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Bifurcation analysis on a two-neuron system with distributed delays in the frequency domain

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Neural Networks 17 (2004) 545–561

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Bifurcation analysis on a two-neuron system with distributed delays in the frequency domain
Xiaofeng Liaoa,b,*, Shaowen Lib,c, Guanrong Chend
aDepartment of Computer Science and Engineering, Chongqing University, Chongqing 400044, China bCollege of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu 610054, China
cDepartment of Mathematics, Southwestern University of Finance and Economics, China dDepartment of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, China
Received 7 January 2003; accepted 8 October 2003
Abstract In this paper, a general two-neuron model with distributed delays and a strong kernel is investigated. By applying the frequency domain
approach and analyzing the associated characteristic equation, the existence of bifurcation parameter for the model is determined. Furthermore, if the mean delay used as a bifurcation parameter, it is found that Hopf bifurcation occurs for the strong kernel. This means that a family of periodic solutions bifurcates from the equilibrium when the bifurcation parameter exceeds a critical value. The direction and stability of the bifurcating periodic solutions are determined by the Nyquist criterion and the graphical Hopf bifurcation theorem. Some numerical simulations are given to justify the theoretical analysis results. q 2004 Elsevier Ltd. All rights reserved.
Keywords: Neuron; Distributed delays; Hopf bifurcation; Graphical hopf bifurcation theorem; Periodic solution; Nyquist criterion

1. Introduction
It is well known that neural networks are complex and large-scale nonlinear dynamical systems (Hop?eld, 1984). Lacking the ability in tackling the intrinsic complexities, neural network models under investigation today have been dramatically simpli?ed (An der Heiden, 1979; Babcock & Westervelt, 1986, 1987; Belair & Dufour, 1998; Campbell, 1999; Destxhe, 1994; Destexhe & Gaspard, 1993; Gopalsamy & Leung, 1996, 1997; Gopalsamy, Leung, & Liu, 1998; Liao, Wu, & Yu, 1999a,b; Liao et al., 2001a,b; Marcus & Westervelt, 1989; Majee & Roy, 1997; Moiola & Chen, 1996; Olien & Belair, 1997; Wei & Ruan, 1999; Willson & Cowan, 1972). Yet, these studies of simpli?ed models are still very useful and insightful, since the complexities found in simple models can often be carried over to large-scale networks in some way thereby yielding much better understanding of the latter from a careful study of the former.
* Corresponding author. Address: Department of Computer Science and Engineering, Chongqing University, Chongqing, 400044, China.
E-mail address: x?iao@cqu.edu.cn (X. Liao).
0893-6080/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.neunet.2003.10.001

Recently, several simple neuron models, with discrete or distributed delays, are proposed, for example (Gopalsamy & Leung, 1997),

dx?t? ? 2x?t? ? a tanh?x?t? 2 bx?t 2 t? 2 c?;

?1?

dt

dx?t?

?1 ? 2x?t? ? a tanh?x?t? 2 b F?s?x?t 2 s?ds 2 c?;

dt

0

?2?

where a denotes the range of the continuous variable x?·?; while b can be considered as a measure of the inhibitory in?uence from the past history, c is a off-set constant, t is the time delay, and F?·? is a kernel function. Using Lyapunov functionals, Gopalsamy and Leung (1997) obtained some necessary and suf?cient conditions for the existence of a globally asymptotically stable equilibrium of Eqs. (1) and (2). Liao et al. (1999a) discussed Eq. (2) with a weak kernel and found that this model does not lead to any stability switching. Furthermore, Liao et al. (1999b) studied Eq. (2) with a strong kernel and found that the stability of the equilibrium may be lost when the mean delay is increased. However, a further increase of the mean delay

546

X. Liao et al. / Neural Networks 17 (2004) 545–561

may cause the system coming back to a stable state again.

Liao et al. (2001b) also studied Eq. (1) and found that the

Hopf bifurcation occurs when the inhibitory in?uence from

the past state varies and then passes through a sequence of

critical values. Moreover, chaotic behavior of Eq. (1) with

non-monotonously increasing transfer function has been

observed in computer simulations.

Gopalsamy and Leung (1996) considered the following

neural network of two neurons constituting an activator –

inhibitor assembly modeled by the delay differential

system

8 >><

dx?t? dt

?

2x?t? ? a tanh?c1y?t 2 t??;

>>:

dy?t? dt

?

2y?t? ? a tanh?2c2x?t 2 t??;

?3?

where a; c1; c2 and t are positive constants, y denotes the activating potential of x; and x is the inhibiting potential.

Gopalsamy and Leung (1996) showed that if the delay has

a suf?ciently large magnitude, the network is excited to

exhibit a temporally periodic behavior, where the analyti-

cal mechanism for the onset of cyclic behavior is through a

Hopf-type bifurcation. Approximate solutions to the

periodic output of the netlet were calculated, and the

stability of the temporally periodic cyclic was investigated

(Gopalsamy and Leung, 1996). For a number of two-

neuron models and their linear stability analysis, the reader

is referred to the work of Babcock and Westervelt (1986,

1987) and Marcus and Westervelt (1989), and some

references cited therein.

Gopalsamy et al. (1998) considered an analogue of

model (3) containing continuously distributed delays in the

following form

8 >>>< >>>:

dx?t? dt
dy?t?

? 2x?t? ? a tanh ? 2y?t? ? a tanh

?t

!

k?t 2 s?y?s?ds ;

21
?t

!

k?t 2 s?x?s?ds ;

?4?

dt

21

in which a is a positive constant and the delay kernel k is

assumed to satisfy the following

8

>< k : ?0; ?1? ! ?0; ?1?; k is piecewise continuous and

>: ?1 k?s?ds ? 1;

?1 sk?s?ds , ?1:

?5?

0

0

Some suf?cient conditions were obtained for the global

Hopf-bifurcation of periodic solutions of Eq. (4), and the

orbital asymptotic stability of the bifurcating periodic

solutions was also investigated (Gopalsamy et al., 1998).

In the case of two delays, Babcock and Westervelt (1986,

1987) studied the following two-neuron network model

8 >><

dx1?t? dt

? 2x1?t? ? a1tanh?x2?t 2 t1??;

>>:

dx2?t? dt

? 2x2?t? ? a2tanh?x1?t 2 t2??;

?6?

where a1; a2; t2 and t2 are positive constants. They showed that system (6) exhibits very interesting and rich dynamics, including under-damped ringing transients, stable and unstable limit cycles, etc. Equations similar to Eq. (6) have been used by An der Heiden (1979) and Willson and Cowan (1972) to model the neuron interactions, where the delays re?ect the ?nite signal propagation speeds along the dendrites and axons.
Olien and Belair (1997), on the other hand, investigated the following system with two delays

8 >><

dx1?t? dt

? 2x1?t? ? a11f ?x1?t 2 t1?? ? a12f ?x2?t 2 t2??;

>>:

dx2?t? dt

? 2x2?t? ? a21f ?x1?t 2 t1?? ? a22f ?x2?t 2 t2??;

?7?

for which several cases, such as t1 ? t2; a11 ? a22 ? 0; etc. were discussed. They obtained some suf?cient conditions for the stability of the stationary point of model (7), and showed that (7) may undergo some bifurcations at certain values of the parameters. Wei and Ruan (1999) analyzed model (7) 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. A similar model representing a single pair of neurons with self-connections was studied by Destexhe and Gaspard (1993). The reader is referred to Campbell (1999) and Majee and Roy (1997), and the references cited therein, for related work on two-neuron networks with delays.
Recently, Liao et al. (2001a) studied the following twoneuron system with distributed delays

8 >>><

dxp1?t? dt

?

2xp1?t?

?

ap1f

?1

!

xp2?t? 2 b2 F?r?xp2?t 2 r?dr 2 c1 ;

0

>>>:

dxp2?t? dt

?

2xp2?t?

