Cogn Neurodyn (2016) 10:149–164 DOI 10.1007/s11571-015-9364-y

RESEARCH ARTICLE

High codimensional bifurcation analysis to a six-neuron BAM neural network Yanwei Liu1 • Shanshan Li2 • Zengrong Liu1 • Ruiqi Wang1

Received: 18 May 2015 / Revised: 15 October 2015 / Accepted: 28 October 2015 / Published online: 14 November 2015 Ó Springer Science+Business Media Dordrecht 2015

Abstract In this article, the high codimension bifurcations of a six-neuron BAM neural network system with multiple delays are addressed. We first deduce the existence conditions under which the origin of the system is a Bogdanov–Takens singularity with multiplicities two or three. By choosing the connection coefficients as bifurcation parameters and using the formula derived from the normal form theory and the center manifold, the normal forms of Bogdanov–Takens and triple zero bifurcations are presented. Some numerical examples are shown to support our main results. Keywords Neural networks  Bogdanov–Takens bifurcation  Triple zero bifurcation

Introduction It is believed that the successful applications of neural network in many fields such as optimization solvers, pattern recognition, automatic control and encryption of image (Cochoki and Unbehauen 1993; Ripley 1996; He et al. 2013; Kadone and Nakamutra 2005; Hoppensteadt and Izhikevich 1997) heavily depend on the theoretical studies about the neuron systems, especially the & Yanwei Liu [email protected] Ruiqi Wang [email protected] 1

Department of Mathematics of Shanghai University, Shanghai 200444, China

2

Institute of Systems Biology, Shanghai University, Shanghai 200444, China

investigation of the dynamical behaviors of neural network models has became focus over the past decade years. Following currently available literatures, the interest in studying the dynamics of the neural network has focused on two main aspects. One topic is aimed at the study of the local and global stability of the equilibrium, the existence and stability of periodic solutions yielded from Hopf bifurcation, as reported in (Xu and Li 2012; Yang and Ye 2009; Xu et al. 2011; Cao 2003; Sun et al. 2007; Zheng et al. 2008; Liu and Cao 2011; Yang et al. 2014), and another topic is aimed at the study of more complex dynamics mainly involving some degenerate bifurcations existing in the neighborhood of the codimensional singularity. Such issues have been addressed by several articles (Ding et al. 2012; He et al. 2012a, b, 2013, 2014; Li and Wei 2005; Dong and Liao 2013; Song and Xu 2012; Campbell and Yuan 2008; Guo et al. 2008; Yang 2008; Liu 2014; Garliauskas 1998; Kepler et al. 1990). For instance, Ding et al. (2012) have considered the zero–Hopf bifurcation of a generalized Gopalsamy neural network model, the normal forms of near a zero–Hopf critical point were deduced by using multiple time scales and center manifold reduction methods respectively. The similar work has been carried out by He et al. (2014), but contraposing a different neural network. The authors in (Dong and Liao 2013; He et al. 2012a; Li and Wei 2005; Song and Xu 2012, 2013; Campbell and Yuan 2008; Guo et al. 2008; Yang 2008) have devoted to the analysis of the Bogdanov–Takens (B–T) bifurcation with codimension two. Guo et al. have deduced the existence of the double Hopf (Guo et al. 2008). Campbell and Yuan (2008) and Liu (2014) have investigated triple zero bifurcation induced by the delays and the connection topologies existing among the neurons. For the cases of Hopf–Pitchfork bifurcation and chaos see (Yang 2008; Garliauskas 1998; Kepler et al. 1990).

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In application of neural networks, some dynamical phenomena of neural networks have been explained and shown by simulations and applied to memory systems and algorithmic designs (Kadone and Nakamutra 2005; Hoppensteadt and Izhikevich 1997; He et al. 2013; Chen and Aihara 1999; Wang and Shi 2006). For example, attractors, homoclinic and heteroclinic orbits can be interpreted as a single storage or memory pattern, or an optimal object, and various complex patterns in the application and design of neural networks. Chaotic behavior of neural networks also has been confirmed to be benefit to design the efficient searching algorithm for solving optimization problems due to its bidirectional structure (Chen and Aihara 1999; Wang and Shi 2006). Undoubtedly, the deeper investigations to dynamic of neural networks facilitates the extensive applications and optimum design of some searching algorithms. This also motivate us to study the more complicated bifurcation behaviors of delayed multi-neuron memory systems, especially, the codimension-2 and codimension-3 bifurcations. Due to the complexity of the problem, more attentions on the investigations of multiple-delay neural network focus on small sized networks or on some special architecture (Sun et al. 2007; Zheng et al. 2008; Dong and Liao 2013; He et al. 2012a). Although, it is convinced that the dynamics investigation to neural networks with a few neurons contributes to understanding large-scale networks, there are inevitably some complicated and pivotal problems that may be neglected if large-scale networks is simplified. On the other hand, from the practical point of the view, improving the designs of the neural networks and extending its application in more fields also need consider the more complex and larger-scale associative models because the real neural networks are complex and large-scale nonlinear dynamical systems, and commonly involve interactions among multiple neurons. For example, the exact control to the roughness of categorization by the proportion between excitatory signal and inhibitory signal depends on the complex layered design (Carpenter and Grossberg 1987). Meanwhile it also is an important mathematical subject to investigate the dynamics of the larger scale neuron systems (Xu and Li 2012; Xu et al. 2011; Yang and Ye 2009; Cao 2003; Liu 2014). As Xu and Li (2012) have studied a bidirectional associative memory (BAM) six-neuron networks, which is described by 8 6 P > > _ x ðtÞ ¼ l x ðtÞ þ cj1 f1 ðxj ðt  s2 ÞÞ; > 1 1 1 > > j¼2 < x_2 ðtÞ ¼ l2 x2 ðtÞ þ c12 f2 ðx1 ðt  s1 ÞÞ; ð1Þ > .. > > > > :. x_6 ðtÞ ¼ l6 x6 ðtÞ þ c16 f6 ðx1 ðt  s1 ÞÞ; where xi ðtÞði ¼ 1; 2; . . .; 6Þ denotes the state of the neuron at time t; li characterizes the attenuation rate of internal neurons processing on the I-layer and the J-layer and

