STABILITY AND HOPF BIFURCATION ON A TWO-NEURON SYSTEM WITH TIME DELAY IN THE FREQUENCY DOMAIN *

Transcription

1 International Journal of Bifurcation and Chaos, Vol. 7, No. 4 (27) c World Scientific Publishing Company STABILITY AND HOPF BIFURCATION ON A TWO-NEURON SYSTEM WITH TIME DELAY IN THE FREQUENCY DOMAIN * WENWU YU and JINDE CAO Department of Mathematics, Southeast University, Nanjing 296, P. R. China yuwenwu2@63.com jdcao@seu.edu.cn Received August 3, 25; Revised February 24, 26 In this paper, a general two-neuron model with time delay is considered, where the time delay is regarded as a parameter. It is found that Hopf bifurcation occurs when this delay passes through a sequence of critical value. By analyzing the characteristic equation and using the frequency domain approach, the existence of Hopf bifurcation is determined. The stability of bifurcating periodic solutions are determined by the harmonic balance approach, Nyquist criterion and the graphic Hopf bifurcation theorem. Numerical results are given to justify the theoretical analysis. Keywords: Time delay; Hopf bifurcation; periodic solutions; harmonic balance; Nyquist criterion; graphic Hopf bifurcation theorem.. Introduction In recent years, the dynamical characteristics (including stable, unstable, oscillatory, and chaotic behavior) of neural networks [Cao & Chen, 24; Cao et al., 25; Cao & Li, 25; Cao & Liang, 24; Cao et al., 24; Cao & Wang, 24, 25; Liao et al., 2a, 2b; Ruan & Wei, 2, 23; Yu & Cao, 25] have attracted the attention of many researchers, and much efforts have been expended. It is well known that neural networks are complex and large-scale nonlinear systems, neural networks under study today have been dramatically simplified [Guo et al., 24; Liao et al., 2a, 2b; Ruan & Wei, 2, 23; Song et al., 25; Song & Wei, 25; Yu & Cao, 25; Yu & Cao, in press]. These investigations of simplified models are still very useful, since the dynamical characteristics found in simple models can be carried over to large-scale networks in some way. So in order to know better the large-scale networks, we should study the simplified networks first. In 946, Tsypkin published his classical paper [Tsypkin, 946] on feedback systems with delay. It is a major extension of the Nyquist criterion in which the problem of delay was solved in a single stroke simply and elegantly. An English translation of this paper was published in a volume in commemoration of Harry Nyquist edited by MacFarlane [Tsypkin, 946]. It is fitting that this paper appeared immediately following that of Nyquist s original paper. The method of Tsypkin is of major significance considering that the analytical formulation of the problem of stability with delay is very complicated. This work was supported by the National Natural Science Foundation of China under Grants and , and the Natural Science Foundation of Jiangsu Province, China under Grants BK2353 and BK

2 356 W. Yu & J. Cao There has been increasing interest in investigating the dynamics of neural networks since Hopfield [984] constructed a simplified neural network model. Based on the Hopfield neural network model, Marcus and Westervelt [989] argued that time delays always occur in the singal transmission and proposed a neural network model with delay. Afterward, a variety of artificial models have been estabished to describe neural networks with delays [Babcock & Westervelt, 987; Baldi & Atiya, 994; Hopfield, 984; Kosko, 988]. Many researchers [Gopalsamy & Leung, 996, 997; Liao et al., 999] focus their attention on the neural networks with time delays and study the dynamical characteristics of neural networks with time delays. It is known to all that periodic solutions can cause a Hopf bifurcation. This occurs