Frequency domain approach to computational analysis of bifurcation and periodic solution in a two-neuron network model with distributed delays and self-feedbacks

Neurocomputing 99 (2013) 206–213

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Neurocomputing journal homepage: www.elsevier.com/locate/neucom

Frequency domain approach to computational analysis of bifurcation and periodic solution in a two-neuron network model with distributed delays and self-feedbacks Min Xiao a,b,c,n, Wei Xing Zheng b, Jinde Cao c a

School of Mathematics and Information Technology, Nanjing Xiaozhuang University, Nanjing 210017, PR China School of Computing, Engineering and Mathematics, University of Western Sydney, Penrith NSW 2751, Australia c Department of Mathematics, Southeast University, Nanjing 210096, PR China b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 14 October 2011 Received in revised form 8 January 2012 Accepted 17 March 2012 Communicated by H. Jiang Available online 28 July 2012

In this paper, a general two-neuron model with distributed delays, self-feedbacks and a weak kernel is studied. It is shown that the Hopf bifurcation occurs as the bifurcation parameter, the mean delay, passes a critical value where a family of periodic solutions emanate from the equilibrium. By applying the frequency domain approach and analyzing the associated characteristic equation, the existence of the bifurcation critical point of the mean delay is determined. The direction and the stability of bifurcating periodic solutions are determined by the Nyquist criterion and the graphical Hopf bifurcation theorem. Numerical simulation results supporting the theoretical analysis are also given. & 2012 Elsevier B.V. All rights reserved.

Keywords: Neuron Distributed delays Stability Periodic solutions Hopf bifurcation Graphical Hopf bifurcation theorem Nyquist criterion Frequency domain

1. Introduction

behaviors of like neurons [5–12]. Olien and Belair [10] investigated the following system with two discrete time delays:

It is well known that neural networks play a key role in such areas as signal processing, pattern recognition, optimization and associative memories since Hopfield [1] introduced a simplified neural network model with instantaneous feedback controls. Thus, understanding the dynamical characteristics (including stable, unstable, oscillatory, and chaotic behavior [2–19]) of neural networks will fundamentally advance the study of core neural problems. For simplicity, many researches have suggested studying dynamical behaviors of simple systems. This is still useful since the complexity found may be carried over to large networks. Marcus and Westervelt [6] argued that time delays should be incorporated into the models of network models due to the signal transmission in order to be more realistic and proposed a neural network model with delays. Afterwards, a great many systems of delayed differential equations representing neural networks have been proposed and studied extensively to understand the dynamical

dx1 ðtÞ ¼ x1 ðtÞ þ a11 f ðx1 ðtt1 ÞÞ þa12 f ðx2 ðtt2 ÞÞ, dt

n Corresponding author at: School of Mathematics and Information Technology, Nanjing Xiaozhuang University, Nanjing 210017, PR China. E-mail address: [email protected] (M. Xiao).

0925-2312/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.neucom.2012.03.020

dx2 ðtÞ ¼ x2 ðtÞ þ a21 f ðx1 ðtt1 ÞÞ þa22 f ðx2 ðtt2 ÞÞ: dt

ð1Þ

Some sufficient conditions for the stability of the stationary point of (1) are obtained in several cases, such as t1 ¼ t2 , a11 ¼ a22 ¼ 0, etc. Moreover, some bifurcations at certain values of the parameters are also showed for (1). For the case without self-connections, Wei and Ruan [12] found that a 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 selfconnections was studied by Destexhe [7]. Related works on twoneuron networks with delays can be referred to [9,19] and the references cited therein. Although the use of constant discrete delays in models with delayed feedbacks provides a good approximation to simple circuits consisting of a small number of neurons, neural networks

M. Xiao et al. / Neurocomputing 99 (2013) 206–213

usually have a spatial extent due to the presence of a multitude of parallel pathways with a variety of axon sizes and lengths [13]. Thus, there will be a distribution of propagation delays. In this case, the signal propagation is no longer instantaneous and cannot be modeled with discrete delays. A more appropriate way is to incorporate distributed delays. However, in the above-mentioned works, only networks with discrete time delays were investigated. Tank and Hopfield [20] have proposed a neural circuit with distributed delays, which solves a general problem of recognizing patterns in a time-dependent signal. For the application of neural networks with distributed delays as described in [20], the readers may also refer to [13–15,17,18,21]. Liao et al. [13–15] studied the following two-neuron system with distributed delays:   Z 1 dxn1 ðtÞ ¼ xn1 ðtÞ þ an1 f xn2 ðtÞb2 FðrÞxn2 ðtrÞ drc1 , dt 0   Z 1 dxn2 ðtÞ ¼ xn2 ðtÞ þ an2 f xn1 ðtÞb1 FðrÞxn1 ðtrÞ drc2 , dt 0

determined. In Section 3, the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by the Nyquist criterion and the graphical Hopf bifurcation theorem [23]. Some numerical simulation results and the frequency-domain graphs are given to illustrate the theoretical analysis result in Section 4. Finally, some conclusions are drawn in Section 5.

2. Existence of the Hopf bifurcation Consider system (3) with the weak kernel FðrÞ ¼ memr ,

m 4 0:

ð4Þ

For simplicity, we set c1 ¼ c2 ¼ 0. Similar results can be obtained when c1 a 0, c2 a 0. Making variable changes Z 1 FðrÞxn1 ðtrÞ dr, x1 ðtÞ ¼ xn1 ðtÞb1 0

ð2Þ

where xni ði ¼ 1,2Þ is the mean soma potential of the neuron, ani denotes to the range of the continuous variable xni , bi measures the inhibitory influence of the past history, ci corresponds to the neuronal threshold and the term xni in the argument of the function f represents a local positive feedback. Moreover, FðÞ is a kernel function. For the case of model (2) with a weak kernel, Liao et al. [13] studied the Hopf bifurcation of model (2) in the time domain developed in [22] by regarding the mean delay as a bifurcation parameter. In other two papers, Liao et al. considered the same model with the weak kernel [14] and the strong kernel [15] in the frequency domain established and later developed by [23–25]. Again, they showed that as the mean delay exceeds a critical value, model (2) undergoes the Hopf bifurcation. The frequency domain approach has some advantages over the classical time domain methods. A typical one is its pictorial characteristic that utilizes advanced computer graphical capabilities thereby bypassing quite a lot of profound and difficult mathematical analysis. It is highly applicable to the study of the closed-loop systems when one applies feedback controllers. In this paper, we consider system (2) with self-feedbacks, that is,   Z 1 dxn1 ðtÞ ¼ xn1 ðtÞ þ an11 f xn1 ðtÞb1 FðrÞxn1 ðtrÞ drc2 dt 0   Z 1 n n þ a12 f x2 ðtÞb2 FðrÞxn2 ðtrÞ drc1 , 0