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li [ 0; the real constants cj1 ðj ¼ 2; 3; . . .; 6Þ and c1j show the connected weights between the neurons in two layers: the I-layer and the J-layer. Xu and Li (2012) have considered the stability of the nondegenerate origin and the existence of Hopf bifurcation of (1) induced by the delays. Compared with (Xu and Li 2012), the contributions of this work are to address Bogdanov–Takens and triple zero bifurcation computations of the BAM six-neuron network, and show the curves of different bifurcation occurrences. The Hopf bifurcation is corresponding the characteristic equation has a pair of purely imaginary roots. While the B–T or triple zero bifurcation is corresponding the characteristic equation has a double-zero or triple zero root. We choose two or three parameters as the bifurcation parameters, by using the center manifold reduction and the normal form method, one can compute the normal form which is equivalent to (1). Finally, different kinds of bifurcation curves used to explain the dynamical behaviors of (1) can be drawn. Note that the authors in (Xiao et al. 2013; Liu and Yang 2014) have considered the stability and Hopf bifurcation of some delayed neural network models with n þ 1 or n neurons, especially model (1) as a special case of the model in (Xiao et al. 2013) has been studied detailedly by Xu and Li (2012), nevertheless, the degenerated dynamics of system (1) at the neighborhood of the origin has never been addressed. Thus, in this article, we devote to investigate the high codimension properties of model (1). The study to the high codimension properties of the model in (Xiao et al. 2013) will be left as our future work. The organization of this paper is: in ‘‘Existence of B–T bifurcation’’ section, the existence conditions under which the origin is B–T singularity are given. In ‘‘B–T bifurcation’’ section, by applying the center manifold reduction and normal form computation, we obtain the normal form and bifurcation curves near the origin of the neural network. In ‘‘Triple zero bifurcation’’ section, triple zero bifurcation is also investigated. In ‘‘Numerical examples and simulations’’ section, some numerical examples and simulations are shown to verify our main results. Finally, a conclusion is given to sum up our works.

Existence of B–T bifurcation Let u1 ðtÞ ¼ x1 ðt  s1 Þ; uj ðtÞ ¼ xj ðtÞ ðj ¼ 2; . . .; 6Þ and s ¼ s1 þs2 , then system (1) is equivalent to the following system 8 6 P > > u_1 ðtÞ ¼ l1 u1 ðtÞ þ cj1 f1 ðuj ðt  sÞÞ; > > > j¼2 < u_2 ðtÞ ¼ l2 u2 ðtÞ þ c12 f2 ðu1 ðtÞÞ; ð2Þ > > > ... > > : u_6 ðtÞ ¼ l6 u6 ðtÞ þ c16 f6 ðu1 ðtÞÞ:

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To obtain our main results, we make the following assumptions fi0 ð0Þ ¼ 1;

fi ð0Þ ¼ 0;

(H1)

 2 l34 l s l l3 l3 l 3 3 2l6 l5 ðl3  l4 Þðl2  l4 Þc14 2 3 5 6 1 þ 2l35 l36 ðl2 l1 þ l2 l3 þ l3 l1 Þs  2c15 c51 l36 ðl3  l5 Þðl2  l5 Þ  2c16 c61 l35 ðl3  l6 Þðl2  l6 Þ  þ 2l35 l36 ðl2 þ l3 þ l1 Þ ;

c041 ¼

i ¼ 1; 2; . . .; 6.

From (H1), we know that the origin is always the equilibrium of system (2). Linearizing system (2) at the zero equilibrium, then it becomes 8 6 P > > > u_1 ðtÞ ¼ l1 u1 ðtÞ þ cj1 uj ðt  sÞ; > > j¼2 < u_2 ðtÞ ¼ l2 u2 ðtÞ þ c12 u1 ðtÞ; ð3Þ > > > ... > > : u_6 ðtÞ ¼ l6 u6 ðtÞ þ c16 u1 ðtÞ:

l45 6l4 c15 ðl4  l5 Þðl3  l5 Þðl2  l5 Þ  36  s l2 l4 l3 l1 l46 þ 3l46 ðl2 l3 l4 þ l1 l2 l3

c051 ¼ 

þ l1 l2 l4 þ l1 l3 l4 Þs2 þ 6c16 c61 ðl4  l6 Þ

FðkÞ ¼  e

4

3

þ l3 Þ þ 6l46 ðl3 l4 þ l1 l4 þ l2 l4 þ l1 l2 þ l1 l3 þ l2 l3 Þs:

2

ðb1 k þ b2 k þ b3 k þ b4 k þ b5 Þ

6

þ k þ a1 k5 þ a2 k4 þ a3 k3 þ a4 k2

ð4Þ (H2) (H3)

þ a5 k þ a6 ¼ 0; where X

ak ¼

li1 li2 li3 . . .lik ;

k ¼ 1; . . .6;

b3 ¼

j¼2 6 X

b4 ¼

j¼2 6 X

b5 ¼

j¼2

i1 ¼2;i1 6¼j

"

#

X

c1j cj1 "

X

c1j cj1

Proof

li1 li2 ;

2  i1 \i2  6;i1 ;i2 6¼j

c1j cj1

li1 li2 li3 li4 :

It follows from (H1) and (H2) that Fð0Þ ¼ F 0 ð0Þ ¼ 0 and  F 00 ð0Þ ¼ l4 l5 l6 s2 l2 l3 l1 þ 2ðl2 þ l1 þ l3 Þ

Let c021 ¼ 



þc14 c41 l25 l26 ðl3  l4 Þ l26 l25 l24 ðl3  l2 Þc12 þ l24 l25 l26 ðl1 þ l3 þ l3 s l1 Þ þ c15 c51 l24 l26 ðl3  l5 Þ þ c16 c61 l24 l25 ðl3  l6 Þ;

c031 ¼

 2 2 2 l23 l l l ðl þ l2 þ sl1 l2 Þ 2 2 2 c13 l6 l5 l4 ðl3  l2 Þ 4 5 6 1

þ c14 c41 l25 l26 ðl2  l4 Þ

ð9Þ

F 000 ð0Þ ¼ 6a3 þ s3 b5  3s2 b4 þ 6sb3  6b2 :

#

2  i1 \i2 \i3 \i4  6;i1 ;i2 ;i3 ;i4 6¼j

l22

By (4), we have

F 00 ð0Þ ¼ 2a4  s2 b5 þ 2sb4  2b3 ;

li1 li2 li3 ;

X

l2 6¼ l3 6¼ l4 , c51 6¼ c051 ;

Fð0Þ ¼ a6  b5 ; F 0 ð0Þ ¼ a5 þ sb5  b4 ;

#

2  i1 \i2 \i3  6;i1 ;i2 ;i3 6¼j

"

c41 ¼ 6 c041 ; c41 ¼ c041 ;

Lemma 1 The characteristic Eq. (4) has the double zero eigenvalues if (H1) and (H2) hold, and has the triple zero eigenvalues if (H1) and (H3) hold.

b1 ¼ c13 c31 þ c14 c41 þ c15 c51 þ c16 c61 þ c12 c21 ; " # 6 6 X X c1j cj1 li1 ; b2 ¼ j¼2

c21 ¼ c021 ; c31 ¼ c031 ; c21 ¼ c021 ; c31 ¼ c031 ; l2 6¼ l3 6¼ l4 6¼ l5 .