when an eigenvalue crosses the imaginary axis from left to right as a real parameter in the equation passing through a critical value. Recently, stability and Hopf bifurcation analysis have been studied in many neural network models [Guo et al., 24; Liao et al., 2a, 2b; Ruan & Wei, 2, 23; Song et al., 25; Song & Wei, 25; Yu & Cao, 25; Yu & Cao, in press]. Since the general models are more complex and we cannot investigate the bifurcation analysis of them. Thus, networks of two neurons have been used as a prototype to understand the dynamics of large-scale neural networks. Hopf bifurcation and stability of bifurcating periodic solutions are often studied using the approach in [Hassard et al., 98] (see, for example [Guo et al., 24; Liao et al., 2a, 2b; Ruan & Wei, 2, 23; Song et al., 25; Song & Wei, 25; Yu & Cao, 25, 26]). In this paper, we will study a more general neural network model with time delay, using Nyquist criterion and the graphical Hopf bifurcation theorem stated in [Mees, 98; Moiola & Chen, 996] to determine the existence of Hopf bifurcation and stability of bifurcating periodic solutions. Gopalsamy and Leung [996] considered the following neural network of two neurons constituting an activator-inhibitor assembly by the delay differential system: dx(t) x(t)+atanh[c y(t τ)], dt () dy(t) y(t)+atanh[ c 2 x(t τ)], dt where a, c, c 2 and τ are positive constants, y denotes the activating potential of x, andx is the inhibiting potential. Gopalsamy and Leung showed that if the delay has a sufficiently large magnitude, the network is excited to exhibit a temporally periodic behavior, where the analytical mechanism for the onset of cyclic behavior is through a Hopf bifurcation. Approximate solutions to the periodic output of the netlet were calculated, and the stability of the temporally periodic cyclic was investigated. Olien and Bèlair [997], on the other hand, investigated the following system with two delays dx (t) x (t)+a f(x (t τ )) dt + a 2 f(x 2 (t τ 2 )), (2) dx 2 (t) x 2 (t)+a 2 f(x (t τ )) dt + a 22 f(x 2 (t τ 2 )), for which several cases, such as τ τ 2, a a 22, etc. were discussed. They obtained some sufficient conditions for the stability of the stationary point of model (2), and showed that (2) undergoes some bifurcations at certain values of the parameters. Wei and Ruan [999] analyzed model (2) with two discrete delays. For the case without self-connections, they found that Hopf bifurcation occurs when the sum of the two delays passes through a sequence of critical values. The stability and direction of the Hopf bifurcation were also determined. In this paper, we will consider a more general equation with a discrete delay, and study the existence of a Hopf bifurcation and the stability of bifurcating periodic solutions of equation. The organization of this paper is as follows: In Sec. 2, we will discuss the stability of the trivial solutions and the existence of Hopf bifurcation. In Sec. 3, a formula for determining the stability of bifurcating periodic solutions will be given by using harmonic balance approach, Nyquist criterion and the graphic Hopf bifurcation theorem introduced at [Allwright, 977; MacFarlane & Postlethwaite, 977; Mees, 98; Moiola & Chen, 993a, 993b, 996]. In Sec. 4, numerical simulations aimed at justifying the theoretical analysis will be reported. 2. Existence of Hopf Bifurcation The neural networks with single delay considered in this paper are described by the following differential