  Z 1 dxn2 ðtÞ ¼ xn2 ðtÞ þ an21 f xn1 ðtÞb1 FðrÞxn1 ðtrÞ drc2 dt 0   Z 1 n n FðrÞxn2 ðtrÞ drc1 , þ a22 f x2 ðtÞb2

207

x2 ðtÞ ¼ xn2 ðtÞb2

Z

1

FðrÞxn2 ðtrÞ dr

0

ð5Þ

we can rewrite (3) as the following equivalent system: Z 0 dx1 ðtÞ FðrÞf ½x1 ðt þ rÞ dr ¼ x1 ðtÞ þ an11 f ½x1 ðtÞ þ an12 f ½x2 ðtÞan11 b1 dt 1 Z 0 an12 b1 FðrÞf ½x2 ðt þ rÞ dr, 1

dx2 ðtÞ ¼ x2 ðtÞ þ an21 f ½x1 ðtÞ þ an22 f ½x2 ðtÞan21 b2 dt Z 0 an22 b2 FðrÞf ½x2 ðt þ rÞ dr:

Z

0

FðrÞf ½x1 ðt þ rÞ dr

1

ð6Þ

1

We make the following assumption on function f: ðH1 Þ f A C 1 ðR, RÞ,

f ð0Þ ¼ 0:

Suppose that ðx 1 ,x 2 Þ is the equilibrium of system (6), then we have x 1 þ an11 f ðx 1 Þ þ an12 f ðx 2 Þan11 b1 f ðx 1 Þan12 b1 f ðx 2 Þ ¼ 0, x 2 þ an21 f ðx 1 Þ þ an22 f ðx 2 Þan21 b2 f ðx 1 Þan22 b2 f ðx 2 Þ ¼ 0:

ð7Þ

According to the inverse function theorem, the equilibrium (0,0) of system (6) is isolated if 9det Jð0,0Þ9 40,

ð8Þ

where J is the Jacobian matrix of the nonlinear system of Eq. (7) with respect to x 1 and x 2 . Inequality (8) is satisfied if ð3Þ

0

where a1i , a2i , bi and ci ði ¼ 1,2Þ are nonnegative constants, and FðÞ is the weak kernel. By using the frequency domain approach, we devote our attention to the properties of the Hopf bifurcation for model (3). It is found that if the mean delay is used as a bifurcation parameter, then model (3) displays the Hopf bifurcation. This means that a family of periodic solutions bifurcate 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. The remainder of the paper is organized as follows. In Section 2, by means of the frequency domain approach introduced by Moiola and Chen [23], the existence of the Hopf bifurcation is

0

0

9½1þ an11 f ð0Þð1b1 Þ½1þ an22 f ð0Þð1b2 Þ9 0

4 an12 an21 ½f ð0Þ2 9ð1b1 Þð1b2 Þ9:

ð9Þ

By applying weak kernel (4), system (6) can be rewritten as follows: Z t dx1 ðtÞ mems f ½x1 ðsÞ ds ¼ x1 ðtÞ þ an11 f ½x1 ðtÞþ an12 f ½x2 ðtÞan11 b1 emt dt 1 Z t an12 b1 emt mems f ½x2 ðsÞ ds, 1

dx2 ðtÞ ¼ x2 ðtÞ þ an21 f ½x1 ðtÞþ an22 f ½x2 ðtÞan21 b2 emt dt Z t an22 b2 emt mems f ½x2 ðsÞ ds: 1

Z

t

mems f ½x1 ðsÞ ds

1

ð10Þ

208

M. Xiao et al. / Neurocomputing 99 (2013) 206–213

Taking the derivative with respect to t on both sides of (10), we can achieve 2

d x1 ðtÞ dx1 ðtÞ dx1 ðtÞ 0 þ an11 f ½x1 ðtÞ ¼ dt dt dt 2 dx2 ðtÞ n 0 n a11 b1 mf ½x1 ðtÞ þ a12 f ½x2 ðtÞ dt   dx1 ðtÞ þ x1 ðtÞan11 f ½x1 ðtÞan12 f ½x2 ðtÞ , an12 b1 mf ½x2 ðtÞm dt

where 0 0 B0 B B¼B @1 0

0

1

0C C C, 0A

C ¼ I44 :

ð17Þ

1 !

gðy; mÞ ¼

mð1b1 Þ½an11 f ðy1 Þ þ an12 f ðy2 Þan11 f 0 ðy1 Þy3 an12 f 0 ðy2 Þy4 : mð1b2 Þ½an21 f ðy1 Þ þ an22 f ðy2 Þan21 f 0 ðy1 Þy3 an22 f 0 ðy2 Þy4 ð18Þ

2

d x2 ðtÞ dx2 ðtÞ dx1 ðtÞ 0 þ an21 f ½x1 ðtÞ ¼ dt dt dt 2 dx2 ðtÞ n 0 n a21 b2 mf ½x1 ðtÞ þ a22 f ½x2 ðtÞ dt   dx2 ðtÞ þ x2 ðtÞan21 f ½x1 ðtÞan22 f ½x2 ðtÞ : an22 b2 mf ½x2 ðtÞm dt ð11Þ Setting x3 ðtÞ ¼ dx1 ðtÞ=dt and x4 ðtÞ ¼ dx2 ðtÞ=dt, then we get the following system: dx1 ðtÞ ¼ x3 ðtÞ, dt dx2 ðtÞ ¼ x4 ðtÞ, dt

0

s

The linearization of the feedback (18) at the equilibrium y¼0 is given by !  ma11 ð1b1 Þ ma12 ð1b1 Þ a11 a12 @g  , JðmÞ ¼  ¼ ma21 ð1b2 Þ ma22 ð1b2 Þ a21 a22 @y y ¼ 0 ð20Þ n 0

where aij ¼ aij f ð0Þ, i,j ¼1,2. So, we have

dx3 ðtÞ ¼ mx1 ðtÞðm þ1Þx3 ðtÞ þ man11 ð1b1 Þf ½x1 ðtÞ dt 0 0 þ man12 ð1b1 Þf ½x2 ðtÞ þan11 f ½x1 ðtÞx3 ðtÞ þ an12 f ½x2 ðtÞx4 ðtÞ,