Then we can get the following result.

1  i1 \i2 \i3 \\ik  6

6 X

ð8Þ

 ðl3  l6 Þðl2  l6 Þ þ 6l46 ðl1 þ l4 þ l2

The characteristic equation of system (3) is ks

ð7Þ

 þ c15 c51 l24 l26 ðl2  l5 Þ þ c16 c61 l24 l25 ðl2  l6 Þ ;

ð5Þ

þ 2ðl2 l1 þ l2 l3 þ l3 l1 Þs 2   2 2 2 l36 l35 c14 c41 ðl3  l4 Þðl2  l4 Þ l4 l5 l6

ð10Þ

þ l36 l34 c15 c51 ðl3  l5 Þðl2  l5 Þ

 þ l35 l34 c16 c61 ðl3  l6 Þðl2  l6 Þ ¼ 6 0:

ð6Þ

Thus, k ¼ 0 is a double root of Eq. (4). Similarly, that (H1) and (H3) hold can lead to Fð0Þ ¼ F 0 ð0Þ ¼ F 00 ð0Þ ¼ 0 and

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" 000

X

3

F ð0Þ ¼ l5 l6 l1 l2 l3 l4 s þ 3

li1 li2 li3 s

2

1  i1 \i2 \i3  4

þ6

X

li1 li2 s þ 6

1  i1 \i2  4

4 X

#

li

 s2 b5 b3 þ 2a2 b5  2a3 a5 þ 2b2 sb5 þ 2a5 b2 þ s2 a25

i¼1

6  4 l c15 c51 ðl4  l5 Þðl3  l5 Þðl2  l5 Þ 3 3 l5 l6 6  þ l45 c16 c61 ðl4  l6 Þðl3  l6 Þðl6 þ l2 Þ 6¼ 0:

þ

This completes the proof. Next, under the conditions (H1)–(H2) or (H1)–(H3), we will find the conditions under which the other eigenvalues of Eq. (4) have negative real parts except for zero roots. Let k ¼ iw be a root of Eq. (4) and substitute it into Eq. (4), then separating the real and imaginary parts, we have w6  a2 w4 þ a4 w2  a6 ¼ ðb1 w4 þ b3 w2  b5 Þ cosðwsÞ þ ðb2 w3  b4 wÞ sinðwsÞ;

s4 b25 Þ ¼ 0: 4

ð14Þ

Similarly, if Eq. (14) has no positive root, we take s0 ¼ þ1, or else, Eq. (14) has at most four positive roots, i ði ¼ 1; 2; 3; 4Þ. Following (11) we have denote them by w si ¼

 1 D arccos ; 4 2 i w i  b3 w þ b5 Þ2 þ ðb2 w3  b4 wÞ2 ðb1 w ð15Þ

where  ¼ ðw 6i  a2 w 4i þ a4 w 2i  b5 Þðb1 w 4i  b3 w2 þ b5 Þ D 2i ðb2 w 2i  b4 Þða3 w 2i  a1 w 4i  a5 Þ; w

Lemma 2

þ ðb2 w3  b4 wÞ cosðwsÞ:

min f si g.

i2f1;2;...;4g

[See Ruan and Wei (2003)] Consider the expoð0Þ

nential polynomial Pðk; eks0 . . .; eksm Þ ¼ kn þ P1 kn1 þ ð11Þ

Let (H1) and (H2) hold, then one can obtain      w2 w10 þ a21  2a2 w8 þ 2a4  2a1 a3  b21 þ a22 w6   þ 2b1 b3  2b5 þ 2a5 a1  b22  2a2 a4 þ a23 w4   þ b23  2b1 b5 þ 2a2 b5 þ 2b2 a5 þ 2b2 sb5 þ a24  2a3 a5 w2   þ 2b3 b5  2a4 b5  2a5 sb5  s2 b25 ¼ 0: ð12Þ Without loss of generality, we assume that Eq. (12) has at most five positive roots, denoting them as wi ði ¼ 1; 2; 3; 4; 5Þ. By (11) we have 1 D arccos ; 4 2 wi ðb1 wi  b3 wi þ b5 Þ2 þ ðb2 w3i  b4 wi Þ2

ð0Þ

where    D ¼ w6i  a2 w4i þ a4 w2i  b5 b1 w4i  b3 w2 þ b5     w2i b2 w2i  b4 a3 w2i  a1 w4i  a5 ; define s0 ¼ mini2f1;2;...;5g fsi g. If Eq. (12) has no positive root, we take s0 ¼ þ1: Let (H1) and (H3) hold, it follows from (11) that

ð0Þ

   þ Pn1 k þ Pn

ð1Þ

ð1Þ

ð1Þ

þ½P1 kn1 þ    þ Pn1 k þ Pn eks0

ðmÞ

ðmÞ

ðmÞ

þ    þ ½P1 kn1 þ    þ Pn1 k þ Pn eksm ; where si  0 ðiÞ

ði ¼ 1; 2; . . .; mÞ and Pj ðj ¼ 1; 2; . . .; mÞ are constants. As ðs1 ; s2 ; . . .; sm Þ vary, the sum of the order of the zeros of Pðk; eks0 . . .; eksm Þ on the open right half plane can change only if a zero appears on or crosses the imaginary axis. Lemma 3 Let ðH1Þ and ðH3Þ hold. All the roots of Eq. (4), except for the double zero or triple zero roots, have negative real parts if one set of the following conditions holds (H4) (H5)

ð13Þ

123

 2b1 b5 þ

define s0 ¼

a3 w3  a1 w5  a5 w ¼ þ ðb1 w4  b3 w2 þ b5 Þ sinðwsÞ

si ¼

    w4 w8 þ a21  2a2 w6 þ b21  2a1 a3  s2 b5  2sa5  þ 2b3 þ a22 Þw4 þ a2 s2 b5 þ 2a2 sa5 þ 2b1 b3  b22   2a2 b3  2b5 þ 2a1 a5 þ a23 Þw2 þ s3 b5 a5  2sa5 b3