3 Stability and Hopf Bifurcation on a Two-Neuron System 357 equations with delay: ẋ (t) a x + b f (x (t τ)) + b 2 f 2 (x 2 (t τ)), (3) ẋ 2 (t) a 2 x 2 + b 2 f (x (t τ)) + b 22 f 2 (x 2 (t τ)), where a i (i, 2) are real and positive, x (t) and x 2 (t) denote the activations of two neurons, τ denote the synaptic transmission delay, b ij ( i, j 2) are the synaptic weights, f i (i, 2) is the activation function and f i : R R is a C 3 smooth function with f i (). In a more simplified case, (3) can be written as ẋ(t) Ax(t)+Bf (x(t τ)), (4) where x (t) a x(t), A, x 2 (t) a 2 b b 2 f (x(t τ)) B, f. b 2 b 22 f 2 (x(t τ)) By introducing a state-feedback control u g(y), one obtains a linear system with a nonlinear feedback, as follows ẋ(t) Ax(t)+Bu, y(t) Cx(t), (5) u g(y(t τ)), where y (t) y(t), C, y 2 (t) g (y (t τ)) f ( y (t τ)) g(y(t τ)). g 2 (y 2 (t τ)) f 2 ( y 2 (t τ)) Next, taking a Laplace transform L( ) on(5), yields L(x) [si + A] BL(g(y)), and so L(y) CL(x) C[sI + A] BL(g(y)) G(s)L(g(y)), (6) where G(s) C[sI + A] B (7) is the standard transfer matrix of the linear part of the system. It follows from (6) that we may only deal with y(t) in the frequency domain, without directly considering x(t). In so doing, we first observe that if x is an equilibrium solution of the first equation of (5), then From (7), we have y (t) G()g(y ). (8) G(s) C[sI + A] B s + a b b 2 s + a 2 b 2 b 22 s + a b b 2 b 2 b 22 s + a 2 b b 2 s + a s + a b 2 b 22. (9) s + a 2 s + a 2 Clearly, y is the equilibrium of the linearized feedback system, then the Jacobian is given by ( g g g 2 g J, () g 2 g 22 g 22 where y) y g ij g i y j (i, j, 2), y so one has b b 2 s + a s + a g G(s)J b 2 b 22 g 22 s + a 2 s + a 2 b g b 2 g 22 s + a s + a b 2 g b 22 g 22. () s + a 2 s + a 2 Set h(λ, s; τ) det λi G(s)Je sτ λ b g e sτ b 2g 22 e sτ s + a s + a b 2g e sτ λ b 22g 22 e sτ s + a 2 s + a 2

4 358 W. Yu & J. Cao ( λ 2 b g + b ) 22g 22 λe sτ s + a s + a 2 + (s + a )(s + a 2 ) (b b 22 b 2 b 2 )g g 22 e 2sτ. (2) Then applying the generalized Nyquist stability criterion, the following results stated in [Mees, 98; Moiola & Chen, 993a, 993b, 996] can be established. Lemma 2. [Moiola & Chen, 996]. If an eigenvalue of the corresponding Jacobian of the nonlinear system, in the time domain, assumes a purely imaginary value iω at a particular value τ τ, then the corresponding eigenvalue of the constant matrix [G(iω )Je iω τ ] in the frequency domain must assume the value +i at τ τ. To apply Lemma 2., let ˆλ ˆλ(iω; τ) bethe eigenvalue of G(iω)Je iωτ that satisfies ˆλ(iω ; τ ) +i. Then ( b g h(,iω ; τ )+ + b ) 22g 22 e sτ s + a s + a 2 + Thus, we obtained (s + a )(s + a 2 ) (b b 22 b 2 b 2 ) g g 22 e 2sτ. (3) s 2 +(a + a 2 )s + a a 2 +[(b g + b 22 g 22 )s + a 2 b g + a b 22 g 22 ]e sτ +[(b b 22,b 2 b 2 )g g 22 ]e 2sτ, (4) and it can be written as s 2 + d s + d 2 +(d 3 s + d 4 )e sτ + d 5 e 2sτ, (5) where d a + a 2, d 2 a a 2, d 3 b g + b 22 g 22, d 4 a 2 b g + a b 22 g 22, d 5 (b b 22 b 2 b 2 )g g 22. It is easy to see that (4) is equivalent to the characteristic equation of (3). Multiplying e sτ on both sides of (5), we have (s 2 + d s + d 2 )e sτ +(d 3 s + d 4 )+d 5 e sτ. (6) Let s iω,τ τ, and substituting these into (6), for the sake of simplicity, we denote ω and τ by ω,τ, respectively, then (6) becomes (cos(ωτ)+isin(ωτ))( ω 2 + d iω + d 2 )+d 3 iω + d 4 + d 5 (cos(ωτ) i sin(ωτ)). (7) Separating the real and imaginary parts, we have { (ω 2 d 2 d 5 )cos(ωτ)+d ω sin(ωτ) d 4, (ω 2 d 2 + d 5 )sin(ωτ) d ω cos(ωτ) d 3 ω. (8) By simple calculation, we obtained sin(ωτ) ω(d 3ω 2 + d d 4 d 2 d 3 d 3 d 5 ) ω 4 +(d 2 2d 2)ω 2 + d 2 2, (9) d2 5 and cos(ωτ) (d 4 d d 3 )ω 2 +(d 5 d 4 d 2 d 4 ) ω 4 +(d 2 2d 2)ω 2 + d 2 2. (2) d2 5 Let e d 2 2d 2, e 2 d 2 2 d2 5, e 3 d 3, e 4 d d 4 d 2 d 3 d 3 d 5, e 5 d 4 d d 3, e 6 d 5 d 4 d 2 d 4, and sin(ωτ), cos(ωτ) can be written as and sin(ωτ) ω(e 3ω 2 + e 4 ) ω 4 + e ω 2 + e 2, (2) cos(ωτ) e 5ω 2 + e 6 ω 4 + e ω 2 + e 2. (22) As is known to all that sin 2 (ωτ)+cos 2 (ωτ),we have ω 8 + f 