?

ap2f

?1

!

xp1?t? 2 b1 F?r?xp1?t 2 r?dr 2 c2 ;

0

?8?

where api ; bi and ci ?i ? 1;2? are nonnegative constants. In this model, xpi ?i ? 1;2? denote the mean soma potential of the neuron, api corresponds to the range of the continuous variable xpi ; bi are measures of the inhibitory in?uence of the past history, ci denote the neuronal threshold, and xpi in the argument of the function f represent local positive feedback.
For the case of model (8) with a weak kernel, its local linear stability was analyzed by using the Routh-Hurwitz criterion (Liao et al., 2001a). If the mean delay is used as a bifurcation parameter, it was found that Hopf bifurcation occurs, meaning that a family of periodic solutions bifurcates from the equilibrium when the bifurcating parameter exceeds a critical value. The direction and

X. Liao et al. / Neural Networks 17 (2004) 545–561

547

stability of the bifurcating periodic solutions were also determined, by employing the normal form and the center manifold theorem. However, only the weak kernel case was discussed (Liao et al., 2001a). In this paper, model (8) with a strong kernel is investigated instead.
At this point, it should be notice that all the aforementioned work used the state-space formulation for delayed differential equations, known as the ‘time domain’ approach (Engelborghs, Lemaire, Belair, & Roose, 2001; Gopalsamy, 1992; Moiola, 1996). Yet, there is another interesting formulation for studying delayed differential equations in the literature. This alternative representation applies the familiar engineering feedback systems theory and methodology: an approach described in the ‘frequency domain’—the complex domain after the standard Laplace transforms have been taken on the state-space system in the time domain. This frequency-domain approach was initiated and developed by Allwright (1977), Mees and Chua (1979) and then Moiola and Chen (1993a,b, 1996), and the ?rst application with the frequency domain approach was given in Moiola, Chiacchiarini, and Desages (1996). This new methodology has some advantages over the classical time-domain methods. This is especially prominent for the case of model (8) with a strong kernel, since it is very dif?cult to determine the stability of the bifurcating periodic solutions by applying the time-domain approach in this case where some bottleneck problems in the analytical study will be encountered. For numerical bifurcation analysis of delay differential equations, the reader is referred to the work of Engelborghs et al. (2001) and some references cited therein.
In this paper, the main interest is in the direction and stability of the bifurcating periodic solutions for model (8) with a strong kernel, and the main methodology of study is by means of the frequency-domain approach. It is found that if the mean delay used as a bifurcation parameter, then Hopf bifurcation occurs for this model with a strong kernel. This means that a family of periodic solutions bifurcates from the equilibrium when the bifurcation parameter exceeds a critical value. The direction and stability of the bifurcating periodic solutions are determined by the Nyquist criterion and the graphical Hopf bifurcation theorem. Some numerical simulations are given to justify the theoretical analysis results.
The organization of this paper is as follows. In Section 2, by means of the frequency-domain approach formulated by Moiola and Chen (1996), the existence of Hopf bifurcation parameter is determined showing that Hopf bifurcation occurs when the bifurcation parameter exceeds a critical value. In Section 3, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are analyzed by means of the Graphical Hopf Bifurcation Theorem (Moiola and Chen, 1996). Some numerical simulation

results and the frequency-domain graphs are given in Section 4, verifying the theoretical analysis results. Finally, some conclusions are given in Section 5.

2. Existence of Hopf bifurcation

Consider model (8) with strong kernels

F?r? ? m2re2mr; m . 0:

?9?

For simplicity, set c1 ? c2 ? 0: In system (8), let

8 >>><

x1?t?

?

xp1?t?

2

b1

?1

F?r?xp1?t

2

r?dr;

0
>>>: x2?t? ? xp2?t? 2 b2 ?1 F?r?xp2?t 2 r?dr:

?10?

0

Then, system (8) is equivalent to the following model

8 >>><

dx1?t? dt

?2x1

?t??ap1f

?x2?t??2ap1b1

?0
21

F?2r?f

?x2?t?r??dr;

>>>:

dx2?t? dt

?2x2

?t??ap2f

?x1?t??2ap2b2

?0
21

F?2r?f

?x1?t?r??dr:

?11?

Assume that

f [ C4?R?; f ?0? ? 0; and uf ?u? . 0 for u – 0:

?12?

Then, by a result of Liao et al. (2001a), the equilibrium (0,0) of Eq. (11) exists if

ap1ap2l?1 2 b1??1 2 b2?l ,

?f

1 0?0??2

:

Since F?r? ? m2re2mr; one has

?0 F?2r?f ?x2?t?r??dr
21

?t ? m2?t2s?em?s2t?f ?x2?s??ds
21

& ?t

?t

'

?m2e2mt t emsf ?x2?s??ds2 semsf ?x2?s??ds: ?13?

21

21

Taking the derivative with respective to t on both sides of

Eq. (11), and using Eq. (13), one obtains

8

>>>>>>>>>>>><2d2dmxt1&2?dt?xd?1t?2t? ?dxxd11t??tt???2aap1p1ff0 ??xx22 ??tt????'d2xd2ta?tp1?b1

m2

e2mt

?t
21

ems

f

?x2

?s??ds;

>>>>>>>>>>>>:2d2dmxt2&2?dt?xd?2t?2t? ?dxxd22t??tt???2aap2p2ff0 ??xx11 ??tt????'d2xd1ta?tp2?b2

m2

e2mt

?t
21

ems

f

?x1

?s??ds:

?14?

548

X. Liao et al. / Neural Networks 17 (2004) 545–561

Then, taking the time derivative again on both sides of Eq. (14) gives

Now, the mean delay m can be used as a bifurcation parameter. By introducing a ‘state-feedback control’, u?

8 >>>><d3dxt13?t??2m2x1?t?2?m2?2m?dxd1t?t?2?2m?1?d2dxt12?t??ap1?12b1?m2f ?x2?t???2ap1mf 0?x2?t??dxd2t?t??ap1f 0?x2?t??d2dxt22?t??ap1f 00?x2?t??

dx2

?t?

!2 ;

dt

>>>>:d3dxt23?t??2m2x2?t?2?m2?2m?dxd2t?t?2?2m?1?d2dxt22?t??ap2?12b2?m2f ?x1?t???2ap2mf 0?x1?t??dxd1t?t??ap2f 0?x1?t??d2dxt12?t??ap2f 00?x1?t??

dx1

?t?

!2 :

dt

?15?

By setting x3?t? ? dx1?t?=dt; x4?t? ? dx2?t?=dt; x5?t? ? d2x1?t?=dt2 and x6?t? ? d2x2?t?=dt2; one arrives at the

following ODE system

8

>>>>>>>>>>>>>>>>>>>>>>>>>< >>>>>>>>>>>>>>>>>>>>>>>>>:

dx1 dt dx2 dt dx3 dt dx4 dt dx5 dt
dx6 dt

? x3; ? x4; ? x5; ? x6; ? 2m2x1 2 ?m2 ? 2m?x3 2 ?2m ? 1?x5 ? ap1?1 2 b1?m2f
?2ap1mf 0?x2?x4 ? ap1f 0?x2?x6 ? ap1f 00?x2?x24; ? 2m2x2 2 ?m2 ? 2m?x4 2 ?2m ? 1?x6 ? ap2?1 2 b2?m2f
?2ap2mf 0?x1?x3 ? ap2f 0?x1?x5 ? ap2f 00?x1?x23

: ?x2?
?x1?

?16?

The non-linear system (16) can be rewritten in a matrix form as

dx ? A?m?x ? H?x?;

?17?

dt

where x ? ?x1; x2; x3; x4; x5; x6?T;

00 0

1

0

A?m??BBBBBBBBBBBB@2000m2

0 0 0 0

0 0 0 2?m2 ?2m?

1 0 0 0

0

01

0 1 0 2?2m?1?

0 0 1 0

CCCCCCCCCCCCA;

0 2m2 0 2?m2 ?2m? 0 2?2m?1?