0\s\s0 ; A1 A2  A3 [ 0; A1 ðA2 A3  A1 A4 Þ  A23 [ 0; A4 [ 0; 0\s\ s0 ; A1 [ 0; A1 A2  A3 [ 0; A3 [ 0,

with A1 ¼ a1 , A2 ¼ a2  b1 , A3 ¼ a3  b2 , A4 ¼ a4  b3 . Proof

Let ðH1Þ–ðH2Þ. For s ¼ 0, Eq. (4) becomes

k2 ½k4 þ A1 k3 þ A2 k2 þ A3 k þ A4  ¼ 0;

ð16Þ

By Routh-Hurwiz criterion, we have if A1 A2  A3 [ 0;

A1 ðA2 A3  A1 A4 Þ  A23 [ 0;

A4 [ 0; ð17Þ

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153

then the rest of roots of Eq. (16) have negative real parts except k1;2 ¼ 0. Let ðH1Þ–ðH2Þ. For s ¼ 0, Eq. (4) becomes k3 ½k3 þ A1 k2 þ A2 k þ A3  ¼ 0;

ð18Þ

By Routh–Hurwize criterion, we know if A1 [ 0;

A1 A2  A3 [ 0;

A3 [ 0;

ð19Þ

then the rest roots of Eq. (18) have negative real parts except the triple zero root. We complete the proof. From Lemmas 1–3 we have the following Theorem.

 c61  00 3f1 ð0Þu26 ðt  1Þ þ f1000 ð0Þu36 ðt  1Þ ; 6  1  00 Hj ¼ c1j 3fj ð0Þu21 ðtÞ þ fj000 ð0Þu31 ðtÞ : 6 þ

Then following (Xu and Huang 2006), the second term of the Taylor expansion of system (21) also can be expressed in the form 1b F 2 ðxt ; aÞ ¼ A1 uðtÞa1 þ A2 uðtÞa2 þ B1 uðt  1Þa1 2 6 X þ B2 uðt  1Þa2 þ Ei ui ðtÞuðt  1Þ i¼1

Theorem 1 Assume ðH1Þ holds. Then system (2) at the origin undergoes B–T or triple zero bifurcation if ðH2Þ and ðH4Þ or ðH3Þ and ðH5Þ hold.

þ

6 X

Fi ui ðtÞuðtÞ þ

6 X

i¼1

Gi ui ðt  1Þuðt  1Þ;

i¼1

where A1 ¼ A2 ¼ O66 ;

B–T bifurcation In this section, we will give the normal form of B–T bifurcation by using the methods introduced in Hale and Verduyn (1993), Xu and Huang (2006). Rescaling the time by t ! st to normalize the delay, and rewriting c21 and c31 as c21 ¼ c021 þ a1 and c31 ¼ c031 þ a2 where ða1 ; a2 Þ near (0, 0), then system (2) is translated to   8 u_1 ðtÞ ¼ s½l1 u1 ðtÞ þ c021 þ a1 f1 ðu2 ðt  1ÞÞ > > >  6 >  P > > > þ c031 þ a2 f1 ðu3 ðt  1ÞÞ þ cj1 f1 ðuj ðt  1ÞÞ; < j¼4 ð20Þ _ u ðtÞ ¼ s½l u ðtÞ þ c f ðu 2 2 12 2 1 ðtÞÞ; > 2 > > > .. > > > :. u_6 ðtÞ ¼ s½l6 u6 ðtÞ þ c16 f6 ðu1 ðtÞÞ: Taking the Taylor expansion of fi ðxÞ, i ¼ 1; 2; . . .; 6, then system (20) takes the form

    8 u_1 ðtÞ ¼ s½l1 u1 ðtÞ þ c021 þ a1 u2 ðt  1Þ þ c031 þ a2 u3 ðt  1Þ > > > > > þc41 u4 ðt  1Þ þ c51 u5 ðt  1Þ þ c61 u6 ðt  1Þ þ H1 þ h:o:t:; > > < u_2 ðtÞ ¼ s½l2 u2 ðtÞ þ c12 u1 ðtÞ þ H2 þ h:o:t:; > > .. > > > . > > : u_6 ðtÞ ¼ s½l6 u6 ðtÞ þ c16 u1 ðtÞ þ H6 þ h:o:t:;

ð21Þ where H1 ¼

 c021 þ a1  00 3f1 ð0Þu22 ðt  1Þ þ f1000 ð0Þu32 ðt  1Þ 6  c031 þ a2  00 þ 3f1 ð0Þu23 ðt  1Þ þ f1000 ð0Þu33 ðt  1Þ 6  c41  00 3f1 ð0Þu24 ðt  1Þ þ f1000 ð0Þu34 ðt  1Þ þ 6  c51  00 3f1 ð0Þu25 ðt  1Þ þ f1000 ð0Þu35 ðt  1Þ þ 6

Ei ¼ O66 ; i ¼ 1; 2; . . .; 6;

G1 ¼ F 2 ¼ F 3 ¼ F 4 0 B0 1 0 B B0 0 0 B1 ¼ sB B. . . B. . . @. . .

¼ F5 ¼ F6 ¼ O66 ; 1 . 0 .. 0 C C . 0 .. 0 C C ; .. .. .. C C . . .A 0 0 0 0 0 0 66 1 0 0 0 1 0 0 0 B0 0 0 0 0 0C C B C B2 ¼ sB B .. .. .. .. .. .. C ; @. . . . . .A 0

0

0 0 0

0 0 0 66 1 0  0 C 0  0C C C C 0  0C C C C 0  0C C C C 0  0C C C A 0  0

B 1 00 B f2 ð0Þc12 B2 B B 1 00 B f ð0Þc13 B2 3 B F1 ¼ sB B 1 f 00 ð0Þc14 B2 4 B B1 B f 00 ð0Þc 15 B2 5 B @1 f 00 ð0Þc16 2 6 0 f100 ð0Þ 0 c B 0 2 21 B B 0 0 G2 ¼ sB B  @ 0

0

B 0 B B G3 ¼ sB 0 B @ 0

;

66



0 C C 0  0 C C ; C   A 0 0    0 66 1 00 0 f1 ð0Þc31 0 0 0 0 C 2 C 0 0 0 0 0 C C C     A 0

0

1

0

0

0

0

;

66

123

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Cogn Neurodyn (2016) 10:149–164

0

B B B G4 ¼ sB 0 B .. @.