Gðs; mÞJðmÞ ¼ 0

ð12Þ The nonlinear system (12) can be rewritten in a matrix form as dx ¼ AðmÞx þ HðxÞ, dt

ð13Þ

0 1 0 ðm þ1Þ

1 C C C, C A

1 ðs þ1Þðs þ mÞ

ma11 ð1b1 Þ

B B ma21 ð1b2 Þ B B ma11 sð1b1 Þ @ ma21 sð1b2 Þ

dx4 ðtÞ ¼ mx2 ðtÞðm þ1Þx4 ðtÞ þ man21 ð1b2 Þf ½x1 ðtÞ dt 0 0 þ man22 ð1b2 Þf ½x2 ðtÞ þan21 f ½x1 ðtÞx3 ðtÞ þ an22 f ½x2 ðtÞx4 ðtÞ:

where x ¼ ðx1 ,x2 ,x3 ,x4 Þ, 0 0 0 1 B 0 0 B 0 AðmÞ ¼ B B m 0 ðm þ 1Þ @ 0 0 m

Next, taking Laplace transform on (16), we obtain the standard transfer matrix of the linear part of system (16): 0 1 1 0 B0 1C 1 B C Gðs; mÞ ¼ C½sIAðmÞ1 B ¼ ð19Þ B C: ðsþ 1Þðs þ mÞ @ s 0 A

ð14Þ

0

1 0 B C 0 B C B n C 0 0 n n n HðxÞ ¼ B ma11 ð1b1 Þf ðx1 Þ þ ma12 ð1b1 Þf ðx2 Þ þ a11 f ðx1 Þx3 þ a12 f ðx2 Þx4 C: B n C @ ma21 ð1b2 Þf ðx1 Þ þ man22 ð1b2 Þf ðx2 Þ þ an21 f 0 ðx1 Þx3 þ an22 f 0 ðx2 Þx4 A

dx ¼ AðmÞx þ Bu, dt

a11

ma22 ð1b2 Þ

a21

ma12 sð1b1 Þ

a11 s

ma22 sð1b2 Þ

a21 s

a12

1

C a22 C C: a12 s C A

ð21Þ

a22 s

Set hðl,s; mÞ ¼ det½lIGðs; mÞJðmÞ  a11 s þa22 s þa11 m þa22 ma11 mb1 a22 mb2 2 2 ¼l l þ l ðs þ 1Þðsþ mÞ # ðsb2 m þ mÞðsb1 m þ mÞða11 a22 a12 a21 Þ þ ¼ 0: ðs þ 1Þ2 ðs þ mÞ2

ð22Þ

Then, we obtain the following results by applying the generalized Nyquist stability criterion with s ¼ io. Lemma 1 (Moiola and Chen [23]). If an eigenvalue of the corresponding Jacobian of the nonlinear system, in the time domain, assumes a purely imaginary value io0 at a particular m ¼ m0 , then the corresponding eigenvalue of the constant matrix Gðio0 ; mÞJðm0 Þ in the frequency domain must assume the value 1þ i0 at m ¼ m0 . Let l^ ¼ l^ ðio; mÞ be the eigenvalue of Gðio; mÞJðmÞ that satisfies ^l ðio ; m Þ ¼ 1 þ i0. Then 0 0

ð15Þ Choosing the mean delay m as the bifurcation parameter and introducing a ‘‘state-feedback control’’ u ¼ gðy; mÞ, we obtain a linear system with a nonlinear feedback as follows:

ma12 ð1b1 Þ

hð1,io0 ; m0 Þ ¼ 1

þ

a11 io0 þ a22 io0 þ a11 m0 þ a22 m0 a11 m0 b1 a22 m0 b2 ðio0 þ 1Þðio0 þ m0 Þ

ðio0 b2 m0 þ m0 Þðio0 b1 m0 þ m0 Þða11 a22 a12 a21 Þ ðio0 þ 1Þ2 ðio0 þ m0 Þ2

¼ 0:

ð23Þ

Hence, we have ða11 io0 þa22 io0 þ a11 m0 þ a22 m0 a11 m0 b1 a22 m0 b2 Þðio0 þ 1Þðio0 þ m0 Þ

ðio0 b2 m0 þ m0 Þðio0 b1 m0 þ m0 Þða11 a22 a12 a21 Þ

y ¼ Cx, u ¼ gðy; mÞ,

ð16Þ

¼ ðio0 þ 1Þ2 ðio0 þ m0 Þ2 :

ð24Þ

M. Xiao et al. / Neurocomputing 99 (2013) 206–213

By (29), we have

Separating the real and imaginary parts, we obtain 4 0 þ c3

c4 m

m



3 0 þc 2

o20 ¼ m0 1 þ

2 0 þ c1

m

209

m0 þc0 ¼ 0,

ð25Þ

 d1 ðm0 Þ , d2 ðm0 Þ

dlðm0 Þ ¼ 4m30 þ 3pm20 þ2qm0 þr: dm0

ð31Þ

For convenience, let m0 ¼ yð1=4Þp; then ð26Þ

4m30 þ 3pm20 þ2qm0 þr ¼ 0

where

becomes

c4 ¼ 2ða11 þ a22 a11 b1 a22 b2 2Þ,

y3 þmy þ n ¼ 0,

ð32Þ

ð33Þ 2

c3 ¼ 7a11 a22 ðb1 þb2 Þ þ20ða11 þ a22 1Þða11 b1 þ a22 b2 Þ2 4ða211 þa222 Þ þ 5ða211 b1 þ a222 b2 Þþ 2a12 a21 ð2b1 b2 Þ12ða11 b1 þa22 b2 þ a11 a22 Þ,

2

where m ¼ ð1=2Þqð3=16Þp , n ¼ ð1=32Þp ð1=8Þpq þð1=4Þr. Obviously, (33) has three roots qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi q pffiffiffiffi 3 3 yð1Þ ¼ 12n þ D þ 12n D,

yð2Þ ¼ o1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi pffiffiffiffi 3 3 12n þ D þ o2 12n D,

yð3Þ ¼ o2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi pffiffiffiffi 3 3 12n þ D þ o1 12n D,