?18?

00

1

H?x??BBBBBBBBBBBB@000ap1?12b1?m2f ?x2??2ap1mf 0?x2?x4?ap1f 0?x2?x6?ap1f 00?x2?x24CCCCCCCCCCCCA:

ap2?12b2?m2f ?x1??2ap2mf 0?x1?x3?ap2f 0?x1?x5?ap2f 00?x1?x23

?19?

g?y;m?; one obtains a linear system with a non-linear

feedback, as follows

8 >>><

dx dt

?A?m?x?Bu;

>>>:uy??2g?Cy;xm; ?;

?20?

00 01

B

?

BBBBBBBBBBBB@

0 0 0 1

0 0 0 0

CCCCCCCCCCCCA;

C ? I;

?21?

01

u?g?y;m?

?

0BBBBBB@aap1p2??11?22abbp112f??0mm0?222ff ??y222?yyy2124??2222aap1p2mmff

0 0

?2y2 ?2y1

?y4 ?y3

2ap1 2ap2

f f

0?2y2 0?2y1

?y6 ?y5

1 CCCCCCA:

?22?

?ap2f 00?2y1?y23

Next, taking a Laplace transform on Eq. (20) yields a standard transfer matrix of the linear part of the system

G?s;m??C?sI 2A?m??21B

0s 0

?BBBBBBBBBBBB@0m00 2

s 0 0 0

21 0

0

21

s

0

0

s

m2 ?2m 0

0

0

0

0

21

0

0

21

s?2m?1 0

12100 01

CCCCCCCCCCCCA

BBBBBBBBBBBB@0001 0000CCCCCCCCCCCCA

0 m2 0

m2 ?2m 0

s?2m?1 0 1

01 0 1

?

?s?

1 m?2?s

?

1?

BBBBBBBBBBBB@0s0s2

1 0 s 0

CCCCCCCCCCCCA:

0 s2

?23?

X. Liao et al. / Neural Networks 17 (2004) 545–561

549

To this end, if this feedback system is linearized

about the equilibrium y?0; then the Jacobian is

given by

J?m?? ??gyy?0

?

0

2a1?12b1?m2

0

! 22a1m 0 2a1 ;

2a2?12b2?m2

0

22a2m 0 2a2 0

?24?

where ai?api f 0?0?; i?1;2: So, one has G?s;m?J?m?

0

?

?s

?

1 m?2?s

?

1?

BBBBBBBBBBBBB@

0 2a2 0 2a2 0

?1 ?1

2 2

b2 b2

?m2 ?m2

s

2a2?1 2 b2?m2s2

2a1?1 2 b1?m2 0 2a1?1 2 b1?m2s 0 2a1?1 2 b1?m2s2 0

0 22a2m 0 22a2ms 0 22a2ms2

a1a2m0 ?m20 2v20 ?m0?12v20?:

?30?

Consequently,

?1?m0?4 2a1a2b1?1?m0?3 2a1a2?22b1??1?m0?2 ??a1a2?2 ?0;

?31?





v20 ?m0

1

2

a1 a2 1 ? m0

:

?32?

Theorem 1. (Existence of Hopf bifurcation parameter) If b2 ? 0 and the following conditions hold

1

22a1m 0 22a1ms 0 22a1ms2

0 2a2 0 2a2s 0

2a1 0 2a1 0 2a1

s s2

CCCCCCCCCCCCCA:

?25?

0

2a2s2 0

Set

h?l;s;m??detllI 2G?s;m?J ?m?l

(

)

?l4

l2

2

a1

a2??12b1

?m2

?s2 ?2ms???12b2 ?s?m?4?s?1?2

?m2

?s2

?2ms?

?0: ?26?
Then, by applying the generalized Nyquist stability criterion, with s?iv; the following results can be established.

Lemma 1. (Moiola & Chen, 1996) If an eigenvalue of the
corresponding Jacobian of the non-linear system, in the
time domain, assumes a purely imaginary value iv0 at a particular m ? m0; then the corresponding eigenvalue of the constant matrix ?G?iv0; m0?J?m0?? in the frequency domain must assume the value 21 ? i0 at m ? m0:

To apply Lemma 1, let l^ ? l^?iv; m? be the eigenvalue of ?G?iv; m?J?m?? that satis?es l^?iv0; m0? ? 21 ? i0: Then
h?21;iv0;m0?

?

1

2

a1

a2

??1

2

b1

?m20

2

v20 ?2im0v0???12b2?m20 ?m0 ?iv0?4?1?iv0?2

2

v20

?

2im0

v0

?

? 0:

?27?

First consider the case of b2 ?0: Eq. (27) becomes

a1a2??12b1?m20 2v20 ?2im0v0???m0 ?iv0?2?1?iv0?2:

?28?

By separating this equation into real and imaginary parts, one obtains

a1a2??12b1?m20 2v20???m20 2v20??12v20?24m0v20;

?29?

(i) a1a2?4 ? b1? . 4

?33?

(ii) a1a2?4 ? b1? # 4; 3a1a2b1 . 8;

9a21a22b21 ? 32a1a2?2 2 b1? . 0;

?34?

(

)

i:e: max

32?b1 2 9b21

2?

;

8 3b1

, a1a2

#

4

4 ?

b1

;

then

q????????????????????????????????

m? ? 3a1a2b1 2 8 ?

9a21a22b21 ? 32a1a2?2 2 b1? . 0: 8

If f ?m?? , 0; the two nonnegative real roots of Eq. (31) exist. m1 [ ?0; m?? and m2 [ ?m?; ?1? are the roots of Eq. (31). Moreover, if a1a2 , 1; m1 and m2 are the Hopf bifurcations of system (8). If a1a2 $ 1 and b1 . 1 2 ?1=?a1a2??; m2 is the unique Hopf bifurcation of system (8). If a1a2 $ 1 and b1 # 1 2 ?1=a1a2?; the Hopf bifurcations of
system (8) do not exist.

Proof. In order to prove the existence of a positive zero m0 . 0; in Eq. (31), de?ne the following function
f ?m? ? ?1 ? m?4 2 a1a2b1?1 ? m?3 2 a1a2?2 2 b1??1 ? m?2 ? ?a1a2?2: ?35?
So, f ?0? ? 1 2 a1a2b1 2 2a1a2 ? a1a2b1 ? ?a1a2?2 ? ?1 2 a1a2?2
$ 0:

550

X. Liao et al. / Neural Networks 17 (2004) 545–561

Taking the derivation on Eq. (33), we have f 0?m??4?1?m?323a1a2b1?1?m?222a1a2?22b1??1?m?
??1?m??4m2 ??823a1a2b1?m??424a1a2 2a1a2b1??; ?36?
Letting f 0?m??0; we obtain the roots of Eq. (36)

mc ?21;

q??????????????????????????????

m^ ? 3a1a2b1 28^

9a21a22b21 ?32a1a2?22b1?: 8

?37?

If a1a2?4?b1?.4; then m2 ,0 and m? .0: If a1a2?4 ? b1? # 4; 3a1a2b1 . 8; 9a21a22b21 ? 32a1a2 ?2 2 b1? . 0; then m2 . 0 and m? . 0: When m , m?; f 0?m? , 0; and m . m?; f 0?m? . 0: So we
have f ?m?? is the local minimal of f ?m?:

f ?m???a21a22b1

1 124b1

9 216a1a2

b1?392a1a2

b21

2

52172a21a22

b31

 21 121
82

b1?694a1a2

b21

q?????????????????????????????! 9a21a22b21?32a1a2?22b1? :

?38?