3 ω 6 + f 2 ω 4 + f ω 2 + f, (23) where f 3 2e e 2 3,f 2 e 2 +2e 2 2e 3 e 4 e 2 5,f 2e e 2 e 2 4 2e 5e 6, f e 2 2 e2 6.Denotez ω2, (23) becomes Let z 4 + f 3 z 3 + f 2 z 2 + f z + f. (24) l(z) z 4 + f 3 z 3 + f 2 z 2 + f z + f. Suppose (H) (24) has at least one positive root. If A, B, f of the system (4) are given, we can use the computer to calculate the roots of (24) easily. Since lim z l(z) +, we conclude that if f <, then (24) has at least one positive root. Without loss of generality, we assume that it has four positive roots, defined by z, z 2, z 3, z 4, respectively. Then (23) will have four positive roots ω z, ω 2 z 2, ω 3 z 3, ω 4 z 4. By (22), we have cos(ω k τ) e 5ωk 2 + e 6 ωk 4 + e ωk 2 + e. (25) 2

5 Stability and Hopf Bifurcation on a Two-Neuron System 359 Thus, we denote τ j k ( e5 ωk {±arccos 2 + e ) } 6 ω k ωk 4 + e ωk 2 + e +2jπ, (26) 2 where k, 2, 3, 4; j,,...,then±iω k is a pair of purely imaginary roots of (4) with τ j k. Define τ τk min {τ k : τ k }, ω ω k. (27) k {,2,3,4} Note that when τ, (5) becomes s 2 + ps + q, (28) where p d + d 3, q d 2 + d 4 + d 5.If (H2): p>andq> holds, (28) has two roots with negative real parts and system (3) is stable near the equilibrium. Till now, we can employ a result from [Ruan & Wei, 2] to analyze (5), which is, for the convenience of the reader, stated as follows: Lemma 2.2 [Ruan & Wei, 2]. exponential polynomial Consider the P (λ, e λτ,...,e λτm ) λ n + p () λn + + p () n λ + p() n +[p () λn + + p () n λ + p() n ]e λτ + +[p (m) λ n + + p (m) n λ + p(m) n ]e λτm, where τ i (i, 2,...,m) and p (i) j (i,,..., m; j, 2,...,n) are constants. As (τ,τ 2,...,τ m ) vary, the sum of the order of the zeros of P (λ, e λτ,...,e λτm ) on the open right half plane can change only if a zero appears on or crosses the imaginary axis. From Lemmas 2. and 2.2, we have the following: Theorem 2.3. Suppose that (H) and (H2) holds, then the following results hold: (I) For Eq. (3), its zero solution is asymptotically stable for τ [,τ ), (II) Eq. (3) undergoes a Hopf bifurcation at the origin when τ τ. That is, system (3) has a branch of periodic solutions bifurcating from the zero solution near τ τ. Remark 2.4. Yu and Cao [25] studied a van der Pol equation. If we choose a A, B, J, and the characteristic equation is λ 2 + aλe λτ + e 2λτ, this is a special case in our characteristic Eq. (5). Remark 2.5. In [Guo et al., 24], though Guo, Huang and Wang studied a two-neuron network model with three delays, the coefficients of the system must satisfy some conditions. We choose a a 2 A, B, a 2 a ( f ) () J f, β a g, () a 2 a 2 g 22, a 2 a 2 g and the characteristic equation discussed in [Guo et al., 24] is [λ + βe λτ ] 2 a 2 a 2 e 2λτ. It is also a special case in our characteristic Eq. (5). Remark 2.6. Song et al. [25] studied a simplified BAM neural network with three delays, but through a simple transformation the model can be changed into one time delay since the BAM neural network do not have self-connections. By the method studied in [Ruan & Wei, 2], Song, Han and Wei studied the following characteristic equation λ 3 + a 2 λ 2 + a λ + a +(b λ + b )e λτ, and this characteristic equation is more simple than ours, also the approach used in that paper is more difficult than ours since it involves much mathematical analysis in that paper. We can also develop our model to a third degree exponential polynomial. Remark 2.7. Song and Wei [25] studied a delayed predator prey system, the characteristic equation is λ 2 + pλ + r +(sλ + q)e λτ, clearly, it is a special case in our characteristic Eq. (5). For the coefficients given in the above characteristic equations, the reader may refer to the references. In this paper, we have a method to solve characteristic Eq. (5).