2

c2 ¼ ða11 þ a22 Þð488a12 a21 þ 14a11 a22 þ 2a11 b1 a22 b2 Þ2ða211 b1 þ a222 b2 Þ þ ðb1 þ b2 Þð23a22 a11 6a12 a21 Þ þ ð3a12 a21 6a11 a22 Þða11 b2 þ a22 b1 Þ 2

2

þ a11 a22 ða11 b1 þ a22 b2 Þ þ ð5a12 a21 2011a11 a22 Þða11 b1 þ a22 b2 Þ þ 16a12 a21 4a11 b1 a22 b2 20ða211 þ a222 Þ þ2ða311 þ a322 Þ32 2

2

þ 17ða211 b1 þ a222 b2 Þ3ða311 b1 þ a322 b2 Þ þ a311 b1 þa322 b2 56a11 a22 ,

c1 ¼ ð5a211 a222 þ 20a11 a22 5a12 a21 2a212 a221 3a11 a22 a12 a21 Þðb1 þ b2 Þ

3

where

pffiffiffi

ð34Þ pffiffiffi

D ¼ ð1=4Þn2 þð1=27Þm3 , o1 ¼ ð1 þ 3iÞ=2, o2 ¼ ð1 3iÞ=2. ðiÞ Then we have mðiÞ 0 ¼ y ð1=4Þp, i ¼ 1,2,3. From Cardano’s formula for the third degree algebra equation, we can introduce the following lemma about the roots of Eq. (32).

þ 19a12 a21 þð35a11 a22 þ 2a11 b1 a22 b2 þ3920a12 a21 Þða11 þ a22 Þ þ ð10a12 a21 20a11 a22 11ÞÞða11 b1 þ a22 b2 Þa211 b1 a222 b2 19

Lemma 2. For Eq. (32), the following results hold.

þ ð6a12 a21 11a11 a22 Þða11 b2 þ a22 b1 Þ þ ð5a12 a21 25Þða211 þa222 Þ

ð2Þ (i) If D o 0, then Eq. (32) has three unequal real roots mð3Þ 0 o m0 o ð1Þ m0 . (ii) If D 40, then Eq. (32) has one simple real root mð1Þ 0 and a pair of conjugate complex roots. (iii) If D ¼ 0 and m ¼ n ¼ 0, then Eq. (32) has one triple real root mð1Þ 0 ¼ ð1=4Þp. (iv) If D ¼ 0 and ð1=4Þn2 ¼ ð1=27Þm3 a0, then Eq. (32) has one ð2Þ simple real root mð1Þ 0 and one double real root m0 .

2

2

69a11 a22 þ ð15 þ5a11 a22 5a12 a21 Þða211 b1 þ a222 b2 Þa11 b1 a22 b2 2

2

10a211 a222 þ 10a11 a22 a12 a21 þ 2a212 a221 b1 b2 þ a211 b1 a22 þ a222 b2 a11 2

2

þ 2a11 a322 b1 þ 2a311 a22 b2 þ ða212 a221 a11 a22 a12 a21 Þðb1 þ b2 Þ 2 2 þ 5ða311 þa322 Þ þ a311 ðb1 5b1 5a22 Þ þ a322 ðb2 5b2 5a11 Þ 2

2

2

2

2a12 a21 ða211 b2 þ a222 b1 Þa311 b1 a22 a322 b2 a11 þ a12 a21 ða211 b1 þ a222 b2 Þ

b1 b2 ða211 þ a222 Þða12 a21 a11 a22 Þa12 a21 b2 b1 2a11 a22 b1 b2 a12 a21 ,

c0 ¼ ð5a12 a21 7a11 a22 2Þða11 b1 þ a22 b2 Þ þ ð4a12 a21 8Þða211 þ a222 Þ4 þ ð5a11 a22 þ 5a211 a222 þ2a212 a221 a211 a322 a311 a222 2a12 a21 Þðb1 þ b2 Þ

7a11 a22 a12 a21 ðb1 þb2 Þ þ ð3a12 a21 4a11 a22 Þða11 b2 þ a22 b1 Þ þ 8a12 a21 þð18a11 a22 þ 2a212 a221 4a11 a22 a12 a21 12a12 a21 þ 10Þða11 þ a22 Þ þ ð2a11 a22 2a12 a21 þ 3Þða211 b1 þ a222 b2 Þ4a311 a22 þ 2a211 a322 24a11 a22 þa311 ð2b1 Þ þ a322 ð2b2 Þ12a211 a222 4a11 a322

þ 16a11 a22 a12 a21 þ 2a311 a222 þ ð2a11 a22 a12 a21 a212 a221 Þðb1 þ b2 Þða11 þa22 Þ4a212 a221 þða11 a22 a12 a21 Þða211 b2 þ a222 b1 Þ ð27Þ and d1 ðm0 Þ ¼ ða11 þ a22 a11 b1 a22 b2 Þðm0 þ 1Þða11 a22 a12 a21 Þð2b1 b2 Þ,

d2 ðm0 Þ ¼ a11 þa22 2m0 2:

c3 , c4

q¼

c2 , c4

r¼

c1 , c4

k¼

c0 : c4

ð3Þ kZ 0, D o 0, mð3Þ 0 4 0, and hðm0 Þ r 0; ð3Þ kZ 0, D o 0, m0 4 0, and hðmð1Þ 0 Þ r 0; ð3Þ ð1Þ kZ 0, D o 0, mð3Þ 0 4 0, hðm0 Þ 4 0, and hðm0 Þ 4 0; ð1Þ ð1Þ kZ 0, D o 0, mð3Þ r 0, m 40, and hð m Þ 0 0 0 r 0; ð1Þ ð1Þ kZ 0, D o 0, mð3Þ 0 r 0, m0 40, and hðm0 Þ 4 0; kZ 0, D o 0, and mð1Þ r 0; 0 ð1Þ kZ 0, D 4 0, mð1Þ 0 4 0, and hðm0 Þ r 0; ð1Þ kZ 0, D 4 0, m0 4 0, and hðmð1Þ 0 Þ 4 0; kZ 0, D 4 0, and mð1Þ 0 r 0; ð1Þ kZ 0, m ¼ n ¼ 0, mð1Þ 0 4 0, and hðm0 Þ r0; ð1Þ kZ 0, m ¼ n ¼ 0, mð1Þ 4 0, and hð m 0 0 Þ 40; ð1Þ kZ 0, m ¼ n ¼ 0, and m0 r 0; ð1Þ kZ 0, D ¼ 0,ð1=4Þn2 ¼ ð1=27Þm3 a 0, mð1Þ 0 4 0, and hðm0 Þ r 0; ð1Þ kZ 0, D ¼ 0,ð1=4Þn2 ¼ ð1=27Þm3 a 0, mð1Þ 4 0, and hð m 0 0 Þ 4 0; ð1Þ 2 3 kZ 0, D ¼ 0,ð1=4Þn ¼ ð1=27Þm a 0, and m0 r 0.