As limm!?1f ?m???1 and f ?0???12a1a2?2$0; if f ?m??,

0; the two nonnegative real roots of Eq. (31) exist. m1[

?0;m?? and m2[?m?;?? 1? Considering v20 ? m0 1 2 m0 . a1a2 2 1:

a1ar?1eam2t0h?e.ro0o;tsthoef

Eq. (31). Hopf bifurcation

If a1a2 , 1; so m1 . a1a2 2 1 and m2 . a1a2 2 1; then m1

and m2 are the Hopf bifurcations of system (1). If a1a2 ? 1; so f ?0? ? ?1 2 a1a2?2 ? 0; i.e. m1 ? 0 and m2 . a1a2 2 1; then m2 is the unique Hopf bifurcation of

system (1).

If a1a2 . 1 and b1 . 1 2 ?1=a1a2?; so f ?a1a2 2 1? ? a21a22?a1a2 2 1??a1a2 2 1 2 a1a2b1? , 0; i.e. m1 , a1a2 2 1 and m2 . a1a2 2 1; then m2 is the unique Hopf bifurcation

of system (1). p2If?aa?1a21?a?a1?2?a22b221.?1p1,?a?1?a?am?2?n22d#1ba?11,#a20122a1n;d?1th=fae?1naa1t2ah?2e;

so f ?p?a?1?a??2? 2 1? 2 1? $ 0; i.e. m1
Hopf bifurcations

? , of

system (1) do not exist. The proof is complete. A

Next, consider the case of b2 – 0: Eq. (27) becomes

a1a2??1 2 b1?m20 2 v20 ? 2im0v0???1 2 b2?m20 2 v20 ? 2im0v0?

? ?m0 ? iv0?4?1 ? iv0?2:

?39?

By separating this equation into real and imaginary parts, one obtains a1a2{??1 2 b1?m20 2 v20???1 2 b2?m20 2 v20? 2 4m20v20}
? ??m20 2 v20?2 2 4m20v20??1 2 v20? 2 8m0v20?m20 2 v20?; ?40?

a1a2m0??2 2 b1 2 b2?m20 2 2v20?

? ?m20 2 v20?2 2 4m20v20 ? 2m0?m20 2 v20??1 2 v20?:

?41?

Therefore,

16?m0 ? 1?8 ? c7?m0 ? 1?7 ? c6?m0 ? 1?6 ? c5?m0 ? 1?5

? c4?m0 ? 1?4 ? c3?m0 ? 1?3 ? c2?m0 ? 1?2

? c1?m0 ? 1? ? c0 ? 0;

?42?

v20

?

m20

d1?m0? d2?m0?

;

?43?

where

c7 ? 16a1a2?2b1b2 2 b1 2 b2?; c6 ? 4a1a2?213b1b2 ? 6?b1 ? b2? 2 16?; c5 ? 4?a1a2?2?22?b1b2?2 2 7b1b2?b1 ? b2? ? 8b1b2
2 2?b1 ? b2?2 ? 8?b1 ? b2?? ? 32a1a2b1b2;

c4 ? ?a1a2?2?28?b1b2?2 ? 62b1b2?b1 ? b2? 2 156b1b2

? 9?b1 ? b2?2 2 56?b1 ? b2? ? 96? 2 8a1a2b1b2;

c3 ? ?a1a2?3?b1 ? b2??4b1b2 2 ?b1 ? b2?2 ? 8?b1 ? b2? 2 16?

? ?a1a2?2b1b2?238b1b2 2 44?b1 ? b2? ? 160?; c2 ? ?a1a2?3?22b1b2?b1 ? b2? ? 20b1b2 ? ?b1 ? b2?3

2 10?b1 ? b2?2 ? 40?b1 ? b2? 2 64?

? ?a1a2?2b1b2?25b1b2 ? 10?b1 ? b2? 2 48?;

c1 ? 24?a1a2?3b1b2?b1 ? b2? 2 8?a1a2b1b2?2;

c0 ? ?a1a2?4?24b1b2 ? ?b1 ? b2?2 2 8?b1 ? b2? ? 16?

? 2?a1a2?3b1b2??b1 ? b2? 2 4? ? ?a1a2b1b2?2; d1?m0? ? ?a1a2?2?2 2 b1 2 b2? ? a1a2b1b2m0?2m0 ? 1?2

? a1a2?b1 ? b2??6m30 ? 12m20 ? 7m0 ? 1?

2 a1a2?16m30 ? 28m20 ? 16m0 ? 4? ? ?10m40

? 32m30 ? 36m20 ? 16m0 ? 2?;

?44?

d2?m0? ? 2?a1a2?2 2 a1a2?b1 ? b2??2m30 ? 3m20 ? m0?

2 a1a2?12m20 ? 16m0 ? 4? ? ?16m50 ? 58m40 ? 80m30

? 52m20 ? 16m0 ? 2?:

Similar to the above discussion, one can easily obtain the following results.

Theorem 2. b2 – 0 and

P(E7i?xi0sctei n?ce16of,H0o;pfthbeinfurtchaetiHonoppfarbaifmuercteart)ioInf

X. Liao et al. / Neural Networks 17 (2004) 545–561

551

parameters m0 of system (8) are the roots of Eq. (42), satisfying ?d1?m0??=?d2?m0?? . 0:

b?1; 44? ? a2?1 2 b2?m~2; b?1; 46? ? 2a2m~; b?1; 48? ? a2; b?1; 56? ? 2a2m~; b?1; 58? ? 2a2;

3. Stability of bifurcating periodic solutions

In order to study the stability of bifurcating periodic solutions, the frequency-domain formulation of Moiola and Chen (1996) is applied.
First, de?ne an auxiliary vector of the form

j1?v~? ?

2wT

?G?iv~; wT v

m~??p1

;

?45?

where m~ is a ?xed value of the parameter m; wT and v are the
left and right eigenvectors of ?G?iv~; m~?J?m~??; respectively, associated with the value l^?iv~; m~?; and





!

p1 ?

D2

V02^v ?

1 2

v^V22

?

1 8

D3v^v^v

;

?46?

where v~ is the frequency of the intersection between the l^ locus and the negative real axis closest to the point ?21 ? i0?; and

D2

?

?2

g?y; ?y2

m~?

y?0

;

D3

?

?3

g?y; ?y3

m~?


y?0

;

V02

?

2

1 4

?I

?

G?0;

m~?J?m~??21G?0;

m~?D2v^v;

V22

?

2

1 4

?I

?

G?2iv~;

m~?J?m~??21G?2iv~;

m~?D2v^v:

?47?

Then, one has

D2

?

f f

00?0? 0?0?

?a?i;

j??2?36;

D3

?

2

f 000?0? f 0?0?

?b?i;

j??2?216;

where

a?1; 8? ? a1?1 2 b1?m~2; a?1; 10? ? 2a1m~; a?1; 12? ? a1;

a?1; 20? ? 2a1m~; a?1; 22? ? 2a1; a?1; 32? ? a1;

a?2; 1? ? a2?1 2 b2?m~2; a?2; 3? ? 2a2m~; a?2; 5? ? a2; a?2; 13? ? 2a2m~; a?2; 15? ? 2a2; a?2; 25? ? a2;

b?1; 68? ? a2; b?1; 116? ? 2a2m~; b?1;118? ? 2a2; b?1; 128? ? 2a2; b?1; 188? ? a2;

b?2; 1? ? a2?1 2 b2?m~2; b?2; 3? ? 2a2m~; b?2; 5? ? a2; b?2; 13? ? 2a2m~; b?2;15? ? 2a2;

b?2; 25? ? a2; b?2; 73? ? 2a2m~; b?2;75? ? 2a2; b?2; 85? ? 2a2; b?2; 145? ? a2;

and the others are zero. Also,

0m 1

v

?

1 d

BBBBBBBBBBBB@

1 imv~ iv~ 2mv~2

CCCCCCCCCCCCA;

0

1

w

?

1 l

BBBBBBBBBBBBB@

ma2?1 2 b2?m~2 a1?1 2 b1?m~2 2ma2m~ 2a1m~ ma2

CCCCCCCCCCCCCA;

?48?

2v~2

a1

where

m

?