6 36 W. Yu & J. Cao 3. Stability of Bifurcating Periodic Solutions Based on the Lemma 2. and the results in [Allwright, 977; Mees, 98; Moiola & Chen, 993a, 993b, 996], we just give some conclusions for simplicity. By applying a second-order harmonic balance approximation in [Mees, 98; Moiola & Chen, 996] to the output, we have { 2 } y(t) y + R Y k e ikωt, (29) k where y is the equilibrium point, R{ } is the real part of the complex constant, and the complex coefficients Y k are determined by the approximation as shown below: we first define an auxiliary vector ξ ( ω) wt [G(i ω)]p e i ω τ w T v, (3) where τ is the fixed value of the parameter τ, w T and v are the left and right eigenvectors of [G(i ω)]je i ωτ, respectively, associated with the value ˆλ(i ω; τ), and p ( [D 2 V 2 v + ) 2 v V 22 + ] 8 D 3v v v, (3) in which denotes the complex conjugate as usual, ω is the frequency of the intersection between the ˆλ locus and the negative real axis closest to the point ( +i), is the tensor product operator, and D 2 2 g(y; τ ) y 2 D 3 3 g(y; τ ) y 3, (32) y, (33) y V 2 4 [I + G()J] G()D 2 v v, (34) V 22 4 [I + G(2i ω)je 2i ω τ ] G(2i ω)d 2 v ve 2i ω τ. (35) Then, the following Hopf bifurcation theorem formulated in the frequency domain can be established [Moiola & Chen, 996]: Theorem 3.. (The Graphical Hopf Bifurcation Theorem). Suppose that when ω varies, the vector ξ ( ω), where ξ ( ω) is defined in (3), and that the half-line, starting from + i and pointing to the direction parallel to that of ξ ( ω), first intersects the locus of the eigenvalue ˆλ(iω; τ) at the point ˆP ˆλ(ˆω; τ) +ξ ( ω)θ 2, (36) at which ω ˆω and the constant θ θ(ˆω). Suppose furthermore, that the above intersection is transversal, namely, R{ξ (ˆω)} I{ξ (ˆω)} det { } { } d R dω ˆλ(ω; τ) d I ωˆω dω ˆλ(ω; τ) ωˆω. (37) Then we have the following conclusions: () The nonlinear system (5) has a periodic solution y(t) y(t;ŷ). Consequently, there exists a unique limit cycle for the nonlinear equation (3); (2) If the half-line L first intersects the locus of ˆλ(iω) when τ > τ (<τ ), then the bifurcating periodic solution exists and the Hopf bifurcation is supercritical (subcritical); (3) If the total number of anticlockwise encirclements of the point P ˆP + εξ ( ω), for a small enough ε >, is equal to the number of poles of λ(s) that have positive real parts, then the limit cycle is stable; otherwise, it is unstable. In the above, as usual, R{ } and I{ } are the real and imaginary parts of the complex number, respectively. From (32), one has D 2 2 g(y; τ) y 2 where y g g 2 g 2 g 22 g 2 g 22 g 22 g 222 g, (38) g 222 g ijk 2 g i y j y k (i, j, k, 2).