0

0

0 .. .

0 .. .

0

0

0

0

0

f100 ð0Þc41 2 0 .. . 0

0 B B B G5 ¼ sB 0 B .. @.



0

 .. .

0 .. .

0 .. .

0



0

0

0

f100 ð0Þ



2

0 B B B G6 ¼ sB 0 B .. @.

 .. .

0 .. .

0 .. .

0



0

0

2

c51

c61

0

1

0 1 0 C C 0C C .. C .A

0 1 C C C C C A

1 1  h B c12 c12 1 C C B B l l h  sl C 2 C B 2 2 C; UðhÞ ¼ B .. C B .. C B . .  C B @ c16 c16 1 A h sl6 l6 l6 0 1 n1 n2 n3 n4 n5 n6 c0 l3 l c41 l c51 l c61 A; Wð0Þ ¼ @ l3 n 21 n0 n0 03 n0 03 n0 03 n0 0 0 0 c31 c31 l2 c31 l4 c31 l5 c31 l6 0

C C 0C C .. C .A

0 .. . 0

f100 ð0Þ

0

0

;

66

;

66

where n0 ¼ :

þ c14 c41 l25 l26 ðl2  l4 Þ þ c15 c51 l24 l26 ðl2  l5 Þ þ c16 c61 l24 l25 ðl2  l6 Þ;

66

To obtain the normal form of system (21) on its center manifold, we need the following lemma.

with n ¼ l36 l35 l34 ½l2 l3 s2 l1 þ 2ðl2 l1 þ l3 l1 þ l2 l3 Þs þ 2ðl1 þ l2 þ l3 Þ  2c14 c41 l35 l36 ðl3  l4 Þðl2  l4 Þ

Lemma 4 [See Xu and Huang (2006)] The bases of P and its dual space P have the following representations

 2c15 c51 l34 l36 ðl3  l5 Þðl2  l5 Þ  2c16 c61 l34 l35 ðl3  l6 Þðl2  l6 Þ;

P ¼ spanU, UðhÞ ¼ ðu1 ðhÞ; u2 ðhÞÞ, 1  h  0, P ¼ spanW, WðsÞ ¼ colðw1 ðsÞ; w2 ðsÞÞ, 0  s  1, where u1 ðhÞ ¼ u01 2 Rn nf0g, u2 ðhÞ ¼ u02 þ u01 h, u02 2 Rn and w2 ðsÞ ¼ w02 2 Rn nf0g, w1 ðsÞ ¼ w01  sw02 ; w01 2 Rn , which satisfy

and

ð1Þ ðA þ BÞu01 ¼ 0;

n2 ¼

ð2Þ ðA þ BÞu02 ¼ ðB þ IÞu01 ;

ð3Þ w02 ðA þ BÞ ¼ 0; ð4Þ w01 ðA þ BÞ ¼ w02 ðB þ IÞ; 1 ð5Þ w02 u02  w02 Bu01 þ w02 Bu02 ¼ 1; 2 1 1 1 ð6Þ w01 u02  w01 Bu01 þ w01 Bu02 þ w02 Bu01  w02 Bu02 ¼ 0: 2 6 2

For system (21), we have 1 0 l1 0 0 0 0 0 B c12  l2 0 0 0 0 C C B B c13 0  l3 0 0 0 C C; A ¼ sB B c14 0 0  l4 0 0 C C B @ c15 0 0 0  l5 0 A c 0 0 0 0  l6 016 1 0 c021 c031 c41 c51 c61 B0 0 0 0 0 0 C B ¼ sB : .. C .. .. .. .. @ ... . A . . . . 0 0 0 0 0 0 66 By Lemma 4, we have

123

2sl2 l23 l4 l5 l6 2 2 2 ½l l l ðl þ l2 þ sl1 l2 Þ c13 ðl2  l3 Þ n 4 5 6 1

n1 ¼

n3 sl3 þ ð1 þ sl3 Þn0 ; sc031

c021 ðl2 n3 sl3 þ ðl2  l3 Þn0 Þ ; sl22 c031   cj1 lj n3 sl3 þ ðlj  l3 Þn0 2l23 m1 m2 n3 ¼ ; nj ¼ ; 3c13 m23 sl2j c031 j ¼ 4; 5; 6; m1 ¼ l24 l25 l26 ðl1 þ l2 þ sl1 l2 Þ þ c14 c41 l25 l26 ðl2  l4 Þ þ c15 c51 l24 l26 ðl2  l5 Þ þ c16 c61 l24 l25 ðl2  l6 Þ;    m2 ¼ l44 l45 l46 l22 l3 s3 l1 þ 3l22 ðl1 þ l3 Þs2 þ 6 l22  l3 l1 s  6ðl1 þ l3 Þ þ 6c14 c41 l45 l46 ðl2  l4 Þðl2 þ l4 Þ  ðl3  l4 Þ þ 6c15 c51 l44 l46 ðl2  l5 Þðl2 þ l5 Þðl3  l5 Þ þ 6c16 c61 l44 l45 ðl2  l6 Þðl2 þ l6 Þðl3  l6 Þ;  m3 ¼ l34 l35 l36 l2 l3 s2 l1 þ 2ðl2 l1 þ l3 l1 þ l2 l3 Þs þ 2ðl1 þ l2 þ l3 Þ  2c14 c41 l35 l36 ðl3  l4 Þðl2  l4 Þ  2c15 c51 l34 l36 ðl3  l5 Þðl2  l5 Þ  2c16 c61 l34 l35 ðl3  l6 Þðl2  l6 Þ:

Following the formulas in (Xu and Huang 2006), we have the following Lemma.