2

a1??1 2 b1?m~2 ?m~ ? iv~?2

2 ?1

v~2 ? 2im~v~? ? iv~?l~

;

q??????????????????????????? d ? ?1 ? mm ??1 ? v~2 ? v~4?;

l ? 2a1??1 2 b1?m~2 2 v~2 ? 2im~v~? : d

Moreover,

V02

?

4d2f

f 00?0? 0?0??a1a2?1 2 b1??1

2

b2?

2

1?

0 a1?1 2 b1? ? mm a1a2?1 2 b1??1 2 b2? 1

?

BBBBBBBBBBBB@

a1 0 0 0

a2

?1

2

b1

??1

2

b2?

?

mm a2

?1

2

b2

?

CCCCCCCCCCCCA;

0

552

X. Liao et al. / Neural Networks 17 (2004) 545–561

V22

?

4d2f

0?0?{a1a2??1

2

b1?m~2

2

4v~2

?

4im~v~???1

f 00?0? 2 b2?m~2

2

4v~2

?

4im~v~?

2

?1

?

2iv~?2?m~

?

2iv~?4}

0

1

?

BBBBBBBBBBBBBBBBBBBBBBBBBBB@

a1??1 2 b1?m~2 a2??1 2 b2?m~2 2iv~a1??1 2 b1 2iv~a2??1 2 b2 24v~2a1??1 2

2 4v~2 ? 4im~v~?{?1 ? 2iv~??m~ ? 2iv~?2 ? m2a2??1 2 b2?m~2 2 4v~2 ? 4im~v~?{m2?1 ? 2iv~??m~ ? 2iv~?2 ? a1??1 2 b1?m~2 ?m~2 2 4v~2 ? 4im~v~?{?1 ? 2iv~??m~ ? 2iv~?2 ? m2a2??1 2 b2 ?m~2 2 4v~2 ? 4im~v~?{m2?1 ? 2iv~??m~ ? 2iv~?2 ? a1??1 2 b1 b1?m~2 2 4v~2 ? 4im~v~?{?1 ? 2iv~??m~ ? 2iv~?2 ? m2a2??1 2

2 4v~2 ? 4im~v~?} 2 4v~2 ? 4im~v~?} ?m~2 2 4v~2 ? 4im~v~?} ?m~2 2 4v~2 ? 4im~v~?} b2?m~2 2 4v~2 ? 4im~v~?}

CCCCCCCCCCCCCCCCCCCCCCCCCCCA

?49?

24v~2a2??1 2 b2?m~2 2 4v~2 ? 4im~v~?{m2?1 ? 2iv~??m~ ? 2iv~?2 ? a1??1 2 b1?m~2 2 4v~2 ? 4im~v~?}

Then, set

!

p1 ?

D2?V02^v

?

1 2

v^V22?

?

1 8

D3v^v^v

? p?11? ? p?12? ? p?13?;

in which

0

1

p?13?

?

2

f 000?0? 8d3f 0?0?

@

a1??1 2 b1?m~2 2 v~2 ? 2im~v~? m2m a2??1 2 b2?m~2 2 v~2 ? 2im~v~?

A;

?51?

?50? and

j1?v~? ?

2wT ?G?iv~; m~??p1 wT v

? j?11? ? j?12? ? j?13?;

?52?

0

1

p?11?

?

4d3?f

0?0??2?a1

?f 00?0??2 a2?1 2 b1

??1

2

b2?

2

1?

BB@

a1a2?1 2 b2???1 2 b1?m~2 2 v~2 ? 2im~v~??mm ? a1?1 2 ma1a2?1 2 b1???1 2 b2?m~2 2 v~2 ? 2im~v~??1 ? mm a2?1

b1?? 2 b2??

CCA

p?12?

?

?f 00?0??2a1a2??1 2 b1?m~2 2 4v~2 ? 4im~v~???1 2 b2?m~2 2 4v~2 ? 4im~v~? 8d3?f 0?0??2{a1a2??1 2 b1?m~2 2 4v~2 ? 4im~v~???1 2 b2?m~2 2 4v~2 ? 4im~v~? 2 ?1 ? 2iv~?2?m~ ? 2iv~?4}

0

(

)1

?

BBBBBBBB@

a1??1 2 b1?m~2 2 v~2 ? 2im~v~?

m2?1 ? 2iv~??m~ ? 2iv~?2 a1??1 2 b1?m~2 2 4v~2 ? 4im~v~?

?1

(

m a2??1 2 b2?m~2 2 v~2 ? 2im~v~?

?1 ? 2iv~??m~ ? 2iv~?2

?

a2??1 2 b2?m~2 2 4v~2 ? 4im~v~?

) m2

CCCCCCCCA

with

j?11?

?2

?f 00?0??2ma1a2??1 2 b1?m~2 2 v~2 ? 2im~v~???1 2 b2?m~2 2 v~2 ? 2im~v~? 4ld3?f 0?0??2?a1a2?1 2 b1??1 2 b2? 2 1??1 ? iv~??m~ ? iv~?2

? ?mm a2?1 2 b2? ? a1?1 2 b1? ? ?1 ? mm ?a1a2?1 2 b1??1 2 b2??;

X. Liao et al. / Neural Networks 17 (2004) 545–561

553

j?12?

?2

?f 00?0??2a1a2??1 2 b1?m~2 2 v~2 ? 2im~v~???1 2 b2?m~2 2 v~2 ? 2im~v~? 8ld3?f 0?0??2{a1a2??1 2 b1?m~2 2 4v~2 ? 4im~v~???1 2 b2?m~2 2 4v~2 ? 4im~v~? 2 ?1 ? 2iv~?2?m~ ? 2iv~?4}

? ma1a2??1 2 b1?m~2 2 4v~2 ? 4im~v~???1 2 b2?m~2 2 4v~2 ? 4im~v~? ?1 ? iv~??m~ ? iv~?2

(

)

?

m2?1 ? 2iv~??m~ ? 2iv~?2 a1??1 2 b1?m~2 2 4v~2 ? 4im~v~?

?

m ?1 ? 2iv~??m~ ? 2iv~?2 ma2??1 2 b2?m~2 2 4v~2 ? 4im~v~??

? ?1 ? mm ?

;

j?13? ?

f 000?0?ma1a2??1 2 b1?m~2 2 v~2 ? 2im~v~???1 2 b2?m~2 2 v~2 ? 2im~v~? 8ld3f 0?0??1 ? iv~??m~ ? iv~?2

? ?1 ? mm ?:

?53?

Since l~ is the eigenvalue of ?G?iv~; m~?J?m~??; one has

l~2

2

a1

a2??1

2

b1

?m~2

2

v~2 ?2im~v~???12b2 ?1?iv~?2?m~ ?iv~?4

?m~2

2

v~2

?

2im~v~?

? 0:

?54?

Considering

m

?

2

a1

??1 2 b1 ?m~2 ?m~ ?iv~?2

2 ?1

v~2 ?2im~v~? ? iv~?l~

;

q???????????????????????????

d ? ?1?mm ??1?v~2 ?v~4?;

l ? 2a1??12b1?m~2 2v~2 ?2im~v~? ; d

one obtains

j?13?

?

2

16?1

l~f 000?0? ?v~2 ?v~4?f

0?0?

:

?55?

Now, the following Hopf Bifurcation Theorem formulated in the frequency domain can be established.

Lemma 2. (Moiola & Chen, 1996) Suppose that the locus of the distinguished characteristic function l^?s? intersects the negative real axis at l^?iv~? that is closest to the point ?21 ?
i0? when the variable s sweeps on the classical Nyquist
contour. Moreover, suppose that j1?v~? is nonzero and the half-line L1 starting from ?21 ? i0? in the direction de?ned by j1?v~? ?rst intersects the locus of l^?iv? at l^?iv^? ? P^ ? 21 ? j1?v~?u2; where u ? O?lm 2 m0l1=2?: Finally, suppose
that following conditions are satis?ed

(i) The eigenlocus l^ has nonzero rate of change with

respect to its parameterization at the criticality

?v0; m0?; i.e.,

"

M?v0; m0? ? det

?F1=?m ?F1=?v

?F2 ?F2

=?m =?v

#
?v0

;m0

?



