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

Neural Networks 17 (2004) 545–561 www.elsevier.com/locate/neunet

Bifurcation analysis on a two-neuron system with distributed delays in the frequency domain Xiaofeng Liaoa,b,*, Shaowen Lib,c, Guanrong Chend a Department of Computer Science and Engineering, Chongqing University, Chongqing 400044, China College of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu 610054, China c Department of Mathematics, Southwestern University of Finance and Economics, China d Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, China

b

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 (Hopfield, 1984). Lacking the ability in tackling the intrinsic complexities, neural network models under investigation today have been dramatically simplified (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 simplified 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: [email protected] (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), dxðtÞ ¼ 2xðtÞ þ a tanh½xðtÞ 2 bxðt 2 tÞ 2 c; ð1Þ dt ð1 dxðtÞ ¼ 2xðtÞ þ a tanh½xðtÞ 2 b FðsÞxðt 2 sÞds 2 c; dt 0 ð2Þ where a denotes the range of the continuous variable xð·Þ; while b can be considered as a measure of the inhibitory influence from the past history, c is a off-set constant, t is the time delay, and Fð·Þ is a kernel function. Using Lyapunov functionals, Gopalsamy and Leung (1997) obtained some necessary and sufficient 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

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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 influence 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 dxðtÞ > > < dt ¼ 2xðtÞ þ a tanh½c1 yðt 2 tÞ; ð3Þ > > : dyðtÞ ¼ 2yðtÞ þ a tanh½2c2 xðt 2 tÞ; dt 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 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-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 twoneuron 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 ðt  dxðtÞ > > ¼ 2xðtÞ þ a tanh kðt 2 sÞyðsÞds ; > < dt 21 ð4Þ ðt  > > > dyðtÞ ¼ 2yðtÞ þ a tanh kðt 2 sÞxðsÞds ; : dt 21 in which a is a positive constant and the delay kernel k is assumed to satisfy the following 8 > < k : ½0; þ1Þ ! ½0; þ1Þ; k is piecewise continuous and ð1 ð1 ð5Þ > kðsÞds ¼ 1; skðsÞds , þ1: : 0

0

Some sufficient 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 ðtÞ > > < dt ¼ 2x1 ðtÞ þ a1 tanh½x2 ðt 2 t1 Þ; ð6Þ > > : dx2 ðtÞ ¼ 2x2 ðtÞ þ a2 tanh½x1 ðt 2 t2 Þ; dt

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 reflect the finite 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 ðtÞ > > < dt ¼ 2x1 ðtÞ þ a11 f ðx1 ðt 2 t1 ÞÞ þ a12 f ðx2 ðt 2 t2 ÞÞ; > > : dx2 ðtÞ ¼ 2x2 ðtÞ þ a21 f ðx1 ðt 2 t1 ÞÞ þ a22 f ðx2 ðt 2 t2 ÞÞ; dt ð7Þ for which several cases, such as t1 ¼ t2 ; a11 ¼ a22 ¼ 0; etc. were discussed. They obtained some sufficient 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 p   ð1 dx1 ðtÞ > p p p p > ¼ 2x ðtÞ þ a f x ðtÞ 2 b FðrÞx ðt 2 rÞdr 2 c ; > 2 1 1 1 2 2 < dt 0   ð1 > dxp ðtÞ > > : 2 ¼ 2xp2 ðtÞ þ ap2 f xp1 ðtÞ 2 b1 FðrÞxp1 ðt 2 rÞdr 2 c2 ; dt 0 ð8Þ where api ; bi and ci ði ¼ 1;2Þ are nonnegative constants. In this model, xpi ði ¼ 1;2Þ denote the mean soma potential of the neuron, api corresponds to the range of the continuous variable xpi ; bi are measures of the inhibitory influence 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

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 first 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 difficult 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

547

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 FðrÞ ¼ m2 re2mr ; m . 0:

ð9Þ

For simplicity, set c1 ¼ c2 ¼ 0: In system (8), let 8 ð1 > p > FðrÞxp1 ðt 2 rÞdr; > < x1 ðtÞ ¼ x1 ðtÞ 2 b1 0

ð10Þ

ð1 > > p > FðrÞxp2 ðt 2 rÞdr: : x2 ðtÞ ¼ x2 ðtÞ 2 b2 0

Then, system (8) is equivalent to the following model 8 ð0 dx1 ðtÞ > > ¼2x1 ðtÞþap1 f ½x2 ðtÞ2ap1 b1 Fð2rÞf ½x2 ðtþrÞdr; > < dt 21 ð0 > dx ðtÞ > > : 2 ¼2x2 ðtÞþap2 f ½x1 ðtÞ2ap2 b2 Fð2rÞf ½x1 ðtþrÞdr: dt 21 ð11Þ Assume that f [ C4 ðRÞ; f ð0Þ ¼ 0; and uf ðuÞ . 0 for u – 0:

ð12Þ

Then, by a result of Liao et al. (2001a), the equilibrium (0,0) of Eq. (11) exists if ap1 ap2 lð1 2 b1 Þð1 2 b2 Þl ,

1 ½f 0 ð0Þ2

:

Since FðrÞ ¼ m2 re2mr ; one has ð0 21

¼

Fð2rÞf ½x2 ðtþrÞdr

ðt 21

m2 ðt2sÞemðs2tÞ f ½x2 ðsÞds

 ðt ¼ m2 e2mt t 21

ems f ½x2 ðsÞds2

ðt 21

sems f ½x2 ðsÞds:



ð13Þ

Taking the derivative with respective to t on both sides of Eq. (11), and using Eq. (13), one obtains 8 2 > d x1 ðtÞ dx1 ðtÞ p 0 dx2 ðtÞ > > > dt2 ¼2 dt þa1 f ½x2 ðtÞ dt > > >  > ðt > dx1 ðtÞ > p > ems f ½x2 ðsÞds; þx1 ðtÞ2a1 f ½x2 ðtÞ 2ap1 b1 m2 e2mt >2m < dt 21 > > d2 x2 ðtÞ dx ðtÞ dx ðtÞ > > ¼2 2 þap2 f 0 ½x1 ðtÞ 1 > 2 > dt dt dt > >  > ðt > dx2 ðtÞ > > :2m ems f ½x1 ðsÞds: þx2 ðtÞ2ap2 f ½x1 ðtÞ 2ap2 b2 m2 e2mt dt 21 ð14Þ

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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’, u¼

8   > d3 x1 ðtÞ dx1 ðtÞ d2 x1 ðtÞ p dx2 ðtÞ p 0 d2 x2 ðtÞ p 00 dx2 ðtÞ 2 2 2 2 p 0 > > 2ð2 þa ¼2 m x ðtÞ2ð m þ2 m Þ m þ1Þ þa ð12b Þ m f ½x ðtÞþ2a m f ½x ðtÞ f ½x ðtÞ þa f ½x ðtÞ ; 1 1 1 1 1 1 2 2 2 2 > < dt3 dt dt dt dt2 dt2   > > d3 x2 ðtÞ dx2 ðtÞ d2 x2 ðtÞ p dx1 ðtÞ p 0 d2 x1 ðtÞ p 00 dx1 ðtÞ 2 > 2 2 2 p 0 > 2ð2 þa ¼2 m x ðtÞ2ð m þ2 m Þ m þ1Þ þa ð12b Þ m f ½x ðtÞþ2a m f ½x ðtÞ f ½x ðtÞ þa f ½x ðtÞ : : 2 2 2 2 2 2 1 1 1 1 dt dt dt dt3 dt2 dt2 ð15Þ