Cogn Neurodyn (2016) 10:149–164

155

Lemma 5 Let ðH1Þ ðH2Þ and ðH4Þ hold. Then the system (21) can be reduced to the following system on the center manifold at ðut ; aÞ ¼ ð0; 0Þ

z_1 ¼ z2 ; ð22Þ z_2 ¼ c1 z1 þ c2 z2 þ g1 z21 þ g2 z1 z2 þ h:o:t:;

(1)

where n0 sl c12 n0 sc13 c 1 ¼ 0 3 a1 þ 0 a2 ; c31 l2 c31 c12 ðl2 n3 sl3 þ l2 n0  l3 n0 Þ c13 n3 s c2 ¼ a1 þ 0 a2 ; 0 2 l2 c31 c31  l3 n0 sf100 ð0Þ c212 c021 c213 c031 c214 c41 c215 c51 c216 c61 þ þ þ þ g1 ¼ l22 l23 l24 l25 l26 2c031  0 00 n0 l s c21 f2 ð0Þc12 f300 ð0Þc13 c031 c41 f400 ð0Þc14 þ 03 þ þ l2 l3 l4 2c31 00 00 c51 f5 ð0Þc15 c61 f6 ð0Þc16 þ þ ; l5 l6  f100 ð0Þ c212 c021 ðl3 l2 n3 s  l3 n0 þ l2 n0 Þ g2 ¼ 0 c31 l32

(3)

þ

c214 c41 ðl3 l4 n3 s  l3 n0 þ l4 n0 Þ l34

þ

c215 c51 ðl3 l5 n3 s  l3 n0 þ l5 n0 Þ l35

þ

c216 c61 ðl3 l2 n3 s  l3 n0 þ l2 n0 Þ l36

þ



c021 ðl3 l2 n3 s  l3 n0 þ l2 n0 Þf200 ð0Þc12 þ n3 sf300 ð0Þc13 l22 c031

c41 ðl3 l4 n3 s  l3 n0 þ l4 n0 Þf400 ð0Þc14 n3 sc213 c031 þ l3 l24 c031 c51 ðl3 l5 n3 s  l3 n0 þ l5 n0 Þf500 ð0Þc15 þ l25 c031 c61 ðl3 l6 n3 s  l3 n0 þ l6 n0 Þf600 ð0Þc16 þ : l26 c031 þ

Since oðc1 ; c2 Þ n20 sc12 c13 ðl2  l3 Þ 6¼ 0; oða ; a Þ ¼  1 2 ðc031 Þ2 l22

(2)

system (22) undergoes a transcritical bifurcation on the curve S ¼ fða1 ; a2 Þ : c1 ¼ 0; c2 2 Rg, at E1 system (22) undergoes an unstable Hopf bifurcation in the half line H ¼ fða1 ; a2 Þ : c2 ¼ 0; c1 \0g, at E2 , system (22) undergoes a stable Hopf bifurcation in the half line n o T ¼ ða1 ; a2 Þ : c2 ¼ gg2 c1 ; c1 [ 0 . 1

In ‘‘Numerical examples and simulations’’ section, a numerical example is given. Remark From the point of the view of application, the results of Theorem 2 under the conditions (H1), (H2), (H4) and fi00 ð0Þ 6¼ 0; ði ¼ 1; . . .; 6Þ can be explained as the exchanges of various stored patterns or memory patterns occurring in perturbed system (20) when the parameters a1 and a2 vary in the small neighborhood of the ð0; 0Þ. Notice that a stable equilibrium is corresponding to a single storage pattern, thus, when parameters ða1 ; a2 Þ pass the transcritical bifurcation curve determined by c1 ða1 ; a2 Þ ¼ 0, system (20) transforms its stored patten from one to other. Similarly, system (20) can exhibit more complex memory pattern (a periodic pattern) when a1 ; a2 approach the Hopf bifurcation curve defined by c2  g2 c1 =g1 ¼ 0, ðc1 ða1 ; a2 Þ [ 0Þ. Furthermore, if fi00 ð0Þ ¼ 0, i ¼ 1; . . .; 6, to discuss the properties of the origin of system (21), we need to compute the third order normal form of the B–T bifurcation. Let ut ¼ UðhÞzðtÞ þ y, then (21) can be decomposed as

z_ ¼ Bz þ Wð0ÞFðUz þ y; aÞ; ð23Þ y_ ¼ AQ1 y þ ðI  pÞX0 FðUz þ y; aÞ; where z ¼ ðz1 ; z2 ÞT , y ¼ ðy1 ; y2 ÞT and u1 ðhÞ ¼ z1 þ hz2 ;

! c1j c1j 1 uj ðhÞ ¼ z1 þ h z2 ; slj lj lj

j ¼ 2; 3; . . .; 6:

ð24Þ

It follows from (Faria and Magalhaes 1995; Jiang and Yuan 2007) that   g13 ðz; 0; aÞ ¼ I  P1I;3 f~31 ðz; 0; aÞ ¼ ProjImðM31 Þc f~31 ðz; 0; aÞ;

then the map ða1 ; a2 Þ ! ðc1 ; c2 Þ is regular. Hence we can get the following theorem. Theorem 2 Let ðH1Þ ðH2Þ and ðH4Þ hold. If fi00 ð0Þ 6¼ 0, i ¼ 1; . . .; 6, on the center manifold, system (20) is equivalent to system (22).

ð25Þ where

Referring Dong and Liao (2013), system (22) has two equilibrium E1 ¼ ð0; 0Þ and E2 ¼ ð gc1 ; 0Þ, the bifurcation

3  1  Dz f2 ðz; 0; aÞU21 ðz; aÞ f~31 ðz; 0; aÞ ¼ f31 ðz; 0; aÞ þ 2   þ Dy f21 ðz; 0; aÞU22 ðz; aÞ    Dz U21 ðz; aÞg12 ðz; 0; aÞ:

curves near (0, 0) in the a1 and a2 parameter space are the following

By (21) and (24), one can get where

1

123

156

Cogn Neurodyn (2016) 10:149–164

f31 ðz; 0; aÞ 0"

!

6 P

#1

6 P

B n1 h1 c021 u32 ð1Þ þ c031 u33 ð1Þ þ cj1 u3j ð1Þ þ nj hj c1j u31 ð0Þ B j¼4 j¼2 B ! " # ¼B 6 B X n0 l 3 0 3 0 3 3 @ h1 c21 u2 ð1Þ þ c31 u3 ð1Þ þ cj1 uj ð1Þ þ d0 c031 j¼4

c021 h2 c12 c031 h3 c13 þ þ l2 l3

6 X j¼4

!

cj1 c1j hj : lj

Then the canonical basis in V34 ðR2 Þ has forty elements: ððz; aÞ3 ; 0ÞT , ð0; ðz; aÞ3 ÞT , the bases of Im ðM31 Þ and Im ðM31 Þc one can see in (Jiang and Yuan 2007). Together with the definition of M31 , one can obtain that the space of Im ðM31 Þ can be spanned by the following elements z21 z2 0

;



0



 ;

z2 l1 l2 ; ; 0 0 !  z1 z22 z21 li ; ; z32 2z1 z2 li   3 z1 l1 l2 li ; ; z2 l1 l2 0 i ¼ 1; 2; 

z22 li





z1 z22

z32 0



;

z31 3z21 z2 z2 l2i

l21 l2 0

0

!

z1 l2i 

z1 z2 li

;



z21 z2 2z1 z22

; !