0;

?56?

where

F1?v; m? ? R{h?21; iv; m?}; F2?v; m? ?

I{h?21; iv; m?}:

(ii) The intersection is transversal, i.e.

2 N?v^; m~? ? det664 RR({jdd1?vlv^^?v}?v^)

3 II{(jdd1?vlv^^?v}?v^) 775

– 0:

?57?

(iii) There are no other intersections between any of the characteristic loci and the line segment joining the point ?21 ? i0? to P^ ; at least within a small neighborhood of radios d . 0:
Then, the system (17) has a periodic solution x?t? of frequency v ? v^ ? O?u^4?: Moreover, by applying a small perturbation around the intersection P^ and using the generalized Nyquist stability criterion, the stability of the periodic solution x?t? can be determined.

According to Lemma 2, one can determine the direction of Hopf bifurcation and the stability of the bifurcating periodic solution by drawing the ?gure of the half-line L1 and the locus l^?iv?: The analysis is carried out as follows (Moiola & Chen, 1996)

1. If the half-line L1 ?rst intersects the locus of l^?iv? when m~ . m0?, m0?; then the bifurcating periodic solution exists and the Hopf bifurcation is supercritical (sub-
critical).
2. If the total number of anticlockwise encirclements of the point P1 ? P^ ? ej1?v~?; for a small enough e . 0; is equal to the number of poles of l?s? that have positive
real parts, then the limit cycle is stable; otherwise, it is
unstable.

One can perturb the bifurcation parameter m slightly from m0 to m~:. If l~ . 21 and

If

dl^ dv

jv?v^gN?v^;

m~?

.

0;

554

X. Liao et al. / Neural Networks 17 (2004) 545–561

or l~ , 21 and

If

dl^ dv

jv?v^gN?v^;

m~?

,

0;

then the half-line L1 intersects the locus of l^?iv?:. Consequently, one obtains the following result.

Theorem 3. Set

s ? sgn

dl~ dm

m?m0

( I

dl^ dv


v?v0

)! N

?v0

;

m0

?;

?58?

where

dl~ dm


m?m0

?

k1?m0? k2?m0?

;

k1?m0? ?128?m0 ? 1?7 ? 7c7?m0 ? 1?6 ? 6c6?m0 ? 1?5

? 5c5?m0 ? 1?4 ? 4c4?m0 ? 1?3 ? 3c3?m0 ? 1?2

? 2c2?m0 ? 1? ? c1;

?59?

k2?m0? ? 128?m0 ? 1?8 ? 6c7?m0 ? 1?7 ? 6c6?m0 ? 1?6

? ?4c5 ? 64c??m0 ? 1?5 ? ?4c4 2 16c??m0 ? 1?4

? ?2c3 ? 2c32??m0 ? 1?3 ? ?2c2 ? 2c22??m0 ? 1?2

? ?2c1 2 16c2??m0 ? 1? ? ?2c03 ? 4c2?;

c7; c6; c5; c4; c3; c2; c1; c0 are de?ned in Eq. (44), c32 ? ?a1a2?2b1b2?238b1b2 2 44?b1 ? b2? ? 160?; c22 ? ?a1a2?2 b1b2?25b1b2 ? 10?b1 ? b2? 2 48?; c03 ? 2?a1a2?3b1b2??b1 ?

b2? 24?; with c ? a1a2b1b2;

and

R

dl dv


v?v0

!

?

2

v0

(

??1

2

?1 ? b1?m20 b1?m20 2 v20

? ?2

v20 ? 4m20

v20

?

??1

2

?1 ? b2?m20 ? v20 b2?m20 2 v20?2 ? 4m20v20

2

1

1 ? v20

)

2 2 m20 ? v20

;

?60?

I

dl dv


v?v0

!

?

( 2

??1

?1 2 b1 2 b1?m20

?m30 ? m0v20 2 v20?2 ? 4m20

v20

?

??1

?1 2 b2?m30 ? m0v20 2 b2?m20 2 v20?2 ? 4m20v20

2

1

1 ? v20

)

2

2m0 m20 ? v20

:

Then

2. if s , 0; the Hopf bifurcation at m ? m0 of system (8) is subcritical.

For more details, see Appendix A. Now, set f ?u? ? tan h?u?: Then, f 0?0? ? 1; f 00?0? ? 0; f 000?0? ? 22: Therefore, in Eq. (55), j?11? ? 0; j?12? ? 0; and

j1?v? ? j?13?

?

8?1

?

l~ v2

?

v4?

:

?61?

Setting m ? m0; v ? v0; l~ ? 21; one can calculate the

following

N?v0;

m0?

(

?R{j1?v0?}I

dl^ dv

(

2 I{j1?v0?}R



)

ddvvl?^ vv0?v0 )

and s ? sgn

dl~ dm


m?m0

( I

dl^ dv


v?v0

)! N

?v0

;

m0

?:

Considering

j1?v0?

?

2

8?1

?

1 v20

?

v40?

to be a negative real, i.e.

R{j1?v0?}

?

2

8?1

?

1 v20

?

v40?

, 0;

I{j1?v0?} ? 0;

one has

sgn?s? ? 2sgn

dl~ dm


m?m0

!

if

( I

dl dv


v?v0

)



0:

Corollary 1. Let

f ?u? ? tan h?u?

and

( I

dl dv



)

v?v0



0;

with

s1

?

2

dl~ dm



?

m?m0

2

k1?m0? k2?m0?

;

?62?

where k1?m0?; k2?m0? are de?ned in Eq. (59). Then

1. If s1 . 0; the Hopf bifurcation at m ? m0 of system (8) is supercritical;
2. If s1 , 0; the Hopf bifurcation at m ? m0 of system (8) is subcritical.

1. if s . 0; the Hopf bifurcation at m ? m0 of system (8) is supercritical;

According to the following equation, one obtains the intersection between the l^ locus and the negative real axis

X. Liao et al. / Neural Networks 17 (2004) 545–561

555

Fig. 1. a1 ? 1:5; a2 ? 0:5; b1 ? 3; b2 ? 0; m ? 0:045: The half-line L1 does not intersect the locus l^?iv?; so no periodic solution exists.

Fig. 2. a1 ? 1:5; a2 ? 0:5; b1 ? 3; b2 ? 0; m ? 0:055: The half-line L1 intersects the locus l^?iv?; and k ? 0; so a stable periodic solution exists.

closest to the point ?21 ? i0?; i.e. l~ (see Appendix A) 16?m~ ? 1?8l~8 ? ?c7?m~ ? 1?7 ? c6?m~ ? 1?6 ? 32c?m~ ? 1?5
2 8c?m~ ? 1?4?l~6 ? ??c5 2 32c??m~ ? 1?5

v~2

?

m~2

d10?m~?l~4 d20?m~?l~4

? ?

d11?m~?l~2 d21?m~?l~2

? ?

d12 d22

;

?64?

where c7; c6; c5; c4; c3; c2; c1; c0; c32; c22; c03; c are de?ned in Eq. (59), and

? ?c4 ? 8c??m~ ? 1?4 ? c32?m~ ? 1?3 ? c22?m~ ? 1?2

d12 ? ?a1a2?2?2 2 b1 2 b2?;

2 8c2?m~ ? 1? ? c2?l~4 ? ??c3 2 c32??m~ ? 1?3

d11?m~? ? a1a2?b1b2m~?2m~ ? 1?2 ? ?b1 ? b2??6m~3 ? 12m~2

? ?c2 2 c22??m~ ? 1?2 ? ?c1 ? 8c2??m~ ? 1? ? c03?l~2

? 7m~ ? 1? 2 ?16m~3 ? 28m~2 ? 16m~ ? 4??;

? ?c0 2 c03 2 c2? ? 0;

?63?

d10?m~? ? 10m~4 ? 32m~3 ? 36m~2 ? 16m~ ? 2;

Fig. 3. a1 ? 1:5; a2 ? 0:5; b1 ? 3; b2 ? 0; m ? 0:65: The half-line L1 intersects the locus l^?iv?; and k ? 0; so a stable periodic solution exists.