By setting x3 ðtÞ ¼ dx1 ðtÞ=dt; x4 ðtÞ ¼ dx2 ðtÞ=dt; x5 ðtÞ ¼ d2 x1 ðtÞ=dt2 and x6 ðtÞ ¼ d2 x2 ðtÞ=dt2 ; one arrives at the following ODE system 8 dx1 > > > > dt > > > > dx > 2 > > > > dt > > > > dx3 > > > > dt > > > > < dx4 dt > > > > dx > 5 > > > dt > > > > > > > > > > dx6 > > > > dt > > > :

¼ x3 ;

0

¼ x5 ; ¼ x6 ;

: 2

¼ 2m x1 2 ðm

2

þ 2mÞx3 2 ð2m þ 1Þx5 þ ap1 ð1 2 b1 Þm2 f ðx2 Þ

þ2ap1 mf 0 ðx2 Þx4 þ ap1 f 0 ðx2 Þx6 þ ap1 f 00 ðx2 Þx24 ; 2

¼ 2m x2 2 ðm

2

ð20Þ

ð16Þ

The non-linear system (16) can be rewritten in a matrix form as dx ¼ AðmÞx þ HðxÞ; dt

ð17Þ

where x ¼ ðx1 ; x2 ; x3 ; x4 ; x5 ; x6 ÞT ; 0

B B 0 B B B 0 B AðmÞ¼ B B B 0 B B B 2m2 @

0

1

0

0

0

0

0

1

0

0

0 0

0

0

1

0

0

0

0

1

0

2ð2m þ1Þ

0

2

0 2ðm þ2mÞ 2

0 2m

0

2

2ðm þ2mÞ

0

0 0

B B0 B B B0 B B¼B B B0 B B B1 @

1

C 0C C C 0C C C; C 0C C C 0C A

C ¼ I;

ð21Þ

0 1

þ 2mÞx4 2 ð2m þ 1Þx6 þ ap2 ð1 2 b2 Þm2 f ðx1 Þ

þ2ap2 mf 0 ðx1 Þx3 þ ap2 f 0 ðx1 Þx5 þ ap2 f 00 ðx1 Þx23

0

8 dx > > ¼AðmÞxþBu; > < dt y¼2Cx; > > > : u¼gðy;mÞ;

¼ x4 ;

0

gðy;mÞ; one obtains a linear system with a non-linear feedback, as follows

1 C C C C C C C; C C C C C A

2ð2m þ1Þ ð18Þ 1

0 C B C B0 C B C B C B0 C B C: HðxÞ¼B C B C B0 C B C B p Ba1 ð12b1 Þm2 f ðx2 Þþ2ap1 mf 0 ðx2 Þx4 þap1 f 0 ðx2 Þx6 þap1 f 00 ðx2 Þx24 C A @ ap2 ð12b2 Þm2 f ðx1 Þþ2ap2 mf 0 ðx1 Þx3 þap2 f 0 ðx1 Þx5 þap2 f 00 ðx1 Þx23 ð19Þ

u¼gðy; mÞ 1 0 p a1 ð12b1 Þm2 f ð2y2 Þ22ap1 mf 0 ð2y2 Þy4 2ap1 f 0 ð2y2 Þy6 C B C B þap1 f 00 ð2y2 Þy24 C B C: ¼B C B p B a2 ð12b2 Þm2 f ð2y1 Þ22ap2 mf 0 ð2y1 Þy3 2ap2 f 0 ð2y1 Þy5 C A @ þap2 f 00 ð2y1 Þy23

ð22Þ

Next, taking a Laplace transform on Eq. (20) yields a standard transfer matrix of the linear part of the system Gðs; mÞ¼C½sI 2AðmÞ21 B 0

s B B0 B B B0 B ¼B B B0 B B 2 Bm @

0 21

0

0

0

s 0

21

0

0

0 s

0

21

0

s

0

21

0 0 2

0 m þ2m 0 2

sþ2m þ1 0 2

m þ2m 0 1 1 0 C B B0 1 C C B C B Bs 0 C C B 1 C: B ¼ C ðsþ mÞ2 ðsþ1Þ B B0 s C C B B 2 C Bs 0 C A @ 0 m 0

sþ2m þ1

1

121 0

0

C C C C C C C C C C C C A

C 0C C C 0C C C C 0C C C 0C A

0 B B0 B B B0 B B B B0 B B B1 @

01

0

0 s2 ð23Þ

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

To this end, if this feedback system is linearized about the equilibrium y¼0; then the Jacobian is given by  ›g  JðmÞ¼  ›y y¼0 ! 0 2a1 ð12b1 Þm2 0 22a1 m 0 2a1 ; ¼ 2a2 ð12b2 Þm2 0 22a2 m 0 2a2 0 ð24Þ

a1 a2 m0 ¼ m20 2 v20 þ m0 ð12 v20 Þ:

0

B B B 2a2 ð1 2 b2 Þm2 B B B0 1 B ¼ B 2 2 ðs þ mÞ ðs þ 1Þ B B 2a2 ð1 2 b2 Þm s B B B0 @

2 2

2a2 ð1 2 b2 Þm s

ð1þ m0 Þ4 2a1 a2 b1 ð1þ m0 Þ3 2a1 a2 ð22b1 Þð1þ m0 Þ2 þða1 a2 Þ2 ¼0; ð31Þ 

v20 ¼ m0 12



a1 a2 : 1þ m0

0

22a1 m

0

0

22a2 m

0

2a2

2a1 ð1 2 b1 Þm2 s

0

22a1 ms

0

0

22a2 ms

0

2a2 s

2a1 ð1 2 b1 Þm2 s2

0

22a1 ms2

0

0

2a2 s2

(i)

a1 a2 ð4 þ b1 Þ . 4

22a2 ms

ð32Þ

Theorem 1. (Existence of Hopf bifurcation parameter) If b2 ¼ 0 and the following conditions hold

2a1 ð1 2 b1 Þm2

0

ð30Þ

Consequently,

where ai ¼api f 0 ð0Þ; i¼1;2: So, one has Gðs;mÞJðmÞ 0

549

2

2a1

1

C C C C C 2a1 s C C C: C 0 C C 2C 2a1 s C A 0 0

ð25Þ

Set hðl;s;mÞ¼detllI2Gðs;mÞJðmÞl ( ) a a ½ð12b1 Þm2 þs2 þ2ms½ð12b2 Þm2 þs2 þ2ms ¼ l4 l2 2 1 2 ðsþmÞ4 ðsþ1Þ2

(ii) a1 a2 ð4 þ b1 Þ # 4; 3a1 a2 b1 . 8; 9a21 a22 b21 þ 32a1 a2 ð2 2 b1 Þ . 0;

¼0:

(

ð26Þ

Then, by applying the generalized Nyquist stability criterion, with s¼iv; 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 ¼ m0 ; then the corresponding eigenvalue of the constant matrix ½Gðiv0 ; m0 ÞJðm0 Þ in the frequency domain must assume the value 21 þ i0 at m ¼ m0 : To apply Lemma 1, let l^ ¼ l^ðiv; mÞ be the eigenvalue of ½Gðiv; mÞJðmÞ that satisfies l^ðiv0 ; m0 Þ ¼ 21 þ i0: Then hð21;iv0 ; m0 Þ ¼12

a1 a2 ½ð12b1 Þm20 2 v20 þ2im0 v0 ½ð12b2 Þm20 2 v20 þ2im0 v0  ðm0 þiv0 Þ4 ð1þiv0 Þ2

¼0:

32ðb1 2 2Þ 8 ; 3b1 9b21

, a1 a2 #

ð34Þ 4 ; 4 þ b1

then

mþ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3a1 a2 b1 2 8 þ 9a21 a22 b21 þ 32a1 a2 ð2 2 b1 Þ 8

. 0:

If f ðmþ Þ , 0; the two nonnegative real roots of Eq. (31) exist. m1 [ ½0; mþ Þ and m2 [ ðmþ ; þ1Þ are the roots of Eq. (31). Moreover, if a1 a2 , 1; m1 and m2 are the Hopf bifurcations of system (8). If a1 a2 $ 1 and b1 . 1 2 ð1=ða1 a2 ÞÞ; m2 is the unique Hopf bifurcation of system (8). If a1 a2 $ 1 and b1 # 1 2 ð1=a1 a2 Þ; 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), define the following function f ðmÞ ¼ ð1 þ mÞ4 2 a1 a2 b1 ð1 þ mÞ3 2 a1 a2 ð2 2 b1 Þð1 þ mÞ2 þ ða1 a2 Þ2 : ð35Þ

ð28Þ

By separating this equation into real and imaginary parts, one obtains a1 a2 ½ð12b1 Þm20 2 v20 ¼ðm20 2 v20 Þð12 v20 Þ24m0 v20 ;

i:e: max

)

ð27Þ

First consider the case of b2 ¼0: Eq. (27) becomes a1 a2 ½ð12b1 Þm20 2 v20 þ2im0 v0 ¼ðm0 þiv0 Þ2 ð1þiv0 Þ2 :

ð33Þ

ð29Þ

So, f ð0Þ ¼ 1 2 a1 a2 b1 2 2a1 a2 þ a1 a2 b1 þ ða1 a2 Þ2 ¼ ð1 2 a1 a2 Þ2 $ 0:

550

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

Taking the derivation on Eq. (33), we have 0

3

a1 a2 m0 ½ð2 2 b1 2 b2 Þm20 2 2v20 

2

f ðmÞ¼4ð1þ mÞ 23a1 a2 b1 ð1þ mÞ 22a1 a2 ð22b1 Þð1þ mÞ 2

¼ð1þ mÞ½4m þð823a1 a2 b1 Þm þð424a1 a2 2a1 a2 b1 Þ; ð36Þ Letting f 0 ðmÞ¼0; we obtain the roots of Eq. (36)

ð41Þ

Therefore, 16ðm0 þ 1Þ8 þ c7 ðm0 þ 1Þ7 þ c6 ðm0 þ 1Þ6 þ c5 ðm0 þ 1Þ5 þ c4 ðm0 þ 1Þ4 þ c3 ðm0 þ 1Þ3 þ c2 ðm0 þ 1Þ2

mc ¼21; m^ ¼

¼ ðm20 2 v20 Þ2 2 4m20 v20 þ 2m0 ðm20 2 v20 Þð1 2 v20 Þ:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3a1 a2 b1 28^ 9a21 a22 b21 þ32a1 a2 ð22b1 Þ 8

ð37Þ :

v20 ¼ m20

If a1 a2 ð4þb1 Þ.4; then m2 ,0 and mþ .0: If a1 a2 ð4 þ b1 Þ # 4; 3a1 a2 b1 . 8; 9a21 a22 b21 þ 32a1 a2 ð2 2 b1 Þ . 0; then m2 . 0 and mþ . 0: When m , mþ ; f 0 ðmÞ , 0; and m . mþ ; f 0 ðmÞ . 0: So we have f ðmþ Þ is the local minimal of f ðmÞ: 

1 9 9 27 2 2 3 a ab f ðmþ Þ¼a21 a22 b1 12 b1 2 a1 a2 b1 þ a1 a2 b21 2 4 16 32 512 1 2 1  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 9 2 12 b1 þ a1 a2 b21 9a21 a22 b21 þ32a1 a2 ð22b1 Þ : 8 2 64 ð38Þ As limm!þ1 f ðmÞ¼þ1 and f ð0Þ¼ð12a1 a2 Þ2 $0; if f ðmþ Þ, 0; the two nonnegative real roots of Eq. (31) exist. m1 [ ½0;mþ Þ and m2 [ðmþ ;þ1Þ are the roots of Eq. (31).  a1 a2  Considering v20 ¼ m0 1 2 1þ m0 . 0; the Hopf bifurcation m0 . a1 a2 2 1: If a1 a2 , 1; so m1 . a1 a2 2 1 and m2 . a1 a2 2 1; then m1 and m2 are the Hopf bifurcations of system (1). If a1 a2 ¼ 1; so f ð0Þ ¼ ð1 2 a1 a2 Þ2 ¼ 0; i.e. m1 ¼ 0 and m2 . a1 a2 2 1; then m2 is the unique Hopf bifurcation of system (1). If a1 a2 . 1 and b1 . 1 2 ð1=a1 a2 Þ; so f ða1 a2 2 1Þ ¼ a21 a22 ða1 a2 2 1Þða1 a2 2 1 2 a1 a2 b1 Þ , 0; i.e. m1 , a1 a2 2 1 and m2 . a1 a2 2 1; then m2 is the unique Hopf bifurcation of system (1). pffiffiffiffiffiffi If a1 a2 . 1 and b1 # 1 2 ð1=a1 a2 Þ; so f ð a1 a2 2 1Þ ¼ ffiffiffiffiffiffi p 2a21 a22 b1 ð a1 a2 2 1Þ , 0 and f ða1 a2 2 1Þ $ 0; i.e. m1 , pffiffiffiffiffiffi a1 a2 2 1 , m2 # a1 a2 2 1; then the Hopf bifurcations of system (1) do not exist. The proof is complete. A Next, consider the case of b2 – 0: Eq. (27) becomes a1 a2 ½ð1 2 b1 Þm20 2 v20 þ 2im0 v0 ½ð1 2 b2 Þm20 2 v20 þ 2im0 v0  ¼ ðm0 þ iv0 Þ ð1 þ iv0 Þ : 4

2

þ c1 ðm0 þ 1Þ þ c0 ¼ 0;

ð39Þ

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

d1 ðm0 Þ ; d2 ðm0 Þ

ð42Þ ð43Þ

where c7 ¼ 16a1 a2 ð2b1 b2 2 b1 2 b2 Þ; c6 ¼ 4a1 a2 ½213b1 b2 þ 6ðb1 þ b2 Þ 2 16; c5 ¼ 4ða1 a2 Þ2 ½22ðb1 b2 Þ2 2 7b1 b2 ðb1 þ b2 Þ þ 8b1 b2 2 2ðb1 þ b2 Þ2 þ 8ðb1 þ b2 Þ þ 32a1 a2 b1 b2 ; c4 ¼ ða1 a2 Þ2 ½28ðb1 b2 Þ2 þ 62b1 b2 ðb1 þ b2 Þ 2 156b1 b2 þ 9ðb1 þ b2 Þ2 2 56ðb1 þ b2 Þ þ 96 2 8a1 a2 b1 b2 ; c3 ¼ ða1 a2 Þ3 ðb1 þ b2 Þ½4b1 b2 2 ðb1 þ b2 Þ2 þ 8ðb1 þ b2 Þ 2 16 þ ða1 a2 Þ2 b1 b2 ½238b1 b2 2 44ðb1 þ b2 Þ þ 160; c2 ¼ ða1 a2 Þ3 ½22b1 b2 ðb1 þ b2 Þ þ 20b1 b2 þ ðb1 þ b2 Þ3 2 10ðb1 þ b2 Þ2 þ 40ðb1 þ b2 Þ 2 64 þ ða1 a2 Þ2 b1 b2 ½25b1 b2 þ 10ðb1 þ b2 Þ 2 48; c1 ¼ 24ða1 a2 Þ3 b1 b2 ðb1 þ b2 Þ 2 8ða1 a2 b1 b2 Þ2 ; c0 ¼ ða1 a2 Þ4 ½24b1 b2 þ ðb1 þ b2 Þ2 2 8ðb1 þ b2 Þ þ 16 þ 2ða1 a2 Þ3 b1 b2 ½ðb1 þ b2 Þ 2 4 þ ða1 a2 b1 b2 Þ2 ; d1 ðm0 Þ ¼ ða1 a2 Þ2 ð2 2 b1 2 b2 Þ þ a1 a2 b1 b2 m0 ð2m0 þ 1Þ2 þ a1 a2 ðb1 þ b2 Þð6m30 þ 12m20 þ 7m0 þ 1Þ 2 a1 a2 ð16m30 þ 28m20 þ 16m0 þ 4Þ þ ð10m40 þ 32m30 þ 36m20 þ 16m0 þ 2Þ; d2 ðm0 Þ ¼ 2ða1 a2 Þ