From the results for the fourth degree algebra equation in Cao and Xiao [16], we can introduce the following lemma about the roots of Eq. (25).

ð29Þ

Lemma 3. For Eq. (25), we have the following results:

where p¼

ðA1 Þ ðA2 Þ ðA3 Þ ðA4 Þ ðA5 Þ ðA6 Þ ðA7 Þ ðA8 Þ ðA9 Þ ðA10 Þ ðA11 Þ ðA12 Þ ðA13 Þ ðA14 Þ ðA15 Þ

ð28Þ

Denote lðm0 Þ ¼ m40 þ pm30 þ qm20 þ r m0 þ k,

For convenience, we make some hypotheses as follows:

ð30Þ

(i) If ko 0, then Eq. (25) has at least one positive root. (ii) If any one of ðA1 Þ, ðA2 Þ, ðA4 Þ, ðA7 Þ, ðA10 Þ, ðA13 Þ holds, then Eq. (25) has positive roots.

210

M. Xiao et al. / Neurocomputing 99 (2013) 206–213

(iii) If any one of ðA3 Þ, ðA5 Þ, ðA6 Þ, ðA8 Þ, ðA9 Þ, ðA11 Þ, ðA12 Þ, ðA14 Þ, ðA15 Þ holds, then Eq. (25) has no positive root. The proof of this lemma is similar to that of Lemma2 in [16]. Theorem 1 (Existence of Hopf bifurcation). Suppose that ðH1 Þ and c1 ¼ c2 ¼ 0 hold. Then (i) If either k o 0 or any one of ðA1 Þ, ðA2 Þ, ðA4 Þ, ðA7 Þ, ðA10 Þ, ðA13 Þ holds, system (3) undergoes a Hopf bifurcation at m ¼ m0 , which is the positive root of Eq. (25) and satisfies 1 þ ðd1 ðm0 Þ=d2 ðm0 ÞÞ 4 0. (ii) If any one of ðA3 Þ, ðA5 Þ, ðA6 Þ, ðA8 Þ, ðA9 Þ, ðA11 Þ, ðA12 Þ, ðA14 Þ, ðA15 Þ holds, the Hopf bifurcation of system (3) does not exist.

3. Stability of bifurcating periodic solutions In this section we study the stability of bifurcating periodic solutions by the frequency domain formulation of Moiola and Chen [23]. For convenience, we set b ¼ b1 ¼ b2 for system (3). We make a further assumption on the function f as follows: ðH2 Þ : f A C 3 ðR, RÞ,

uf ðuÞ 4 0

0

for u a 0,

00

000

f ð0Þ a0, f ð0Þ ¼ f ð0Þ ¼ 0, and f ð0Þ a0: A special function which has been widely used in the neuron network models as a transfer function given by tanhðuÞ does satisfy the condition ðH2 Þ. First, we define an auxiliary vector of the following form: ~ Þ¼ x1 ðo

~ ; m~ Þp1 wT ½Gðio , wT v

ð35Þ

~ is the frequency where m~ is the fixed value of the parameter m, o ~ ; m~ Þ locus and the negative of the intersection between the l^ ðio real axis closest to the point ð1 þ i0Þ, wT and v are the left and ~ ; m~ ÞJðm~ Þ, respectively, associated with right eigenvectors of ½Gðio ~ ; m~ Þ, and the value l^ ðio p1 ¼ ½D2 ðV 02  v þ 12 v  V 22 Þ þ 18D3 v  v  v,

ð36Þ

where D2 ¼

 @2 gðy; m~ Þ @y2 

,

D3 ¼

y¼0

 @3 gðy; m~ Þ @y3 

,

ð37Þ

y¼0

V 02 ¼ 14½I þ Gð0; m~ ÞJðm~ Þ1 Gð0; m~ ÞD2 v  v,

ð38Þ

~ ; m~ ÞJðm~ Þ1 Gð2io ~ ; m~ ÞD2 v  v: V 22 ¼ 14½I þ Gð2io

ð39Þ

Also, 0

1 e B 1 C B C v¼B C, ~ A @ ieo ~ io

0

1 a21 em~ ð1bÞ B C B a12 m~ ð1bÞ C C, w¼B B C a21 e @ A a12

where e¼

~ a12 ½ð1bÞm~ þ io : ~ ~ ~ ~ ~ ðm þio Þð1 þ io Þl þ a11 ½ð1bÞm~ þ io

Hence,    1 1 p1 ¼ D2 V 02  v þ v  V 22 þ D3 v  v  v 2 8 1 ¼ D3 v  v  v 8 ! 000 ~  a12 þ a11 e2 e f ð0Þ½ð1bÞm~ þ io ¼ 0 a22 þ a21 e2 e 8f ð0Þ

~ ; m~ Þp1 wT ½Gðio wT v 000 ~ f ð0Þ½ð1bÞm~ þio ¼ 0 ~ Þð1 þ io ~ Þða21 e2 þ a12 Þ 8f ð0Þðm~ þ io

~ Þ¼ x1 ðo

½a21 eða12 þ a11 e2 eÞ þ a12 ða22 þ a21 e2 eÞ:

000

f ð0Þ ½bði,jÞ264 , 0 f ð0Þ

Now, the following Hopf bifurcation theorem formulated in the frequency domain can be established: Lemma 4 (Moiola and Chen [23]). Suppose that the locus of the distinguished characteristic function l^ ðsÞ intersects the negative real ~ Þ that is closest to the point ð1 þ i0Þ when the variable s axis at l^ ðio sweeps on the classical Nyquist contour. Moreover, suppose that ~ Þ is a nonzero and the half-line L1 starting from ð1þ i0Þ in the x1 ðo ~ Þ first intersects the locus of l^ ðioÞ at direction defined by x1 ðo ~ Þ ¼ P^ ¼ 1 þ x1 ðo ~ Þy2 , where y ¼ Oð9mm0 91=2 Þ. Finally, suppose l^ ðio that following conditions are satisfied: (i) The eigenlocus l^ has a nonzero rate of change with respect to its parameterization at the criticality ðo0 , m0 Þ, i.e., " # @F 1 =@m @F 2 =@m  Mðo0 , m0 Þ ¼ det a 0,  @F 1 =@o @F 2 =@o  ðo0 , m0 Þ

where F 1 ðo, mÞ ¼ Rfhð1,io; mÞg, F 1 ðo, mÞ ¼ Jfhð1,io; mÞg. (ii) The intersection 2 is transversal, i.e., 3 ^ Þg ^ Þg Rfx1 ðo Jfx1 ðo   n o n o 5 a 0: ^ , m~ Þ ¼ det4 Nðo l^  l^  J ddo R ddo  

bð1,24Þ ¼ a12 ,

bð1,30Þ ¼ a12 ,

bð2,1Þ ¼ m~ a21 ð1bÞ, bð2,24Þ ¼ a22 ,

bð1,3Þ ¼ a11 ,

bð2,3Þ ¼ a21 ,

bð2,30Þ ¼ a22 ,

bð1,9Þ ¼ a11 ,

bð1,33Þ ¼ a11 ,

bð1,22Þ ¼ m~ a12 ð1bÞ,

bð1,54Þ ¼ a12 ,

bð2,9Þ ¼ a21 ,

bð2,33Þ ¼ a21 ,

bð2,22Þ ¼ m~ a22 ð1bÞ,

bð2,54Þ ¼ a22

and the others elements of D3 are zero. Therefore, V 02 ¼ V 22 ¼ 0:

o ¼ o^

ð40Þ

where bð1,1Þ ¼ m~ a11 ð1bÞ,

ð44Þ

(iii) There are no other intersections between any of the character^ at istic loci and the line segment joining the point ð1 þ i0Þ to P, least within a small neighborhood of radios d 4 0.

Then, we have D3 ¼ 

ð43Þ

and

o ¼ o^

D2 ¼ 0216 ,

ð42Þ

ð41Þ

Then, the system (16) has a periodic solution y(t) of frequency 4 o ¼ o^ þ Oðy^ Þ. Moreover, by applying a small perturbation around the intersection P^ and using the generalized Nyquist stability criterion, the stability of the periodic solution y(t) can be determined. According to Lemma 4, we can determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solution by drawing the figure of the half-line L1 and the locus l^ ðioÞ. For more detailed knowledge, one can see Moiola and Chen [23]. If the half-line L1 first intersects the locus of l^ ðioÞ when m~ 4 m0 ð o m0 Þ, then the bifurcating periodic solution exists and the direction of the Hopf bifurcation is supercritical (subcritical). If the total number of anticlockwise encirclements of the point

M. Xiao et al. / Neurocomputing 99 (2013) 206–213

~ Þ, for a small enough e 40, is equal to the number of P 1 ¼ P^ þ ex1 ðo poles of lðsÞ that have positive real parts, then the limit cycle is stable. otherwise, it is unstable. We can perturb the bifurcation parameter m slightly from m0 to m~ . If l~ 4 1 and Jfdl^ =do9o ¼ o^ g Nðo^ , m~ Þ 4 0, or l~ o 1 and ^ , m~ Þ o 0, then the half-line L1 intersects the Jfdl^ =do9o ¼ o^ gNðo locus of l^ ðioÞ. Theorem 2. Suppose that ðH2 Þ 8  0  < dl^  dl~   @ s ¼ sgn J   :do dm m ¼ m0

where  dl~   dm

and b ¼ b1 ¼ b2 hold. Set 91 = A  Nðo0 , m Þ, 0 ;

ð45Þ

211

a3 ðmÞ ¼ ½ða211 þ a222 Þðb1Þðb4Þ þ 2bð2a12 a21 a11 a22 Þðb þ 3Þ þ 8a11 a22 m3 2

2

þ ½ða211 þ a222 Þð2b 13bþ 16Þ þ2a12 a21 ð4b þ14b4Þ 2

2a11 a22 bð11 þ2bÞm2 þ½ða211 þ a222 Þðb 11b þ 20Þ 2

2

þ 2a12 a21 ð2b þ 10b8Þ2a11 a22 ðb þ13b28Þm þ ða211 þa222 Þð83bÞ þ 2a12 a21 ð2b4Þ2a11 a22 ð5b12Þ,

a2 ðmÞ ¼ fða311 þ a322 Þðb2 3b þ 2Þ þ ða11 þa22 Þ½4a12 a21 ðb2 þ 4b1Þ 2 2 a11 a22 ðb þ 9b10Þgm2 þ fða311 þa322 Þðb 4b þ4Þ

o ¼ o0

2

2

þ ða11 þa22 Þ½4a12 a21 ðb þ 6b4Þa11 a22 ðb þ 20b28Þgm ða311 þ a322 Þð2bÞ þ ða11 þ a22 Þ½4a12 a21 ð2b3Þ þ a11 a22 ð1811bÞ,

¼

m ¼ m0

a05 ðm0 Þa04 ðm0 Þ þ a03 ðm0 Þa02 ðm0 Þ þ a01 ðm0 Þa00 ðm0 Þ , 5a5 ðm0 Þ4a4 ðm0 Þ þ3a3 ðm0 Þ2a2 ðm0 Þ þ a1 ðm0 Þ

a1 ðmÞ ¼ fða211 þ a222 Þ½a12 a21 ð9bb2 4Þ þ a11 a22 ðb2 5bþ 4Þ 2

2

2

þ a211 a222 ð86b2b Þ þ 4a212 a221 ðbb Þ þ 2a11 a22 a12 a21 ð3b þ 5b4Þgm 4

3

2

a5 ðmÞ ¼ 4m þ 16m þ 24m þ 16m þ 4,

ða211 þ a222 Þða12 a21 a11 a22 Þð3b4Þ þ 4a212 a221 ð1bÞ þ2a211 a222 ð65bÞ

a4 ðmÞ ¼ 2ða11 þa22 Þð1bÞm4 þ 8ða11 þ a22 Þð2bÞm3 þ 12ða11 þ a22 Þð3bÞm2 þ 8ða11 þa22 Þð4bÞm