 ;

;

z2 l2i 0

!

;

we obtain the third order normal form of system (21) as follows

z_1 ¼ z2 ; ð27Þ z_2 ¼ c1 z1 þ c2 z2 þ a3 z31 þ b3 z21 z2 þ h:o:t:; where the higher terms of a are omitted and

;

z1 z2 li

z22 li  l1 l22 0

p; p 2 ImðM31 Þc ; 0; p 2 ImðM31 Þ;  3  0 z ProjImðM31 Þc 1 ¼ ; 3z21 z2 0  2  0 z 1 ai ProjImðM31 Þc ; ¼ z2 a2i 0   z 1 a1 a2 0 ProjImðM31 Þc ; ¼ a1 a2 z 2 0  2  0 z 1 ai c ProjImðM31 Þ ; ¼ 2z1 z2 ai 0 ProjImðM31 Þc p ¼

To obtain the third-order normal form, it needs the decomposition   V34 ðR2 Þ ¼ Im M31 Im ðM31 Þc :



a3 ¼

;

;

1 ¼ f30



    0 0 0 0 0 ; ; ; ; ; z31 z21 z2 z21 li z1 l2i z1 z2 l i      0 0 0 0 0 ; ; ; ; ; z2 l2i l3i l21 l2 l1 l22 zi l1 l2 i ¼ 1; 2: From (25), (27) and

1 2 f ; 6 30

1 2 1 b3 ¼ ðf21 þ 3f30 Þ; 6 ! 6 c c3 6 X X j1 1j n1 h1 þ nj hj c1j ; l3j j¼2 j¼2

6 c c3 X n0 l3 c021 h2 c12 c031 h3 c13 j1 1j h þ þ 1 l2 l3 c031 l3j j¼2 c41 c14 c51 c15 c61 c16 þ þ þ ; l4 l5 l 6  6 cj1 c3 1 þ 1 X 1j slj 3h1 l3 n0 ¼  : c031 j¼2 l3j

2 f30 ¼

and the complementary space Im ðM31 Þc can be spanned by the elements

123

ð26Þ

(

000

hj ¼ fj ð0Þ; d0 ¼ u31 ð0Þ

C C C C; C A

2 f21

After time rescaling and coordinate transformation given 4 z1 , w2 ¼  by t ¼  jab33 j t, w1 ¼ paffiffiffiffiffi

ja3 j

2

a4 p ffiffiffiffiffi z2 , system (27)

ja3 j

ja3 j

is equivalent to the following system

w_ 1 ¼ w2 ; w_ 2 ¼ v1 w1 þ v2 w2 þ sw31  w21 w2 þ h:o:t:;

ð28Þ

Cogn Neurodyn (2016) 10:149–164

 2 where v1 ¼ ba33 c1 , v2 ¼  jab33 j c2 , s ¼ sgnða3 Þ. From (He et al. 2012a; Dong and Liao 2013) we know the bifurcations of system (28) is related to the sign of s. If s ¼ 1, we have (1)

system (28) undergoes a Pitchfork bifurcation on the curve S ¼ fða1 ; a2 Þ : v1 ¼ 0; v2 2 Rg;

(2)

system (28) undergoes a heteroclinic bifurcation on the curve 1 T ¼ fða1 ; a2 Þ : v2 ¼  v1 þ Oðv21 Þ; v1 \0g; 5

If s ¼ 1, we have (1)

system (28) undergoes a Pitchfork bifurcation on the curve S ¼ fða1 ; a2 Þ : v1 ¼ 0; v2 2 Rg;

(2)

system (28) undergoes a Hopf bifurcation at the trivial equilibrium on the curve H0 ¼ fða1 ; a2 Þ : v2 ¼ 0; v1 \0g;

(3)

system (28) undergoes a Hopf bifurcation at the nontrivial equilibrium on the curve H1 ¼ fða1 ; a2 Þ : v1 ¼ v2 ; v1 [ 0g;

(4)

system (28) undergoes a homoclinic bifurcation on the curve 4 T ¼ fða1 ; a2 Þ : v2 ¼ v1 þ Oðv21 Þ; v1 [ 0g; 5

(5)

periodic memory pattern) is yielded when ða1 ; a2 Þ cross the Hopf bifurcation curve H0 . With ða1 ; a2 Þ moving continuously and passing the Pitchfork bifurcation curve Sþ to the side of (v1 [ 0), two unstable nontrivial equilibria are bifurcated from the trivial equilibrium. Further, the two unstable nontrivial equilibria become locally stable stored pattern when ða1 ; a2 Þ cross the Hopf bifurcation curve H1 which also gives rise to two unstable small limit cycles located inside the surrounding big limit cycle.

system (28) undergoes a Hopf bifurcation H at the trivial equilibrium on the curve H ¼ fða1 ; a2 Þ : v2 ¼ 0; v1 \0g;

(3)

157

system (28) undergoes a double cycle bifurcation on the curve Hd ¼ fða1 ; a2 Þ : v2 ¼ dv1 þ Oðv21 Þ; v1 [ 0; d 0:752g:

To verify above results, two numerical examples and some simulations are shown in ‘‘Numerical examples and simulations’’ section. Remark One can see if the condition f 00 ð0Þ 6¼ 0, ði ¼ 1; . . .; 6Þ in the Theorem (2) does not holds, then with ða1 ; a2 Þ varying in the small neighborhood of ð0; 0Þ, system (20) can exhibit the more complicated stored patterns or memory patterns, including the transition of stored pattern from one to two through a Pitchfork bifurcation. The two stored patterns also may lead to three kinds periodic memory patterns. In fact, a stable limit cycle (a