2

ð44Þ

2 a1 a2 ðb1 þ b2 Þð2m30 þ 3m20 þ m0 Þ

2 a1 a2 ð12m20 þ 16m0 þ 4Þ þ ð16m50 þ 58m40 þ 80m30 þ 52m20 þ 16m0 þ 2Þ: Similar to the above discussion, one can easily obtain the following results.

a1 a2 {½ð1 2 b1 Þm20 2 v20 ½ð1 2 b2 Þm20 2 v20  2 4m20 v20 } ¼ ½ðm20 2 v20 Þ2 2 4m20 v20 ð1 2 v20 Þ 2 8m0 v20 ðm20 2 v20 Þ; ð40Þ

Theorem 2. P (Existence of Hopf bifurcation parameter) If b2 – 0 and 7i¼0 ci þ 16 , 0; then the Hopf bifurcation

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

parameters m0 of system (8) are the roots of Eq. (42), satisfying ðd1 ðm0 ÞÞ=ðd2 ðm0 ÞÞ . 0:

ð45Þ

where m~ is a fixed value of the parameter m; wT and v are the left and right eigenvectors of ½Gðiv~; m~ÞJðm~Þ; respectively, associated with the value l^ðiv~; m~Þ; and     1 1 p1 ¼ D2 V02 ^v þ v ^V22 þ D3 v^v^v ; 2 8

bð1; 56Þ ¼ 2a2 m~; bð1; 58Þ ¼ 2a2 ;

bð1; 128Þ ¼ 2a2 ; bð1; 188Þ ¼ a2 ;

In order to study the stability of bifurcating periodic solutions, the frequency-domain formulation of Moiola and Chen (1996) is applied. First, define an auxiliary vector of the form 2wT ½Gðiv~; m~Þp1 ; wT v

bð1; 44Þ ¼ a2 ð1 2 b2 Þm~2 ; bð1; 46Þ ¼ 2a2 m~; bð1; 48Þ ¼ a2 ;

bð1; 68Þ ¼ a2 ; bð1; 116Þ ¼ 2a2 m~; bð1;118Þ ¼ 2a2 ;

3. Stability of bifurcating periodic solutions

j1 ðv~Þ ¼

ð46Þ

where v~ is the frequency of the intersection between the l^ locus and the negative real axis closest to the point ð21 þ i0Þ; and  ›2 gðy; m~Þ  D2 ¼  ; ›y2 y¼0

bð2; 1Þ ¼ a2 ð1 2 b2 Þm~2 ; bð2; 3Þ ¼ 2a2 m~; bð2;5Þ ¼ a2 ; bð2; 13Þ ¼ 2a2 m~; bð2;15Þ ¼ 2a2 ; bð2; 25Þ ¼ a2 ; bð2; 73Þ ¼ 2a2 m~; bð2;75Þ ¼ 2a2 ; bð2; 85Þ ¼ 2a2 ; bð2; 145Þ ¼ a2 ; and the others are zero. Also, 0

C B C B1 C B C B B imv~ C C 1B C; v¼ B C dB C B iv~ C B C B 2 B 2mv~ C A @

0

ma2 ð1 2 b2 Þm~2

B B B a1 ð1 2 b1 Þm~2 B B 1B B 2ma2 m~ w¼ B lB B 2a1 m~ B B B ma2 @

1 C C C C C C C C; C C C C C A

a1

where

m¼2

1 V02 ¼ 2 ½I þ Gð0; m~ÞJðm~Þ21 Gð0; m~ÞD2 v^v; 4 1 V22 ¼ 2 ½I þ Gð2iv~; m~ÞJðm~Þ21 Gð2iv~; m~ÞD2 v^v: 4

ð47Þ

Then, one has D3 ¼ 2

f 000 ð0Þ ½bði; jÞ2£216 ; f 0 ð0Þ

where að1; 8Þ ¼ a1 ð1 2 b1 Þm~2 ; að1; 10Þ ¼ 2a1 m~; að1; 12Þ ¼ a1 ; að1; 20Þ ¼ 2a1 m~; að1; 22Þ ¼ 2a1 ; að1; 32Þ ¼ a1 ; að2; 1Þ ¼ a2 ð1 2 b2 Þm~2 ; að2; 3Þ ¼ 2a2 m~; að2; 5Þ ¼ a2 ; að2; 13Þ ¼ 2a2 m~; að2; 15Þ ¼ 2a2 ; að2; 25Þ ¼ a2 ;

a1 ½ð1 2 b1 Þm~2 2 v~2 þ 2im~v~ ; ðm~ þ iv~Þ2 ð1 þ iv~Þl~

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d ¼ ð1 þ mmÞð1  þ v~2 þ v~4 Þ;

l¼ f 00 ð0Þ ½aði; jÞ2£36 ; f 0 ð0Þ

1

m

2v~2

 ›3 gðy; m~Þ  D3 ¼  ; ›y3 y¼0

D2 ¼

551

2a1 ½ð1 2 b1 Þm~2 2 v~2 þ 2im~v~ : d

Moreover, V02 ¼

f 00 ð0Þ 4d f ð0Þ½a1 a2 ð1 2 b1 Þð1 2 b2 Þ 2 1 1 0 a1 ð1 2 b1 Þ þ mma  1 a2 ð1 2 b1 Þð1 2 b2 Þ C B B a1 a2 ð1 2 b1 Þð1 2 b2 Þ þ mma  2 ð1 2 b2 Þ C C B C B C B0 C B C; B C B C B0 C B C B C B0 A @ 2 0

0

ð48Þ

552

V22 ¼

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

f 00 ð0Þ 4d2 f 0 ð0Þ{a1 a2 ½ð1 2 b1 Þm~2 2 4v~2 þ 4im~v~½ð1 2 b2 Þm~2 2 4v~2 þ 4im~v~ 2 ð1 þ 2iv~Þ2 ðm~ þ 2iv~Þ4 } 0

1

a1 ½ð1 2 b1 Þm~2 2 4v~2 þ 4im~v~{ð1 þ 2iv~Þðm~ þ 2iv~Þ2 þ m2 a2 ½ð1 2 b2 Þm~2 2 4v~2 þ 4im~v~}

C B C B C B C B C B a ½ð1 2 b Þm~2 2 4v~2 þ 4im~v~{m2 ð1 þ 2iv~Þðm~ þ 2iv~Þ2 þ a ½ð1 2 b Þm~2 2 4v~2 þ 4im~v~} C B 2 2 1 1 C B C B C B C B C B 2 2 2 2 2 2 B 2iv~a1 ½ð1 2 b1 Þm~ 2 4v~ þ 4im~v~{ð1 þ 2iv~Þðm~ þ 2iv~Þ þ m a2 ½ð1 2 b2 Þm~ 2 4v~ þ 4im~v~} C C B C B £B C C B C B 2 2 2 2 2 2 B 2iv~a2 ½ð1 2 b2 Þm~ 2 4v~ þ 4im~v~{m ð1 þ 2iv~Þðm~ þ 2iv~Þ þ a1 ½ð1 2 b1 Þm~ 2 4v~ þ 4im~v~} C C B C B C B C B C B 2 2 2 2 2 2 2 B 24v~ a1 ½ð1 2 b1 Þm~ 2 4v~ þ 4im~v~{ð1 þ 2iv~Þðm~ þ 2iv~Þ þ m a2 ½ð1 2 b2 Þm~ 2 4v~ þ 4im~v~} C C B C B C B A @ 2 2 2 2 2 2 2 24v~ a2 ½ð1 2 b2 Þm~ 2 4v~ þ 4im~v~{m ð1 þ 2iv~Þðm~ þ 2iv~Þ þ a1 ½ð1 2 b1 Þm~ 2 4v~ þ 4im~v~}