7 Stability and Hopf Bifurcation on a Two-Neuron System 36 Also D 3 3 g(y; τ) y 3 g g 2 g 2 g 22 g 2 g 22 g 22 g 222 y g 2 g 22 g 22 g 222 g 22 g 222 g 222 g 2222 ( g g 2222 ), (39) where g ijkl 3 g i y j y k y l (i, j, k, l, 2). As we know w T and v are the left and right eigenvectors of [G(i ω)]je i ωτ, respectively, associated with the value ˆλ(i ω; τ) λ, wehave v i ω + a λe i ω τ b g, (4) v 2 b 2 g 22 b 2 g 22 and w i ω + a 2 b 2 g λe i ω τ i ω + a 2 b 2 g b g i ω + a ( w 2 ). (4) From (34) and (35), we obtained V 2 4 [I + G()J] G()D 2 v v and + b g b 2 g 22 b b 2 a a a a g v 2 4 b 2 g + b 22g 22 b 2 b 22 g 222 v 2 a 2 a 2 a 2 a 2 v 2 v 2 [( 4 + b g a )( + b 22g 22 a 2 ) b ] 2b 2 g g 22 a a 2 b a + (b b 22 b 2 b 2 )g 22 a a 2 b 2 a b 2 b 22 + (b b 22 b 2 b 2 )g a 2 a 2 a a 2 ( g g 222 v 2 v 2 ), (42) V 22 4 [I + G(2i ω)je 2i ω τ ] G(2i ω)d 2 v ve 2i ω τ + b g e 2i ω τ b 2 g 22 e 2i ω τ b b 2 2i ω + a 2i ω + a 2i ω + a 2i ω + a 4 b 2 g e 2i ω τ + b 22g 22 e 2i ω τ b 2 b 22 2i ω + a 2 2i ω + a 2 2i ω + a 2 2i ω + a 2

8 362 W. Yu & J. Cao g g 222 v 2 v 2 v 2 v 2 [( 4 + b g )( e 2i ω τ 2i ω + a + b 22g 22 2i ω + a 2 e 2i ω τ b + (b b 22 b 2 b 2 )g 22 e 2i ω τ 2i ω + a (2i ω + a )(2i ω + a 2 ) g e 2i ω τ. g 222 v 2 v 2 ) b 2b 2 g g 22 e 4i ω τ (2i ω + a )(2i ω + a 2 ) b 2 2i ω + a b 2 b 22 + (b b 22 b 2 b 2 )g e 2i ω τ 2i ω + a 2 2i ω + a 2 (2i ω + a )(2i ω + a 2 ) ] (43) Let V2 () V22 () V 2 and V 22, V 2 (2) V 22 (2) substituting (42) and (43) into (3), we obtained ( p [D 2 V 2 v + ) 2 v V 22 + ] 8 D 3v v v g V 2 () + 2 g V 22 () + 8 g g 222 V 2 (2)v g 222V 22 (2)v g 2222v2v 2 2 (44) Substituting (39) (44) into (3), we can obtain ξ ( ω). Corollary 3.2. Let k be the total number of anticlockwise encirclements of the point P ˆP +εξ ( ω) for a small enough ε>, where ˆP is the intersection of the half-line L and the locus ˆλ(iω). Then () If k, the bifurcating periodic solutions of system (3) are stable; (2) If k, the bifurcating periodic solutions of system (3) are unstable. Remark 3.3. In this paper we study the stability of bifurcating periodic solutions using the harmonic balance approach, Nyquist criterion and the graphic Hopf bifurcation theorem. It is an algebraic and graphical approach and more simple than the normal form method and center manifold theorem introduced by Hassard et al. [98]. It does not involve much mathematical analysis. The stability of bifurcating periodic orbits have been analyzed drawing the amplitude locus, L, and the locus ˆλ(iω) ina neighborhood of the Hopf bifurcation point. 4. Numerical Examples In this section, some numerical results of simulating system (3) are presented. The half-line and locus ˆλ(iω) are shown in the corresponding frequency graphs. If they intersect, a limit cycle exists, or else, no limit cycle exists. Corollary 3.2 implies that the stabilities of the bifurcating periodic solutions are determined by the total number k of the anticlockwise encirclements of the point P ˆP + εξ ( ω) for a small enough ε>. Suppose that the half-line L and the locus ˆλ(iω) intersect.ifk, the bifurcating periodic solutions of system (3) are stable; if k, the bifurcating periodic solutions of system (3) are unstable. In order to verify the theoretical analysis results derivedabove,system(3)issimulatedindifferent cases. 2 (i) A, B, tanh(x) f(x). tanh(x) Equation (28) have two negative roots and 6, Eq. (24) has one positive root , from Eq. (26), we have τ j j (j,,...,), and τ.583.