Fig. 4. a1 ? 1:5; a2 ? 0:5; b1 ? 3; b2 ? 0; m ? 0:75: The half-line L1 does not intersect the locus l^?iv?; so no periodic solution exists.

556

X. Liao et al. / Neural Networks 17 (2004) 545–561

Fig. 5. a1 ? 2; a2 ? 1; b1 ? 3; b2 ? 0; m ? 4:5: The half-line L1 intersects the locus l^?iv? and k ? 0; so a stable periodic solution exists.

Fig. 6. a1 ? 2; a2 ? 1; b1 ? 3; b2 ? 0; m ? 5:. The half-line L1 does not intersect the locus l^?iv?; so no periodic solution exists.

d22 ? 2?a1a2?2; d21 ?m~??2a1 a2 ??b1 ?b2 ??2m~3 ?3m~2 ?m~???12m~2 ?16m~?4??;

d20 ?m~??16m~5 ?58m~4 ?80m~3 ?52m~2 ?16m~ ?2:

?65?

Then, one draws the half-line j1?v~? starting from ?21?i0? and the locus l^?iv?; and obtains the total number k of anticlockwise encirclements of the point P1?P^ ?ej1?v~? for a small enough e.0:

According to Eq. (26), one has

?l?s??2

?

a1

a2

??1

2

b1

?m2

?s2 ?2ms???12 ?s ? m?4 ?s ? 1?2

b2?m2

? s2

?

2ms?

:

?66?

Hence, s?2m and s?21 are the poles of l?s?; and the number of poles of l?s? that have positive real parts is zero.

Corollary 2. Let k be the total number of anticlockwise encirclements of the point P1 ? P^ ? ej1?v~? for a small

Fig. 7. a1 ? 2; a2 ? 1:5; b1 ? 0:8; b2 ? 1:1; m ? 2:. The half-line L1 intersects the locus l^?iv?; and k ? 0; so a stable periodic solution exists.

Fig. 8. a1 ? 2; a2 ? 1:5; b1 ? 0:8; b2 ? 1:1; m ? 2:1: The half-line L1 does not intersect the locus l^?iv?; so no periodic solution exists.

X. Liao et al. / Neural Networks 17 (2004) 545–561

557

Fig. 9. a1 ? 0:5; a2 ? 1:8; b1 ? 0:8; b2 ? 0:3; m ? 0:01: The half-line L1 does not intersect the locus l^?iv?; so no periodic solution exists.

Fig. 10. a1 ? 0:5; a2 ? 1:8; b1 ? 0:8; b2 ? 0:3; m ? 0:025: The half-line L1 intersects the locus l^?iv?; and k ? 0; so a stable periodic solution exists.

enough e . 0; where P^ is the intersection of the half-line L1 and the locus l^?iv?: Then
1. if k ? 0; the bifurcating periodic solutions of system (8) is stable;
2. if k – 0; the bifurcating periodic solutions of system (8) is unstable.
4. Numerical examples
In this section, some numerical examples of system (8), with Eq. (9) at different values of a1; a2; b1 and b2; are

discussed. By Corollary 1, s1 determines the direction of a Hopf bifurcation. If s1 . 0; the Hopf bifurcation is supercritical; if s1 , 0; the Hopf bifurcation is subcritical. The half-line L1 and the locus l^?iv? are shown in the corresponding frequency graphs. If they intersect, a limit
cycle exists, or else, no limit cycle exists. Corollary 2 implies
that the stabilities of the bifurcating periodic solutions are
determined by the total number k of anticlockwise encirclements of the point P1 ? P^ ? ej1?v~? for a small enough e . 0: Suppose that the half-line L1 and the locus l^?iv? intersect. If k ? 0; the bifurcating periodic solutions is stable; if k – 0;
the bifurcating periodic solutions is unstable.

Fig. 11. a1 ? 0:5; a2 ? 1:8; b1 ? 0:8; b2 ? 0:3; m ? 0:14: The half-line L1 intersects the locus l^?iv?; and k ? 0; so a stable periodic solution exists.

Fig. 12. a1 ? 0:5; a2 ? 1:8; b1 ? 0:8; b2 ? 0:3; m ? 0:17: The half-line L1 does not intersect the locus l^?iv?; so no periodic solution exists.

558

X. Liao et al. / Neural Networks 17 (2004) 545–561

Fig. 13. a1 ? 0:1; a2 ? 2; b1 ? 1:01; b2 ? 30; m ? 0:15: The half-line L1 does not intersect the locus l^?iv?; so no periodic solution exists.

Fig. 14. a1 ? 0:1; a2 ? 2; b1 ? 1:01; b2 ? 30; m ? 0:2: The half-line L1 intersects the locus l^?iv?; and k ? 0; so a stable periodic solution exists.

In order to verify the theoretical analysis results derived
above, system (8) is simulated with Eq. (9) in different cases.
(i) Let a1 ? 1:5; a2 ? 0:5; b1 ? 3; b2 ? 0: Then, m0 ? 0:0502 or m0 ? 0:6894:.
Choose m0 ? 0:0502: Then s1 ? 0:9454 . 0: Hence, m0 ? 0:0502 is a supercritical Hopf bifurcation point.
Set m0 ? 0:6894: Then s1 ? 20:1685 , 0: Hence, m0 ? 0:6894 is a subcritical Hopf bifurcation point.

(ii) Let a1 ? 2; a2 ? 1; b1 ? 3; b2 ? 0: Then, m0 ? 4:6217:
Then s1 ? 20:0826 , 0; so m0 ? 4:6217 is a subcritical Hopf bifurcation point.
(iii) Let a1 ? 2; a2 ? 1:5; b1 ? 0:8; b2 ? 1:1: Then m0 ? 2:0521:
Then s1 ? 20:2761 , 0; so m0 ? 2:0521 is a subcritical Hopf bifurcation point.

Fig. 15. a1 ? 0:1; a2 ? 2; b1 ? 1:01; b2 ? 30; m ? 1:5: The half-line L1 intersect the locus l^?iv?; and k ? 0; so a stable periodic solution exists. Fig. 16. a1 ? 0:1; a2 ? 2; b1 ? 1:01; b2 ? 30; m ? 1:7: The half-line L1 does not intersect the locus l^?iv?; so no periodic solution exists.

X. Liao et al. / Neural Networks 17 (2004) 545–561

559

(iv) Let a1 ? 0:5; a2 ? 1:8; b1 ? 0:8; b2 ? 0:3: Then m0 ? 0:0188 or m0 ? 0:1468:
Choose m0 ? 0:0188: Then s1 ? 0:8876 . 0; so m0 ? 0:0188 is a supercritical Hopf bifurcation point.
Set m0 ? 0:1468: Then s1 ? 20:2949 , 0; so m0 ? 0:1468 is a subcritical Hopf bifurcation point.
(v) Let a1 ? 0:1; a2 ? 2; b1 ? 1:01; b2 ? 30: Then m0 ? 0:1640 or m0 ? 1:5726:
Choose m0 ? 0:1640: Then s1 ? 1:3091 . 0; so m0 ? 0:1640 is a supercritical Hopf bifurcation point.
Set m0 ? 1:5726: Then s1 ? 20:2111 , 0; so m0 ? 1:5726 is a subcritical Hopf bifurcation point (Figs. 1– 16)
(Enun 2– 6).