Then, set pð3Þ 1

  1 1 p1 ¼ D2 ðV02 ^v þ v ^V22 Þ þ D3 v^v^v 2 8 ð50Þ

0

pð2Þ 1 ¼

and

j1 ðv~Þ ¼

B ½f ð0Þ B 4d 3 ½f 0 ð0Þ2 ½a1 a2 ð1 2 b1 Þð1 2 b2 Þ 2 1 @

8d 3 ½f 0 ð0Þ2 {a

2

2wT ½Gðiv~; m~Þp1 ð2Þ ð3Þ ¼ jð1Þ 1 þ j1 þ j1 ; wT v

a1 a2 ð1 2 b2 Þ½ð1 2 b1 Þm~2 2 v~2 þ 2im~v~½mm  þ a1 ð1 2 b1 Þ ma1 a2 ð1 2 b1 Þ½ð1 2 b2 Þm~2 2 v~2 þ 2im~v~½1 þ mma  2 ð1 2 b2 Þ

½f 00 ð0Þ2 a1 a2 ½ð1 2 b1 Þm~2 2 4v~2 þ 4im~v~½ð1 2 b2 Þm~2 2 4v~2 þ 4im~v~ 2 2 ~2 2 4v~2 þ 4im~v~ 2 ð1 þ 2iv~Þ2 ðm~ þ 2iv~Þ4 } 1 a2 ½ð1 2 b1 Þm~ 2 4v~ þ 4im~v~½ð1 2 b2 Þm

) 1 m2 ð1 þ 2iv~Þðm~ þ 2iv~Þ2 þ1 C B a1 ½ð1 2 b1 Þm~ 2 v~ þ 2im~v~ C B a1 ½ð1 2 b1 Þm~2 2 4v~2 þ 4im~v~ C B C B C B C B C B ( ) C B 2 ð1 þ 2i v Þð m þ 2i v Þ ~ ~ ~ @ 2 2 2 A ma þ m  2 ½ð1 2 b2 Þm~ 2 v~ þ 2im~v~ a2 ½ð1 2 b2 Þm~2 2 4v~2 þ 4im~v~ 0

(

2

2

with

jð1Þ 1 ¼2

2

ð51Þ

in which

pð1Þ 1 ¼

0 1 2 2 f 000 ð0Þ @ a1 ½ð1 2 b1 Þm~ 2 v~ þ 2im~v~ A; ¼2 3 0 2 2 8d f ð0Þ m2 ma  ½ð1 2 b Þm~ 2 v~ þ 2im~v~ 2

ð2Þ ð3Þ ¼ pð1Þ 1 þ p1 þ p1 ;

00

ð49Þ

½f 00 ð0Þ2 ma1 a2 ½ð1 2 b1 Þm~2 2 v~2 þ 2im~v~½ð1 2 b2 Þm~2 2 v~2 þ 2im~v~ 4ld 3 ½f 0 ð0Þ2 ½a1 a2 ð1 2 b1 Þð1 2 b2 Þ 2 1ð1 þ iv~Þðm~ þ iv~Þ2

£ ½mma  2 ð1 2 b2 Þ þ a1 ð1 2 b1 Þ þ ð1 þ mmÞa  1 a2 ð1 2 b1 Þð1 2 b2 Þ;

1 C C A

ð52Þ

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

jð2Þ 1 ¼2

553

½f 00 ð0Þ2 a1 a2 ½ð1 2 b1 Þm~2 2 v~2 þ 2im~v~½ð1 2 b2 Þm~2 2 v~2 þ 2im~v~ 8ld 3 ½f 0 ð0Þ2 {a1 a2 ½ð1 2 b1 Þm~2 2 4v~2 þ 4im~v~½ð1 2 b2 Þm~2 2 4v~2 þ 4im~v~ 2 ð1 þ 2iv~Þ2 ðm~ þ 2iv~Þ4 }

ma1 a2 ½ð1 2 b1 Þm~2 2 4v~2 þ 4im~v~½ð1 2 b2 Þm~2 2 4v~2 þ 4im~v~ ð1 þ iv~Þðm~ þ iv~Þ2 ( ) mð1 m2 ð1 þ 2iv~Þðm~ þ 2iv~Þ2  þ 2iv~Þðm~ þ 2iv~Þ2 £ þ þ ð1 þ mmÞ  ; a1 ½ð1 2 b1 Þm~2 2 4v~2 þ 4im~v~ ma2 ½ð1 2 b2 Þm~2 2 4v~2 þ 4im~v~ £

jð3Þ 1 ¼

f 000 ð0Þma1 a2 ½ð1 2 b1 Þm~2 2 v~2 þ 2im~v~½ð1 2 b2 Þm~2 2 v~2 þ 2im~v~ £ ð1 þ mmÞ:  8ld 3 f 0 ð0Þð1 þ iv~Þðm~ þ iv~Þ2

Since l~ is the eigenvalue of ½Gðiv~; m~ÞJðm~Þ; one has

(ii) The intersection is transversal, i.e. 2

a a ½ð12b1 Þm~2 2 v~2 þ2im~v~½ð12b2 Þm~2 2 v~2 þ2im~v~ l~2 2 1 2 ð1þiv~Þ2 ðm~ þiv~Þ4 ¼ 0:

R{j1 ðv^Þ} ) 6 (  Nðv^; m~Þ ¼ det6 dl^  4 R  dv v¼v^

ð54Þ

Considering

– 0:

a ½ð12b1 Þm~2 2 v~2 þ2im~v~ m¼2 1 ; ðm~ þiv~Þ2 ð1þiv~Þl~ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v~2 þ v~4 Þ; d ¼ ð1þmmÞð1þ  l¼

one obtains

l~f 000 ð0Þ : 16ð1þ v~2 þ v~4 Þf 0 ð0Þ

ð55Þ

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^ðsÞ intersects the negative real axis at l^ðiv~Þ that is closest to the point ð21 þ i0Þ when the variable s sweeps on the classical Nyquist contour. Moreover, suppose that j1 ðv~Þ is nonzero and the half-line L1 starting from ð21 þ i0Þ in the direction defined by j1 ðv~Þ first intersects the locus of l^ðivÞ at l^ðiv^Þ ¼ P^ ¼ 21 þ j1 ðv~Þu2 ; where u ¼ Oðlm 2 m0 l1=2 Þ: Finally, suppose that following conditions are satisfied (i)

The eigenlocus l^ has nonzero rate of change with respect to its parameterization at the criticality ðv0 ; m0 Þ; i.e., # ›F1 =›m ›F2 =›m  Mðv0 ; m0 Þ ¼ det  ›F =›v ›F =›v 

"

1

2

3 I{j1 ðv^Þ} ( )7  7 dl^  5 I  dv v¼v^ ð57Þ

(iii) There are no other intersections between any of the characteristic loci and the line segment joining the ^ at least within a small point ð21 þ i0Þ to P; neighborhood of radios d . 0:

2a1 ½ð12b1 Þm~2 2 v~2 þ2im~v~ ; d

jð3Þ 1 ¼2

ð53Þ

– 0;

ð56Þ

ðv 0 ; m 0 Þ

where F1 ðv; mÞ ¼ R{hð21; iv; mÞ}; F2 ðv; mÞ ¼ I{hð21;iv; mÞ}:

Then, the system (17) has a periodic solution xðtÞ of frequency v ¼ v^ þ Oðu^4 Þ: Moreover, by applying a small perturbation around the intersection P^ and using the generalized Nyquist stability criterion, the stability of the periodic solution xðtÞ 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 figure of the half-line L1 and the locus l^ðivÞ: The analysis is carried out as follows (Moiola & Chen, 1996) 1. If the half-line L1 first intersects the locus of l^ðivÞ when m~ . m0 ð, m0 Þ; then the bifurcating periodic solution exists and the Hopf bifurcation is supercritical (subcritical). 2. If the total number of anticlockwise encirclements of the point P1 ¼ P^ þ ej1 ðv~Þ; for a small enough e . 0; is equal to the number of poles of lðsÞ 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^ j ^gNðv^; m~Þ . 0; dv v¼v

554

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

or l~ , 21 and If

2. if s , 0; the Hopf bifurcation at m ¼ m0 of system (8) is subcritical.

dl^ j ^gNðv^; m~Þ , 0; dv v¼v

then the half-line L1 intersects the locus of l^ðivÞ:. Consequently, one obtains the following result. Theorem 3. Set ( )!   dl~  dl^  Nðv0 ; m0 Þ; s ¼ sgn I   dm m¼m0 dv v¼v0

j1 ðvÞ ¼ jð3Þ 1 ¼ ð58Þ

where  dl~  k ðm Þ ¼ 1 0 ;  dm m¼m0 k2 ðm0 Þ k1 ðm0 Þ ¼128ðm0 þ 1Þ7 þ 7c7 ðm0 þ 1Þ6 þ 6c6 ðm0 þ 1Þ5

l~ : 8ð1 þ v2 þ v4 Þ

ð61Þ

Setting m ¼ m0 ; v ¼ v0 ; l~ ¼ 21; one can calculate the following ) (  dl^  Nðv0 ; m0 Þ ¼R{j1 ðv0 Þ}I  dv v¼v0 ) (  dl^  2 I{j1 ðv0 Þ}R  dv v¼v 0

and

þ 5c5 ðm0 þ 1Þ4 þ 4c4 ðm0 þ 1Þ3 þ 3c3 ðm0 þ 1Þ2 þ 2c2 ðm0 þ 1Þ þ c1 ;

For more details, see Appendix A. Now, set f ðuÞ ¼ tan hðuÞ: Then, f 0 ð0Þ ¼ 1; f 00 ð0Þ ¼ 0; ð2Þ 000 f ð0Þ ¼ 22: Therefore, in Eq. (55), jð1Þ 1 ¼ 0; j1 ¼ 0; and

ð59Þ

k2 ðm0 Þ ¼ 128ðm0 þ 1Þ8 þ 6c7 ðm0 þ 1Þ7 þ 6c6 ðm0 þ 1Þ6

)! (   dl~  dl^  Nðv0 ; m0 Þ: s ¼ sgn I   d m  m¼ m0 d v  v¼ v0 Considering

þ ð4c5 þ 64cÞðm0 þ 1Þ5 þ ð4c4 2 16cÞðm0 þ 1Þ4 3

þ ð2c3 þ 2c32 Þðm0 þ 1Þ þ ð2c2 þ 2c22 Þðm0 þ 1Þ

2

þ ð2c1 2 16c2 Þðm0 þ 1Þ þ ð2c03 þ 4c2 Þ; c7 ; c6 ; c5 ; c4 ; c3 ; c2 ; c1 ; c0 are defined in Eq. (44), c32 ¼ ða1 a2 Þ2 b1 b2 ½238b1 b2 2 44ðb1 þ b2 Þ þ 160; c22 ¼ ða1 a2 Þ2 b1 b2 ½25b1 b2 þ 10ðb1 þ b2 Þ 2 48; c03 ¼ 2ða1 a2 Þ3 b1 b2 ½ðb1 þ b2 Þ 24; with c ¼ a1 a2 b1 b2 ; and ! (  dl  ð1 þ b1 Þm20 þ v20 R ¼ 2 v0  d v  v¼ v0 ½ð1 2 b1 Þm20 2 v20 2 þ 4m20 v20 ð1 þ b2 Þm20 þ v20 1 2 2 2 2 2 2 ½ð1 2 b2 Þm0 2 v0  þ 4m0 v0 1 þ v20 ) 2 ; ð60Þ 2 2 m0 þ v20 ! (  dl  ð1 2 b1 Þm30 þ m0 v20 ¼2 I  dv v¼v0 ½ð1 2 b1 Þm20 2 v20 2 þ 4m20 v20 þ

ð1 2 b2 Þm30 þ m0 v20 1 2 ½ð1 2 b2 Þm20 2 v20 2 þ 4m20 v20 1 þ v20 ) 2m0 : 2 2 m0 þ v20 þ

j1 ðv0 Þ ¼ 2

1 8ð1 þ v20 þ v40 Þ

to be a negative real, i.e. R{j1 ðv0 Þ} ¼ 2

1 , 0; 8ð1 þ v20 þ v40 Þ

I{j1 ðv0 Þ} ¼ 0;

one has !  dl~  sgnðsÞ ¼ 2sgn  d m  m¼ m0

(

if

)  dl  I – 0:  d v  v¼ v0

Corollary 1. Let (

)  dl  f ðuÞ ¼ tan hðuÞ and I – 0;  d v  v¼ v0 with  dl~  k ðm Þ s1 ¼ 2 ¼2 1 0 ;  k2 ðm0 Þ dm m¼m0

ð62Þ

where k1 ðm0 Þ; k2 ðm0 Þ are defined in Eq. (59). Then

Then

1. If s1 . 0; the Hopf bifurcation at m ¼ m0 of system (8) is supercritical; 2. If s1 , 0; the Hopf bifurcation at m ¼ m0 of system (8) is subcritical.

1. if s . 0; the Hopf bifurcation at m ¼ 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 ¼ 1:5; a2 ¼ 0:5; b1 ¼ 3; b2 ¼ 0; m ¼ 0:045: The half-line L1 does not intersect the locus l^ðivÞ; so no periodic solution exists.

Fig. 2. a1 ¼ 1:5; a2 ¼ 0:5; b1 ¼ 3; b2 ¼ 0; m ¼ 0:055: The half-line L1 intersects the locus l^ðivÞ; and k ¼ 0; so a stable periodic solution exists.

closest to the point ð21 þ i0Þ; i.e. l~ (see Appendix A) 8 ~8

7

6

v~2 ¼ m~2

16ðm~ þ 1Þ l þ ½c7 ðm~ þ 1Þ þ c6 ðm~ þ 1Þ þ 32cðm~ þ 1Þ

5

d10 ðm~Þl~4 þ d11 ðm~Þl~2 þ d12 ; d20 ðm~Þl~4 þ d21 ðm~Þl~2 þ d22

ð64Þ

2 8cðm~ þ 1Þ4 l~6 þ ½ðc5 2 32cÞðm~ þ 1Þ5

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

þ ðc4 þ 8cÞðm~ þ 1Þ4 þ c32 ðm~ þ 1Þ3 þ c22 ðm~ þ 1Þ2

d12 ¼ ða1 a2 Þ2 ð2 2 b1 2 b2 Þ;

2 8c2 ðm~ þ 1Þ þ c2 l~4 þ ½ðc3 2 c32 Þðm~ þ 1Þ3

d11 ðm~Þ ¼ a1 a2 ½b1 b2 m~ð2m~ þ 1Þ2 þ ðb1 þ b2 Þð6m~3 þ 12m~2

þ ðc2 2 c22 Þðm~ þ 1Þ2 þ ðc1 þ 8c2 Þðm~ þ 1Þ þ c03 l~2 þ ðc0 2 c03 2 c2 Þ ¼ 0;

þ 7m~ þ 1Þ 2 ð16m~3 þ 28m~2 þ 16m~ þ 4Þ; ð63Þ

d10 ðm~Þ ¼ 10m~4 þ 32m~3 þ 36m~2 þ 16m~ þ 2;

Fig. 3. a1 ¼ 1:5; a2 ¼ 0:5; b1 ¼ 3; b2 ¼ 0; m ¼ 0:65: The half-line L1 intersects the locus l^ðivÞ; and k ¼ 0; so a stable periodic solution exists.