9 Stability and Hopf Bifurcation on a Two-Neuron System Im Im Re Frequency graph Re Frequency graph x x t Waveform graph t Waveform graph x2.2 x x Phase graph Fig.. τ.45. The half-line L does not intersect the locus ˆλ(iω), so no periodic solution exists. - - x Phase graph Fig. 2. τ.55. The half-line L intersects the locus ˆλ(iω), and k, so a stable periodic solution exists.

10 364 W. Yu & J. Cao Im Im Re..9.8 Re Frequency graph Frequency graph x x t Waveform graph t Waveform graph x2 x x Phase graph Fig. 3. τ.6. The half-line L does not intersect the locus ˆλ(iω), so no periodic solution exists..8 x Phase graph Fig. 4. τ.7. The half-line L intersects the locus ˆλ(iω), and k, so a stable periodic solution exists.

11 Stability and Hopf Bifurcation on a Two-Neuron System 365 We choose τ.45 <τ and τ.55 >τ, respectively. The corresponding frequency, waveform and phase graph are shown in Figs. and 2. By Lemma 2., Theorems 2.3 and 3. we know in Fig. its zero solution is asymptotically stable, in Fig. 2 the bifurcating periodic solution is stable and the system undergoes a Hopf bifurcation at the origin. 2 3 (ii) A, B, tanh(x) f(x). tanh(x) Equation (28) have two negative roots and , Eq. (24) has one positive root.2259, from Eq. (26), we have τ j j (j,,...,), and τ.675. We choose τ.6 <τ and τ.7 >τ, respectively. The corresponding frequency, waveform and phase graph are shown in Figs. 3 and 4. By Lemma 2., Theorems 2.3 and 3. we know in Fig. 3 its zero solution is asymptotically stable, Fig. 4 undergoes a Hopf bifurcation at the origin. 5. Conclusions A more general two-neuron model with time delay studied in this paper from the frequency domain approach turns out to be not so mathematically involved and so difficult as analyzing in the time domain [Guo et al., 24; Liao et al., 2a, 2b; Ruan & Wei, 2, 23; Song et al., 25; Song & Wei, 25; Yu & Cao, 25, 26]. By using the time delay as the bifurcation parameter, it has been shown that a Hopf bifurcation occurs when this parameter passes through a critical value. The stability of bifurcating periodic orbits have been analyzed drawing the amplitude locus, L,andthe locus ˆλ(iω) in a neighborhood of the Hopf bifurcation point. It is very difficult to solve large-scale neural networks with time delays, since the characteristic equation in large-scale neural networks is a more complex transcendental equation. In studying the stability and Hopf bifurcation analysis, there are still much work to be done, we should focus on large-scale neural networks with more time delays. References Allwright, D. J. [977] Harmonic balance and the Hopf bifurcation theorem, Math. Proc. Camb. Philosph. Soc. 82, Babcock, K. L. & Westervelt, R. M. [987] Dynamics of simple electronic neural networks, Physica D 28, Baldi, P. & Atiya, A. [994] How delays affect neural dynamics and learning, IEEE Trans. Neural Networks 5, Cao, J. & Chen, T. [24] Globally exponentially robust stability and periodicity of delayed neural networks, Chaos Solit. Fract. 22, Cao, J. & Liang, J. [24] Boundedness and stability for Cohen Grossberg neural network with time-varying delays, J. Math. Anal. Appl. 296, Cao, J., Liang, J. & Lam, J. [24] Exponential stability of high-order bidirectional associative memory neural networks with time delays, Physica D 99, Cao, J. & Wang, J. [24] Absolute exponential stability of recurrent neural networks with Lipschitzcontinuous activation functions and time delays, Neural Networks 7, Cao, J., Huang, D. & Qu, Y. [25] Global robust stability of delayed recurrent neural networks, Chaos Solit. Fract. 