corresponding characteristic equation of (45) has at least one root with positive real part. (ii) The zero steady state of system (45) is asymptotically stable when τ ∈ [0, τ (0) ), and unstable when τ > τ (0) . (iii) System (45) undergoes a Hopf bifurcation at the zero steady state when τ = τ (j) , j = 0, 1, 2, . . .. Further, using the arithmetic given above, we can determine the direction of Hopf bifurcation and stability of the bifurcating periodic solutions. Theorem 3.2. Suppose that (H3 ) and ν4 < (c12 c21 + c13 c31 )2 f 4 (0) < 3ν4 hold, and τ (j) is defined by (49). Then, the direction of Hopf bifurcation and stability of the bifurcating periodic solutions at τ (j) are determined by sign f  (0)/f  (0). More specifically, if f  (0)/f  (0) < 0(> 0), then the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solutions are orbitally asymptotically stable (unstable) on the center manifold. Proof. Under the condition (H3 ), for system (45), we have c21 f  (0) c21 f  (0) c31 f  (0) c31 f  (0) = 0, l2 = , l3 = = 0, l4 = , 2! 3! 2! 3! c12 f  (0) c12 f  (0) c13 f  (0) = 0 c13 f  (0) m1 = = 0, m2 = , n1 = , n2 = , 2! 3! 2! 3!

l1 =

(50)

and α=

c12 f  (0) c13 f  (0) ,β = , ν + iω0 ν + iω0 c21 f  (0)eiω0 τ c31 f  (0)eiω0 τ , β∗ = . ν + iω0 ν + iω0 (j)

α∗ =

(j)

(51)

From (50), (51) and (32), we can easily obtain g20 = g11 = g02 = 0, and ¯ −iω0 τ ¯ (j) [(l2 α2 α¯ + l4 β2 β)e g21 = 6Dτ

(j)

+ m 2 α∗ + n 2 β ∗ ]

¯ (j) f  (0)f  (0)e−iω0 τ (j) [c12 c21 (|α|2 + 1) + c13 c31 (|β|2 + 1)] Dτ . = ν + iω0

(52)

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Note that (ν + ω0 i)2 = (c12 c21 + c13 c31 )f 2 (0)e−iω0 τ . (j)

Thus, we have ¯ = D

(ν + iω0 )2 eiω0 τ . = (c12 c21 + c13 c31 )(2 + ντ (j) + iω0 τ (j) )f 2 (0) (j)

1 1 + αα∗ + ββ∗ + (αα21 + βα31 )τ (j) e−iω0 τ

(j)

This, together with (52), implies g21 = τ (j)

ν(2 + ντ (j) ) + τ (j) ω02 + 2ω0 i f  (0) 2 2 [c c (|α| + 1) + c c (|β| + 1)] , 12 21 13 31 f  (0) (c12 c21 + c13 c31 )<

where < = (2 + ντ (j) )2 + (τ (j) ω0 )2 > 0. Thus, we obtain Re{c1 (0)} =

ν(2 + ντ (j) ) + τ (j) ω02 f  (0) 1 Re{g21 } = τ (j)  [c12 c21 (|α|2 + 1) + c13 c31 (|β|2 + 1)] . 2 2f (0) (c12 c21 + c13 c31 )<

In addition, noticing that Re{λ (0)} > 0 and ci1 (i = 2, 3) > 0,

c1j (j = 2, 3) < 0,

we can obtain κ2 = −

Re{c1 (0)} > 0(< 0) Re{λ (0)}

if

f  (0) < 0(> 0) f  (0)

and β2 = 2Re{c1 (0)} < 0(> 0)

if

f  (0) < 0(> 0). f  (0)

Thus, by the results due to Hopfied [15], the theorem follows.




Remark 3.3. Note that when τ > τ (0) , the corresponding characteristic equation of system (45) has at least one root with positive real part. Therefore, when τ = τ (j) (j ≥ 1), the nontrivial periodic solutions bifurcating from the positive equilibrium must be unstable in the phase space, even though they are stable on the center manifold. But when τ = τ (0) , the stability of bifurcating periodic solutions on the center manifold implies the same stability of them in the whole phase space.

Y. Song et al. / Physica D 200 (2005) 185–204

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Fig. 2. Behavior of the neuron in system (53) with τ1 = 1.5, τ2 = 1.3. The zero steady state is stable when τ1 + τ2 < 2.2143.