Fig. 4. a1 ¼ 1:5; a2 ¼ 0:5; b1 ¼ 3; b2 ¼ 0; m ¼ 0:75: The half-line L1 does not intersect the locus l^ðivÞ; so no periodic solution exists.

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

Fig. 5. a1 ¼ 2; a2 ¼ 1; b1 ¼ 3; b2 ¼ 0; m ¼ 4:5: The half-line L1 intersects the locus l^ðivÞ and k ¼ 0; so a stable periodic solution exists.

Fig. 6. a1 ¼ 2; a2 ¼ 1; b1 ¼ 3; b2 ¼ 0; m ¼ 5:. The half-line L1 does not intersect the locus l^ðivÞ; so no periodic solution exists.

d22 ¼ 2ða1 a2 Þ2 ;

According to Eq. (26), one has

d21 ðm~Þ¼2a1 a2 ½ðb1 þb2 Þð2m~3 þ3m~2 þm~Þþð12m~2 þ16m~ þ4Þ; d20 ðm~Þ¼16m~5 þ58m~4 þ80m~3 þ52m~2 þ16m~ þ2:

ð65Þ

Then, one draws the half-line j1 ðv~Þ starting from ð21þi0Þ and the locus l^ðivÞ; and obtains the total number k of ^ ej1 ðv~Þ for anticlockwise encirclements of the point P1 ¼ Pþ a small enough e .0:

½lðsÞ2 ¼

a1 a2 ½ð12b1 Þm2 þs2 þ2ms½ð12b2 Þm2 þs2 þ2ms : ðsþ mÞ4 ðsþ1Þ2 ð66Þ

Hence, s¼2m and s¼21 are the poles of lðsÞ; and the number of poles of lðsÞ that have positive real parts is zero. Corollary 2. Let k be the total number of anticlockwise encirclements of the point P1 ¼ P^ þ ej1 ðv~Þ for a small

Fig. 7. a1 ¼ 2; a2 ¼ 1:5; b1 ¼ 0:8; b2 ¼ 1:1; m ¼ 2:. The half-line L1 intersects the locus l^ðivÞ; and k ¼ 0; so a stable periodic solution exists.

Fig. 8. a1 ¼ 2; a2 ¼ 1:5; b1 ¼ 0:8; b2 ¼ 1:1; m ¼ 2:1: The half-line L1 does not intersect the locus l^ðivÞ; so no periodic solution exists.

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

557

Fig. 9. a1 ¼ 0:5; a2 ¼ 1:8; b1 ¼ 0:8; b2 ¼ 0:3; m ¼ 0:01: The half-line L1 does not intersect the locus l^ðivÞ; so no periodic solution exists.

Fig. 10. a1 ¼ 0:5; a2 ¼ 1:8; b1 ¼ 0:8; b2 ¼ 0:3; m ¼ 0:025: The half-line L1 intersects the locus l^ðivÞ; and k ¼ 0; so a stable periodic solution exists.

enough e . 0; where P^ is the intersection of the half-line L1 and the locus l^ðivÞ: Then 1. if k ¼ 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^ðivÞ 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 ¼ P^ þ ej1 ðv~Þ for a small enough e . 0: Suppose that the half-line L1 and the locus l^ðivÞ intersect. If k ¼ 0; the bifurcating periodic solutions is stable; if k – 0; the bifurcating periodic solutions is unstable.

Fig. 11. a1 ¼ 0:5; a2 ¼ 1:8; b1 ¼ 0:8; b2 ¼ 0:3; m ¼ 0:14: The half-line L1 intersects the locus l^ðivÞ; and k ¼ 0; so a stable periodic solution exists.

Fig. 12. a1 ¼ 0:5; a2 ¼ 1:8; b1 ¼ 0:8; b2 ¼ 0:3; m ¼ 0:17: The half-line L1 does not intersect the locus l^ðivÞ; so no periodic solution exists.

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

Fig. 13. a1 ¼ 0:1; a2 ¼ 2; b1 ¼ 1:01; b2 ¼ 30; m ¼ 0:15: The half-line L1 does not intersect the locus l^ðivÞ; so no periodic solution exists.

Fig. 14. a1 ¼ 0:1; a2 ¼ 2; b1 ¼ 1:01; b2 ¼ 30; m ¼ 0:2: The half-line L1 intersects the locus l^ðivÞ; and k ¼ 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 ¼ 1:5; a2 ¼ 0:5; b1 ¼ 3; b2 ¼ 0: Then, m0 ¼ 0:0502 or m0 ¼ 0:6894:. Choose m0 ¼ 0:0502: Then s1 ¼ 0:9454 . 0: Hence, m0 ¼ 0:0502 is a supercritical Hopf bifurcation point. Set m0 ¼ 0:6894: Then s1 ¼ 20:1685 , 0: Hence, m0 ¼ 0:6894 is a subcritical Hopf bifurcation point.

(ii) Let a1 ¼ 2; a2 ¼ 1; b1 ¼ 3; b2 ¼ 0: Then, m0 ¼ 4:6217: Then s1 ¼ 20:0826 , 0; so m0 ¼ 4:6217 is a subcritical Hopf bifurcation point. (iii) Let a1 ¼ 2; a2 ¼ 1:5; b1 ¼ 0:8; b2 ¼ 1:1: Then m0 ¼ 2:0521: Then s1 ¼ 20:2761 , 0; so m0 ¼ 2:0521 is a subcritical Hopf bifurcation point.

Fig. 15. a1 ¼ 0:1; a2 ¼ 2; b1 ¼ 1:01; b2 ¼ 30; m ¼ 1:5: The half-line L1 intersect the locus l^ðivÞ; and k ¼ 0; so a stable periodic solution exists.

Fig. 16. a1 ¼ 0:1; a2 ¼ 2; b1 ¼ 1:01; b2 ¼ 30; m ¼ 1:7: The half-line L1 does not intersect the locus l^ðivÞ; so no periodic solution exists.

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

(iv) Let a1 ¼ 0:5; a2 ¼ 1:8; b1 ¼ 0:8; b2 ¼ 0:3: Then m0 ¼ 0:0188 or m0 ¼ 0:1468: Choose m0 ¼ 0:0188: Then s1 ¼ 0:8876 . 0; so m0 ¼ 0:0188 is a supercritical Hopf bifurcation point. Set m0 ¼ 0:1468: Then s1 ¼ 20:2949 , 0; so m0 ¼ 0:1468 is a subcritical Hopf bifurcation point. (v) Let a1 ¼ 0:1; a2 ¼ 2; b1 ¼ 1:01; b2 ¼ 30: Then m0 ¼ 0:1640 or m0 ¼ 1:5726: Choose m0 ¼ 0:1640: Then s1 ¼ 1:3091 . 0; so m0 ¼ 0:1640 is a supercritical Hopf bifurcation point. Set m0 ¼ 1:5726: Then s1 ¼ 20:2111 , 0; so m0 ¼ 1:5726 is a subcritical Hopf bifurcation point (Figs. 1– 16) (Enun 2– 6).

Acknowledgements

5. Conclusions

Appendix A

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 difficult 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^ðivÞ; 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 ¼ 0; (the H10 degeneracy) one cannot decide the direction of the bifurcating periodic orbits by only using L1 and l^ðivÞ: 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 find the corresponding solutions ðv^2 ; u^2 Þ; ðv^3 ; u^3 Þ and ðv^4 ; u^4 Þ; as carried out by Moiola and Chen (1996). The fundamental equations about the amplitude loci L2 ; L3 and L4 are

l^ðivÞ ¼ 21 þ j1 ðv~Þu2 þ j2 ðv~Þu4 ;

ð67Þ

l^ðivÞ ¼ 21 þ j1 ðv~Þu2 þ j2 ðv~Þu4 þ j3 ðv~Þu6 ;

559

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.