2a11 a22 a12 a21 ð7b8Þ,

þ 2ða11 þ a22 Þð5bÞ,

a0 ðmÞ ¼ 2ða11 þ a22 Þð1bÞða11 a22 a12 a21 Þ2 , Frequency graph

0.6

and (

0.4

R ^ λ

^ ℑ (λ)

0.2

)  dl  RðK 1 1ÞRðK 3 K 1 K 2 Þ þ JðK 1 1ÞJðK 3 K 1 K 2 Þ , ¼ doo ¼ o0 ½RðK 1 1Þ2 þ ½JðK 1 1Þ2

(

)  dl  RðK 1 1ÞJðK 3 K 1 K 2 ÞJðK 1 1ÞRðK 3 K 1 K 2 Þ J , ¼ doo ¼ o0 ½RðK 1 1Þ2 þ½JðK 1 1Þ2

0 −0.2

where

−0.4

K1 ¼

−0.6 −0.8 −1.5

−1

−0.5

0

0.5

1

1.5

^ ℑ (λ)

K2 ¼

Fig. 1. Frequency graph with a11 ¼ a12 ¼ 2, a21 ¼ a22 ¼ 0:4, b1 ¼ b2 ¼ 0:8 and m ¼ 1:41 4 m0 . There is no intersection between the half-line L1 and the eigenlocus l^ ðioÞ. So no periodic solution exists.

Waveform plot

K3 ¼

ða11 þ a22 Þf2o0 m0 ð1bÞ þ i½o20 m0 ð1bÞð1 þ m0 Þ þ m0 g ðm0 þ io0 Þ2 ð1 þio0 Þ2

Waveform plot

0.5

0.6

0.5

0.4

0.4

0.3

0.3

0.2

x2

0 −0.2 −0.4 −0.6 −0.8

0

100

200

300

400 t

500

600

700

800

0.2

0.1

0.1

0

0

−0.1

−0.1

Phase portrait

x2

0.6

0.2

,

2i 2i 2i   : ð1bÞm0 þ io0 1 þio0 m0 þ io0

0.8

0.4

x1

ðm0 þ io0 Þð1 þ io0 Þ , ða11 þ a22 Þ½ð1bÞm0 þ io0 ðm0 þio0 Þð1 þio0 Þ

0

100

200

300

400 t

500

600

700

800

−0.2 −0.8 −0.6 −0.4 −0.2

0 x1

0.2

0.4

0.6

0.8

Fig. 2. Waveform plot and phase portrait of system (3) with a11 ¼ a12 ¼ 2, a21 ¼ a22 ¼ 0:4, b1 ¼ b2 ¼ 0:8 and m ¼ 1:41 4 m0 . The initial value ðx1 ð0Þ,x2 ð0ÞÞ ¼ ð0:5,0:4Þ. The equilibrium (0,0) is asymptotically stable and no periodic solution exists.

212

M. Xiao et al. / Neurocomputing 99 (2013) 206–213

is the intersection of the half-line L1 and the locus l^ ðioÞ. Suppose that ðH2 Þ and b ¼ b1 ¼ b2 hold. Then

Then 1. if s 4 0, the Hopf bifurcation at m ¼ m0 of system (3) is supercritical; 2. if s o0, the Hopf bifurcation at m ¼ m0 of system (3) is subcritical. We can recall the algorithm and process of Appendix B in [15] to derive the above formulas. ~ Þ starting from ð1 þ i0Þ Then, we can draw the half-line x1 ðo and the locus l^ ðioÞ, and obtain the total number k of antic~ Þ for a small lockwise encirclements of the point P 1 ¼ P^ þ ex1 ðo enough e 4 0. According to Eq. (22) with b ¼ b1 ¼ b2 , we have

l2 þ

ða11 þa22 Þðs þ mmbÞ ðs þ mmbÞ2 ða11 a22 a12 a21 Þ lþ ¼ 0: ðs þ 1Þðs þ mÞ ðs þ1Þ2 ðs þ mÞ2

ð46Þ

Hence, s ¼ m and s ¼ 1 are the poles of lðsÞ, and the number of poles of lðsÞ that have positive real parts is zero. Theorem 3. Let k be the total number of anticlockwise encircle~ Þ for a small enough e 40, where P^ ments of the point P1 ¼ P^ þ ex1 ðo

Frequency graph

s ¼ 0:3326 o0:

^ ℑ (λ)

0.3 0.2 0.1 0 −0.1 −0.2 −1.8 −1.6 −1.4 −1.2

−1

−0.8 −0.6 −0.4 −0.2

0

^ ℜ (λ) Fig. 3. Frequency graph with a11 ¼ a12 ¼ 2, a21 ¼ a22 ¼ 0:4, b1 ¼ b2 ¼ 0:8 and m ¼ 1:39 o m0 . The half-line L1 intersects the eigenlocus l^ ðioÞ and k ¼ 0. So a stable periodic solution exists.

Waveform plot

0.4

0

x2

x1

0.2

−0.2 −0.4 −0.6 50

100

150 t

200

Waveform plot

0.6

0.6

250

300

Therefore, m0 ¼ 1:4 is a subcritical Hopf bifurcation. Now, if we focus our attention on the frequency graphs in each figure in the vicinity of the critical point l^ ¼ 1, we can study the existence and the direction of the Hopf bifurcation in system (3) by perturbing the bifurcation parameter m in the neighborhood of m0 . There is no intersection between the amplitude locus L1 and the characteristic eigenlocus l^ ðioÞ when m 4 m0 as is illustrated by computer simulations (Fig. 1: m ¼ 1:41). Thus, according to Theorem 3, there is no periodic solution as the corresponding waveform plot and phase portrait illustrated in Fig. 2. As the mean delay m passes its critical value m0 , a transversal intersection occurs. According to Theorem 3, this gives rise to the emergence of a periodic solution. This intersection is illustrated in Fig. 3. The bifurcating periodic solution can be viewed in the corresponding waveform plot and phase portrait in Fig. 4.