Example 3.4. As an example, we consider the following system:    x˙ 1 (t) = −2x1 (t) + 2 tanh(y1 (t − τ2 )) + tanh(y2 (t − τ2 )), y˙ 1 (t) = −2y1 (t) − 2 tanh(x1 (t − τ1 )),   y˙ 2 (t) = −2y2 (t) − tanh(x1 (t − τ1 )),

(53)

which has a unique steady state (0, 0, 0). From (46) to (48), we have p = 12,

q = 23,

r = −36.

In this case h(z) = z3 + pz2 + qz + r has one positive real root z0 = 1 and two negative real roots, and then ω0 = 1. By (49), we have   3 + 2jπ, τ = arccos − 5

Fig. 3. Phase portrait of system (53) with τ1 = 1.5, τ2 = 1.3. The zero steady state is stable when τ1 + τ2 < 2.2143.

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Fig. 4. Behavior of the neuron in system (53) with τ1 = 1.2, τ2 = 0.9. The zero steady state losses its stability and Hopf bifurcation occurs.

Fig. 5. Phase portrait of system (53) with τ1 = 1.2, τ2 = 0.9. The bifurcating periodic solution is asymptotically stable.

where τ = τ1 + τ2 . From Theorem 3.1, we know that the zero steady state of system (53) is asymptotically stable when τ1 + τ2 ∈ [0, 2.2143). This is illustrated by the numerical simulation shown in Figs. 2 and 3 in which τ1 = 1.5, τ2 = 1.3. Further, when τ1 + τ2 is increased to the critical value 2.2143, the origin losses its stability and Hopf bifurcation occurs. By Theorem 3.2, the bifurcation is supercritical and the bifurcating periodic solution is asymptotically stable (see Figs. 4 and 5).

4. Conclusions Neural networks, as large-scale nonlinear dynamics system, are complex while the dynamics of the delayed neural networks are even richer and more complicated [36]. Neural networks with delays exhibit very rich dynamics. From the point of view of nonlinear dynamics, analyzing these neural networks is useful in solving problems of both theoretical and practical importance. Recently, a variety of neural network models with delays have been studied. A few sufficient criteria have been established for stability and global attractivity of neural network models with delays (see, for example, [21,22,28–30,38,39]). It is the purpose of this article to propose a tri-neuron network

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capable of producing periodic behavior. Periodic sequences of neural impulse are of fundamental significance for the control of dynamic functions of the body such as heart beat which occurs with great regularity and breathing. It is of great interest to understand the mechanism of neural networks which cause and sustain such periodic activities. It is well known that periodic solutions can arise through the Hopf bifurcation in delay differential equations. Thus, there are lots of papers on the stability and Hopf bifurcation of neural network models with delays, but most of them deal with only the neural network models with the same time delay [3,8–11,13,16,35–37] or consider only the two-neuron network models with delays (see, for example, [24,27,32]). There are few papers on the bifurcations of the high dimensional neural network models with multiple delays (see [4,5,23,33,34]) In this paper, we have considered a tri-neuron BAM networks with delays. Although Wang and Zou [31] considered a BAM networks with 2n neurons and delays, the method in [31] cannot be applied to the present model. By using the technique in Ruan and Wei [26], we have studied the distribution of the roots of a general third degree exponential polynomial equation. Applying the obtained results to system (2), we have found that under certain conditions, when the sum of the delays, τ = τ1 + τ2 , varies, the zero solution loses its stability and a Hopf bifurcation occurs, that is a family of periodic solutions bifurcate from the zero solution when τ passes a critical value, say τ (0) . In addition, using the normal form theory and center manifold reduction, the stability and direction of the Hopf bifurcation are determined. We have shown that if (H3 ) and ν4 < (c12 c21 + c13 c31 )2 f 4 (0) < 3ν4 hold, then the direction of Hopf bifurcation and stability of the bifurcating periodic solutions of system (45) at τ (j) are determined by sign{f  (0)/f  (0)}. A numerical example illustrated the obtained results.

Acknowledgements This research was partially supported by the National Nature Science Foundation of PR China (nos. 10371072 and 10471030) and by the National Ministry of Education. We are grateful to the three anonymous references for their careful reading and constructive comments and suggestions which helped improving the presentation of the paper.

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