23, Cao, J. & Li, X. [25] Stability in delayed Cohen Grossberg neural networks: LMI optimization approach, Physica D 22, Cao, J. & Wang, J. [25] Global asymptotic and robust stability of recurrent neural networks with time delays, IEEE Trans. Circuits Syst.-I 52, Gopalsamy, K. & Leung, I. [996] Delay induced periodicity in a neural netlet of excitation and inhibition, Physica D 89, Gopalsamy, K. & Leung, I. [997] Convergence under dynamical thresholds with delays, IEEE Trans. Neural Networks 8, Guo, S., Huang, L. & Wang, L. [24] Linear stability and Hopf bifurcation in a two-neuron network with three delays, Int. J. Bifurcation and Chaos 4, Hassard, B. D., Kazarinoff, N. D. & Wan, Y. H. [98] Theory and Application of Hopf Bifurcation (Cambridge University Press, Cambridge). Hopfield, J. J. [984] Neurons with graded response have collective computational properties like those of two-state neurons, Proc. Nat. Acad. Sci. USA 8, Kosko, B. [988] Bi-directional associative memories, IEEE Trans. Syst. Man Cybern. 8, Liao, X., Wu, Z. & Yu, J. [999] Stability switches and bifurcation analysis of a neural network with continuously delay, IEEE Trans. Syst. Man Cybern. 29,

12 366 W. Yu & J. Cao Liao, X., Wong, K. & Wu, Z. [2a] Hopf bifurcation and stability of periodic solutions for van der Pol equation with distributed delay, Nonlin. Dyn. 26, Liao, X., Wong, K. & Wu, Z. [2b] Bifurcation analysis in a two-neuron system with distributed delay, Physica D 49, MacFarlane, A. G. J. & Postlethwaite, I. [977] The generalized Nyquist stability criterion and multivariable root loci, Int. J. Contr. 25, Marcus, C. M. & Westervelt, R. M. [989] Stability of analog neural network with delay, Phys. Rev. A 39, Mees, A. I. & Chua, L. O. [979] The Hopf bifurcation theorem and its applications to nonlinear oscillations in circuits and systems, IEEE Trans. Circuits Syst. 26, Mees, A. I. [98] Dynamics of Feedback Systems (John Wiley & Sons, Chichester, UK). Moiola, J. L. & Chen, G. [993a] Computation of limit cycles via higher-order harmonic balance approximation, IEEE Trans. Autom. Contr. 38, Moiola, J. L. & Chen, G. [993b] Frequency domain approach to computation and analysis of bifurcations and limit cycles: A tutorial, Int. J. Bifurcation and Chaos 3, Moiola, J. L. & Chen, G. [996] Hopf Bifurcation Analysis: A Frequency Domain Approach (World Scientific, Singapore). Olien, L. & Bélair, J. [997] Bifurcations, stability and monotonicity properties of a delayed neural network model, Physica D 2, Ruan, S. & Wei, J. [2] On the zeros of a third degree exponential polynomial with applications to a delayed model for the control of testosterone secretion, IMA J.Math.Appl.Med.Biol.8, Ruan, S. & Wei, J. [23] On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. Contin. Discr. Impuls. Syst. Ser. A Math. Anal., Song, Y., Han, M. & Wei, J. [25] Stability and Hopf bifurcation analysis on a simplified BAM neural network with delays, Physica D 2, Song, Y. & Wei, J. [25] Local Hopf bifurcation and global periodic solutions in a delayed predator prey system, J. Math. Anal. Appl. 3, 2. Tsypkin, Y. [946] Stability of systems with delayed feedback, Automat. Telemekh 7, 7 29; Also reprinted in MacFarlane, A. G. J. (ed.) [979] Frequency Response Methods in Control Systems (IEEE Press), pp Wei, J. & Ruan, S. [999] Stability and bifurcation in a neural network model with two delays, Physica D 3, Yu, W. & Cao, J. [25] Hopf bifurcation and stability of periodic solutions for van der Pol equation with time delay, Nonlin. Anal. 62, Yu, W. & Cao, J. [26] Stability and Hopf bifurcation analysis on a four-neuron BAM neural network with time delays, Phys. Lett. A 35,