0.5

0.5

0.4

0.4

0.3

0.3

0.2 x2

0.8

0

In this section, we present some numerical results to verify the correctness of our computation and the frequency graph. By Theorem 2, s determines the direction of the Hopf bifurcation, i.e., the Hopf bifurcation is supercritical if s 4 0 and subcritical if s o 0. The half-line L1 and the locus l^ ðioÞ are shown in the frequency graphs. If they intersect, a limit cycle appears. By Theorem 3, the stability of bifurcating periodic solutions is determined by the total number k of anticlockwise encirclements ~ Þ for a sufficiently small e 40. Suppose of the point P 1 ¼ P^ þ ex1 ðo that the half-line L1 and the locus l^ ðioÞ intersect, then the bifurcating periodic solution is stable if k ¼ 0 and unstable if k a 0. As an example, let a11 ¼ a12 ¼ 2, a21 ¼ a22 ¼ 0:4, b1 ¼ b2 ¼ 0:8, f ðxÞ ¼ tanhðxÞ in system (3). By Eqs. (25) and (26), we can determine that

From Eq. (45), it follows that

0.4

−0.8

4. Numerical examples and the frequency graphs

o0 ¼ 0:8532, m0 ¼ 1:4:

0.5

−2

1. if k ¼ 0, then the bifurcating periodic solution of system (3) is stable; 2. if k a0, then the bifurcating periodic solution of system (3) is unstable.

0.2

0.1

0.1

0

0

−0.1

−0.1

0

50

100

150 t

200

250

300

Phase portrait

−0.2 −0.8 −0.6 −0.4 −0.2

0 x1

0.2

0.4

0.6

0.8

Fig. 4. Waveform plot and phase portrait of system (3) with a11 ¼ a12 ¼ 2, a21 ¼ a22 ¼ 0:4, b1 ¼ b2 ¼ 0:8 and m ¼ 1:39 o m0 . The initial value ðx1 ð0Þ,x2 ð0ÞÞ ¼ ð0:5,0:4Þ. The Hopf bifurcation occurs from the equilibrium ð0,0Þ.

M. Xiao et al. / Neurocomputing 99 (2013) 206–213

5. Conclusions A two-neuron model with distributed delays, self-feedbacks and the weak kernel has been studied in the frequency domain. By choosing the mean delay as the bifurcation parameter, we have shown that system (3) undergoes a Hopf bifurcation as this bifurcation parameter passes a critical value. The stability of bifurcating periodic solutions has been analyzed by drawing the amplitude locus L1 and the eigenlocus l^ ðioÞ in a neighborhood of the Hopf bifurcation point ð1 þ0iÞ. Parameter s has been used to determine the direction of the Hopf bifurcation: if s 40, the Hopf bifurcation is supercritical and if s o 0, the Hopf bifurcation is subcritical. Acknowledgments This work was supported in part by the Natural Science Foundation of Jiangsu Province of China (Grant Nos. BK2012072 and BK2012741), the National Natural Science Foundation of China (Grant No. 11072059), the China Postdoctoral Science Foundation funded project (Grant No. 20090461056), the Jiangsu Ordinary University Natural Science Research Project (Grant No. 11KJD120002), and a research grant from the Australian Research Council.

213

[20] D.W. Tank, J.J. Hopfield, Neural computation by concentrating information in time, Proc. Natl. Acad. Sci. U.S.A. 84 (1987) 1896–1991. [21] B. De Vries, J.C. Principle, The gamma model—a new neural model for temporal processing, Neural Networks 5 (1992) 565–576. [22] B. Hassard, N. Kazarinoff, Y. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981. [23] J.L. Moiola, G.R. Chen, Hopf Bifurcation Analysis: A Frequency Domain Approach, World Scientific, Singapore, 1996. [24] A.I. Mees, L.O. Chua, The Hopf bifurcation theorem and its applications to nonlinear oscillations in circuits and systems, IEEE Trans. Circuits Syst. 26 (1979) 235–254. [25] D.J. Allwright, Harmonic balance and the Hopf bifurcation theorem, Math. Proc. Cambridge Philos. Soc. 82 (1977) 453–467.

Min Xiao received the B.S. and M.S. degrees, both in Mathematics/Fundamental Mathematics, from Nanjing Normal University, China, in 1998 and 2001, respectively, and the Ph.D. degree in Applied Mathematics from Southeast University, China, in 2007. He has held various postdoctoral/visiting positions at Southeast University, China, City University of Hong Kong, Hong Kong, and University of Western Sydney, Australia. Currently he is an Associate Professor with the School of Mathematics and Information Technology, Nanjing Xiaozhuang University, China. His current research interests include fractional calculus theory, bifurcation control theory, neural networks, and complex dynamical networks.

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Wei Xing Zheng received the Ph.D. degree in Electrical Engineering from Southeast University, China in 1989. He has held various faculty/research/visiting positions at Southeast University, China, Imperial College of Science, Technology and Medicine, UK, University of Western Australia, Curtin University of Technology, Australia, Munich University of Technology, Germany, University of Virginia, USA, and University of California-Davis, USA. Currently he holds the rank of Full Professor at University of Western Sydney, Australia. Dr. Zheng has been an Associate Editor of several journals, including IEEE Transactions on Automatic Control (2004–2007) and Automatica (2011–present). He has also served as the Chair of IEEE Circuits and Systems Society’s Technical Committee on Neural Systems and Applications and as the Chair of IEEE Circuits and Systems Society’s Technical Committee on Blind Signal Processing.

Jinde Cao (M’07–SM’07) received the B.S. degree from Anhui Normal University, Wuhu, China, the M.S. degree from Yunnan University, Kunming, China, and the Ph.D. degree from Sichuan University, Chengdu, China, all in Mathematics/Applied Mathematics, in 1986, 1989, and 1998, respectively. He was with Yunnan University from 1989 to 2000. Since 2000, he has been with the Department of Mathematics, Southeast University, Nanjing, China. From 2001 to 2002, he was a Post-Doctoral Research Fellow with the Department of Automation and Computer-Aided Engineering, Chinese University of Hong Kong, Shatin, Hong Kong. He was a Visiting Research Fellow and a Visiting Professor with the School of Information Systems, Computing and Mathematics, Brunel University, Middlesex, U.K., from 2006 to 2008. He is the author or co-author of more than 160 research papers and five edited books. His current research interests include nonlinear systems, neural networks, complex systems, complex networks, stability theory, and applied mathematics. Dr. Cao was an Associate Editor of the IEEE Transactions on Neural Networks and Neurocomputing. He is an Associate Editor of the Journal of the Franklin Institute, Mathematics and Computers in Simulation, Abstract and Applied Analysis, International Journal of Differential Equations, Discrete Dynamics in Nature and Society, and Differential Equations and Dynamical Systems. He is a reviewer of Mathematical Reviews and Zentralblatt-Math.