A: Computing the eigenvalue l~: Consider Eq. (54) again a a ½ð12b1 Þm~2 2 v~2 þ2im~v~½ð12b2 Þm~2 2 v~2 þ2im~v~ l~2 2 1 2 ð1þiv~Þ2 ðm~ þiv~Þ4 ¼ 0: Notice that l~ ¼ l^ðiv~; m~Þ is a real number. By separating Eq. (54) into real and imaginary parts, one has a1 a2 {½ð12b1 Þm~2 2 v~2 ½ð12b2 Þm~2 2 v~2 24m~2 v~2 } ¼ l~2 {½ðm~2 2 v~2 Þ2 24m~2 v~2 ð12 v~2 Þ28m~v~2 ðm~2 2 v~2 Þ}; ðA1Þ a1 a2 m~½ð22b1 2b2 Þm~2 22v~2  ¼ l~2 ½ðm~2 2 v~2 Þ2 24m~2 v~2 þ2m~ðm~2 2 v~2 Þð12 v~2 Þ:

ðA2Þ

Therefore, one obtains the following equation 16ðm~ þ1Þ8 l~8 þ½c7 ðm~ þ1Þ7 þc6 ðm~ þ1Þ6 þ32cðm~ þ1Þ5 28cðm~ þ1Þ4 l~6 þ½ðc5 232cÞðm~ þ1Þ5 þðc4 þ8cÞðm~ þ1Þ4

ð68Þ þc32 ðm~ þ1Þ3 þc22 ðm~ þ1Þ2 28c2 ðm~ þ1Þþc2 l~4

l^ðivÞ ¼ 21 þ j1 ðv~Þu þ j2 ðv~Þu þ j3 ðv~Þu 2

þ j4 ðv~Þu : 8

4

6

þ½ðc3 2c32 Þðm~ þ1Þ3 þðc2 2c22 Þðm~ þ1Þ2 ð69Þ

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.

þðc1 þ8c2 Þðm~ þ1Þþc03 l~2 þðc0 2c03 2c2 Þ¼0; ðA3Þ

v~2 ¼ m~2

d10 ðm~Þl~4 þd11 ðm~Þl~2 þd12 : d20 ðm~Þl~4 þd21 ðm~Þl~2 þd22

ðA4Þ

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

where c7 ;c6 ;c5 ;c4 ;c3 ;c2 ;c1 ;c0 are defined in Eq. (44), and 2

c32 ¼ða1 a2 Þ b1 b2 ½238b1 b2 244ðb1 þb2 Þþ160; c22 ¼ða1 a2 Þ2 b1 b2 ½25b1 b2 þ10ðb1 þb2 Þ248; c03 ¼2ða1 a2 Þ3 b1 b2 ½ðb1 þb2 Þ24; c¼a1 a2 b1 b2 ;

Fix m at m~; and take the derivative with respect to v on both sides of Eq. (54). One has   dl  l~ 22v~þ2im~ 22v~ þ2im~ ¼ þ dv v¼v~ 2 ð12b1 Þm~2 2v~2 þ2im~v~ ð12b2 Þm~2 2v~2 þ2im~v~  2i 4i 2 2 : ðA7Þ 1þiv~ m~ þiv~ Consequently, ! (  dl  l~ 2ð1þb1 Þm~2 v~ þ2v~3 ¼ R  dv v¼v~ 2 ½ð12b1 Þm~2 2v~2 2 þ4m~2 v~2

d22 ¼2ða1 a2 Þ2 ; d21 ðm~Þ¼2a1 a2 ½ðb1 þb2 Þð2m~3 þ3m~2 þm~Þ þð12m~2 þ16m~þ4Þ;

ðA5Þ

2ð1þb2 Þm~2 v~ þ2v~3 2v~ 2 2 2 2 2 2 ½ð12b2 Þm~ 2v~  þ4m~ v~ 1þv~2 ) 4v~ ; 2 2 m~ þv~2 ! (  dl  l~ 2ð12b1 Þm~3 þ2m~v~2 I ¼  dv v¼v~ 2 ½ð12b1 Þm~2 2v~2 2 þ4m~2 v~2 þ

d20 ðm~Þ¼16m~5 þ58m~4 þ80m~3 þ52m~2 þ16m~þ2; d12 ¼ða1 a2 Þ2 ð22b1 2b2 Þ; d11 ðm~Þ¼a1 a2 ½b1 b2 m~ð2m~þ1Þ2 þðb1 þb2 Þ ð6m~3 þ12m~2 þ7m~þ1Þ 2ð16m~3 þ28m~2 þ16m~þ4Þ;

2ð12b2 Þm~3 þ2m~v~2 2 2 ½ð12b2 Þm~2 2v~2 2 þ4m~2 v~2 1þv~2 ) 4m~ : 2 2 m~ þv~2

ðA8Þ

þ

d10 ðm~Þ¼10m~4 þ32m~3 þ36m~2 þ16m~þ2: According to (A3), one can compute the eigenvalue l~: B: Computing ðdl~=dmÞm¼m0 : 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 ¼ m0 ; l~ ¼ 21; one obtains  dl~  k ðm Þ ðA6Þ ¼ 1 0 ;   dm m¼m0 k2 ðm0 Þ

ðA9Þ

Then, using m0 instead of m~; v0 instead of v~; and 2 1 instead of l~;, one obtains ! (  dl  ð1þb1 Þm20 þv20 ¼2v0 R  dv v¼v~ ½ð12b1 Þm20 2v20 2 þ4m20 v20 ð1þb2 Þm20 þv20 ½ð12b2 Þm20 2v20 2 þ4m20 v20 ) 1 2 ; 2 2 1þv20 m20 þv20

where

þ

k1 ðm0 Þ ¼ 128ðm0 þ 1Þ7 þ 7c7 ðm0 þ 1Þ6 þ 6c6 ðm0 þ 1Þ5 þ 5c5 ðm0 þ 1Þ4 þ 4c4 ðm0 þ 1Þ3 þ 3c3 ðm0 þ 1Þ2 þ 2c2 ðm0 þ 1Þ þ c1 ; 8

7

6

k2 ðm0 Þ ¼ 128ðm0 þ 1Þ þ 6c7 ðm0 þ 1Þ þ 6c6 ðm0 þ 1Þ

ðA10Þ

! (  dl  ð12b1 Þm30 þm0 v20 I ¼2  dv v¼v~ ½ð12b1 Þm20 2v20 2 þ4m20 v20

þ ð4c5 þ 64cÞðm0 þ 1Þ5 þ ð4c4 2 16cÞðm0 þ 1Þ4

ð12b2 Þm30 þm0 v20 ½ð12b2 Þm20 2v20 2 þ4m20 v20 ) 1 2m0 : 2 2 1þv20 m20 þv20 þ

þ ð2c3 þ 2c32 Þðm0 þ 1Þ3 þ ð2c2 þ 2c22 Þðm0 þ 1Þ2 þ ð2c1 2 16c2 Þðm0 þ 1Þ þ ð2c03 þ 4c2 Þ: C: Computing the real and imaginary parts of ðdl=dvÞv¼v0 : Consider Eq. (54) a a ½ð12b1 Þm~2 2 v2 þ2imv ~ ½ð12b2 Þm~2 2 v2 þ2imv ~  l2 2 1 2 2 4 ð1þivÞ ðm~ þivÞ ¼ 0:

ðA11Þ

References Allwright, D. J. (1977). Harmonic balance and the Hopf bifurcation theorem. Mathematical Proceedings of Cambridge Philosphical Society, 82, 453 –467.

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