Acknowledgements
The authors would like to thank two referees for helpful suggestions and comments. The work described in this paper was supported by grants from the National Natural Science Foundation of China (No. 60271019), the Doctorate Foundation of the Ministry of Education of China (No. 20020611007), the Applied Basic Research Grants the Committee of Science and Technology of Chongqing (No. 7370), and the Hong Kong Research Grants Council under the CERG Grant CityU 1115/03E.

5. Conclusions

A two-neuron model with distributed delay and a strong
kernel has been studied from the frequency-domain
approach, which turns out to be not so mathematically
involved and not so dif?cult as analyzing the model in the
time domain (Liao et al., 2001a; Hale and Kocak, 1991;
Hassard et al., 1981). By using the average 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 and direction of the bifurcating
periodic orbits have also been analyzed by drawing the amplitude locus, L1; and the locus, l^?iv?; in a neighborhood of the Hopf bifurcation point. Parameter s or s1 was used to determine the direction of the Hopf bifurcation: if s . 0; the Hopf bifurcation is supercritical; if s , 0; the Hopf bifurcation is subcritical; but if s1 ? 0; (the H10 degeneracy) one cannot decide the direction of the bifurcating periodic orbits by only using L1 and l^?iv?: In this case, one has to resort to an more advanced algorithm including the
forth-, sixth- and even eighth-order harmonic balance approximations, i.e. the amplitude loci L2; L3 and L4; respectively, to ?nd the corresponding solutions ?v^2; u^2?; ?v^3; u^3? and ?v^4; u^4?; as carried out by Moiola and Chen (1996). The fundamental equations about the amplitude loci L2; L3 and L4 are

l^?iv? ? 21 ? j1?v~?u2 ? j2?v~?u4;

?67?

l^?iv? ? 21 ? j1?v~?u2 ? j2?v~?u4 ? j3?v~?u6;

?68?

l^?iv? ? 21 ? j1?v~?u2 ? j2?v~?u4 ? j3?v~?u6

? j4?v~?u8:

?69?

By applying these high-order Hopf bifurcating formulas, one can expect to obtain more accurate results and the globe bifurcating behavior. Since this task is computationally intensive, it is beyond the scope of the present paper and will be further investigated elsewhere in the near future.

Appendix A

A: Computing the eigenvalue l~: Consider Eq. (54) again

l~2

2

a1

a2

??1

2

b1

?m~2

2

v~2 ?2im~v~???12b2 ?1?iv~?2?m~ ?iv~?4

?m~2

2

v~2

?

2im~v~?

? 0:

Notice that l~? l^?iv~;m~? is a real number. By separating Eq. (54) into real and imaginary parts, one has
a1a2{??12b1?m~2 2v~2???12b2?m~2 2 v~2?24m~2v~2}
? l~2{??m~2 2v~2?2 24m~2v~2??12 v~2?28m~v~2?m~2 2v~2?}; ?A1?

a1a2m~??22b1 2b2?m~2 22v~2? ? l~2??m~2 2v~2?2 24m~2v~2 ? 2m~?m~2 2v~2??12 v~2??:

?A2?

Therefore, one obtains the following equation

16?m~?1?8l~8??c7?m~?1?7 ?c6?m~?1?6?32c?m~?1?5 28c?m~?1?4?l~6???c5 232c??m~?1?5??c4?8c??m~?1?4 ?c32?m~?1?3 ?c22?m~?1?2 28c2?m~?1??c2?l~4 ???c3 2c32??m~?1?3 ??c2 2c22??m~?1?2 ??c1 ?8c2??m~?1??c03?l~2 ??c0 2c03 2c2??0; ?A3?

v~2

?

m~2

d10 d20

?m~?l~4 ?m~?l~4

?d11 ?d21

?m~?l~2 ?m~?l~2

?d12 ?d22

:

?A4?

560

X. Liao et al. / Neural Networks 17 (2004) 545–561

where c7;c6;c5;c4;c3;c2;c1;c0 are de?ned in Eq. (44), and c32??a1a2?2b1b2?238b1b2244?b1?b2??160?;

c22??a1a2?2b1b2?25b1b2?10?b1?b2?248?;

c03?2?a1a2?3b1b2??b1?b2?24?; c?a1a2b1b2;

d22?2?a1a2?2;

d21?m~??2a1a2??b1?b2??2m~3?3m~2?m~?

??12m~2?16m~?4??;

?A5?

d20?m~??16m~5?58m~4?80m~3?52m~2?16m~?2;

d12??a1a2?2?22b12b2?;

d11?m~??a1a2?b1b2m~?2m~?1?2??b1?b2?

??6m~3?12m~2?7m~?1?

2?16m~3?28m~2?16m~?4??;

d10?m~??10m~4?32m~3?36m~2?16m~?2:

According to (A3), one can compute the eigenvalue l~:

B: Computing ?dl~=dm?m?m0 : By using m instead of m~ in Eq. (A3), and by taking the

derivative with respect to m on both sides of Eq. (A3), and

setting m ?

dl~ dm


m?m0

?

m0; l~ ?

k1?m0? k2?m0?

;

21;

one

obtains

?A6?

where

k1?m0? ? 128?m0 ? 1?7 ? 7c7?m0 ? 1?6 ? 6c6?m0 ? 1?5

? 5c5?m0 ? 1?4 ? 4c4?m0 ? 1?3 ? 3c3?m0 ? 1?2

? 2c2?m0 ? 1? ? c1;

k2?m0? ? 128?m0 ? 1?8 ? 6c7?m0 ? 1?7 ? 6c6?m0 ? 1?6

? ?4c5 ? 64c??m0 ? 1?5 ? ?4c4 2 16c??m0 ? 1?4

? ?2c3 ? 2c32??m0 ? 1?3 ? ?2c2 ? 2c22??m0 ? 1?2

? ?2c1 2 16c2??m0 ? 1? ? ?2c03 ? 4c2?:
C: Computing the real and imaginary parts of ?dl=dv?v?v0 :
Consider Eq. (54)

l2

2

a1

a2??1

2

b1

?m~2

2

v2 ?2im~v???12b2 ?1?iv?2?m~ ?iv?4

?m~2

2

v2

?

2im~v?

? 0:

Fix m at m~; and take the derivative with respect to v on

both sides of Eq. (54). One has

dl dv

 ?
v?v~

l~ 2

22v~?2im~

?

22v~?2im~

?12b1?m~22v~2?2im~v~ ?12b2?m~22v~2?2im~v~

!

2 1?2iiv~ 2

4i m~?iv~

:

?A7?

Consequently,

R

ddvl v?v~!?

( l~ 2?1?b1?m~2v~?2v~3 2 ??12b1?m~22v~2?2?4m~2

v~2

?

2?1?b2?m~2v~?2v~3 ??12b2 ?m~2 2v~2 ?2 ?4m~2 v~2

2

2v~ 1?v~2

)

4v~ 2 m~2 ?v~2

;

I

ddvl v?v~!?

( l~ 2

2?12b1?m~3?2m~v~2 ??12b1 ?m~2 2v~2 ?2 ?4m~2

v~2

?

2?12b2?m~3?2m~v~2 ??12b2 ?m~2 2v~2 ?2 ?4m~2 v~2

2

2 1?v~2

)

2m~24?m~v~2 :

?A8? ?A9?

Then, using m0 instead of m~; v0 instead of v~; and 2 1 instead

of l~;, one obtains

R

ddvl v?v~!?2v0

(

?1?b1?m20?v20 ??12b1?m202v20?2?4m20v20

?

?1?b2?m20?v20

??12b2 ?m20 2v20 ?2 ?4m20 v20

)

1

2

21?v20 2m20?v20

;

?A10?

I

ddvl v?v~!?2(

?12b1 ?m30 ?m0 v20 ??12b1 ?m20 2v20 ?2 ?4m20

v20

? ?12b2?m30?m0v20 ??12b2 ?m20 2v20 ?2 ?4m20 v20 )

2

1 1?v20

2

2m0 m20 ?v20

:

?A11?

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