Triple zero bifurcation To discuss the triple zero bifurcation of system (20), we rewrite c21 , c31 and c41 as c21 ¼ c021 þ b1 , c31 ¼ c031 þ b2 and c41 ¼ c041 þ b3 , where ðb1 ; b2 ; b3 Þ vary near (0, 0, 0), taking the Taylor expansion then system (20) becomes     8 u_1 ðtÞ ¼ s½l1 u1 ðtÞ þ c021 þ b1 u2 ðt  1Þ þ c031 þ b2 u3 ðt  1Þ > > >   > > þ c041 þ b3 u4 ðt  1Þ þ c51 u5 ðt  1Þ þ c61 u6 ðt  1Þ þ T1 þ h:o:t:; > > < u_2 ðtÞ ¼ s½l2 u2 ðtÞ þ c12 u1 ðtÞ þ H2 þ h:o:t:; > > .. > > > . > > : u_6 ðtÞ ¼ s½l6 u6 ðtÞ þ c16 u1 ðtÞ þ H6 þ h:o:t:;

ð29Þ where  c021 þ b1  00 3f1 ð0Þu22 ðt  1Þ þ f1000 ð0Þu32 ðt  1Þ 6 c031 þ b2  00 3f1 ð0Þu23 ðt  1Þ þ 6  c0 þ b3  00 3f1 ð0Þu24 ðt  1Þ þ f1000 ð0Þu33 ðt  1Þ þ 41 6  þ f1000 ð0Þu34 ðt  1Þ  c51  00 3f1 ð0Þu25 ðt  1Þ þ f1000 ð0Þu35 ðt  1Þ þ 6  c61  00 3f1 ð0Þu26 ðt  1Þ þ f1000 ð0Þu36 ðt  1Þ ; þ 6  1  00 Hj ¼ c1j 3fj ð0Þu21 ðtÞ þ fj000 ð0Þu31 ðtÞ : 6 T1 ¼

Then following Qiao et al. (2010), the second term of the Taylor expansion of system (29) also can be expressed in the form 1b F 2 ðxt ; bÞ ¼ A1 uðtÞb1 þ A2 uðtÞb2 þ A3 uðtÞb3 2 þ B1 uðt  1Þb1 þ B2 uðt  1Þb2 þ B3 uðt  1Þb3 þ

6 X

Ei ui ðtÞuðt  1Þ

i¼1

þ

6 X i¼1

Fi ui ðtÞuðtÞ þ

6 X

Gi ui ðt  1Þuðt  1Þ;

i¼1

123

158

Cogn Neurodyn (2016) 10:149–164

0

where, 0

A3 ¼ ð0Þ66 ;

0 B0 B 3 ¼ sB @ ... 0

0 0 0 0 .. .. . . 0 0

1 0 .. .

0 0 .. .

1 0 0C .. C .A

0

0

0

;

ð30Þ

66

the other coefficient matrices are the same as them in (22). To obtain the normal form it also needs to compute the corresponding expressions of UðhÞ and WðsÞ of system (29) by using the following Lemma. Lemma 6 [See (Qiao et al. 2010)] The bases of P and its dual space P have the following representations P ¼ spanU; P ¼ spanW;

UðhÞ ¼ ðu1 ðhÞ; u2 ðhÞ; u3 ðhÞÞ; 1  h  0; WðsÞ ¼ colðw1 ðsÞ; w2 ðsÞ; w3 ðsÞÞ; 0  s  1

where u1 ðhÞ ¼ u01 2 Rn nf0g; u2 ðhÞ ¼ u02 þ u01 h; u3 ðhÞ ¼ u03 þ u02 h þ 12 u01 h2 ; u02 , u03 2 Rn and w3 ðsÞ ¼ w03 2 Rn nf0g, w2 ðsÞ ¼ w02  sw03 , w1 ðsÞ ¼ w01  sw02 þ 12 s2 w03 , w01 ; w02 2 Rn , which satisfy ðA þ BÞu02 ¼ ðB þ IÞu01 ; 1 ð3Þ ðA þ BÞu03 ¼ ðB þ IÞu02  Bu01 ; 2 0 0 ð4Þ w3 ðA þ BÞ ¼ 0; ð5Þ w2 ðA þ BÞ ¼ w03 ðB þ IÞ; 1 ð6Þ w01 ðA þ BÞ ¼ w02 ðB þ IÞ  w03 B; 2 1 1 ð7Þ w03 ðB þ IÞu03  w03 Bu02 þ w03 Bu01 ¼ 1; 2 6 1 0 0 1 0 0 0 0 ð8Þ w2 ðB þ IÞu3  w2 Bu2 þ w2 Bu1 2 6 1 0 0 1 0 0 1 0 0  w3 Bu3 þ w3 Bu2  w3 Bu1 ¼ 0; 2 6 24 1 0 0 1 0 0 1 0 0 0 0 ð9Þ w1 ðB þ IÞu3  w1 Bu2 þ w1 Bu1  w2 Bu3 2 6 2 1 0 0 1 0 0 þ w2 Bu2  w2 Bu1 6 24 1 0 0 1 1 0 0 w Bu ¼ 0: þ w3 Bu3  w03 Bu02 þ 6 24 120 3 1

ð1Þ

ðA þ BÞu01 ¼ 0;

123

B B B c12 B Bl 2 UðhÞ ¼ B B B .. B . B @ c16 l6 0 e1 B Wð0Þ ¼ @ r1

1 1 2 h C 2   C c12 1 c12 1 h 1 2 C h  þ h C C sl2 l2 l2 s2 l22 sl2 2 C; C .. .. C C . .   C c16 1 c16 1 h 1 2 A h  þ h sl6 l6 l6 s2 l26 sl6 2 1 e2 e3 e4 e5 e6 C r 2 r 3 r 4 r 5 r 6 A; h

l1 l2 l3 l4 l5 l6 where l1 ¼

6l4 l3 s2 l2 l46 l45 4 4 3 l5 l6 ðl2 s l1 l3 l4 þ d1 s2 þ d2 s

þ d3 Þ þ d4

;

with d1 ¼ 3ðl2 l1 l4 þ l3 l1 l4 þ l2 l1 l3 þ l4 l3 l2 Þ; d2 ¼ 6ðl3 l1 þ l1 l4 þ l2 l4 þ l2 l3 þ 6l2 l1 þ 6l4 l3 Þ; d3 ¼ 6ðl2 þ 6l3 þ 6l1 þ 6l4 Þ; d4 ¼ 6c15 c51 l46 ðl4  l5 Þðl3  l5 Þðl2  l5 Þ þ 6c16 c61 l45 ðl4  l6 Þðl3  l6 Þðl2  l6 Þ;

ð2Þ

The expressions of coefficient matrices A and B see (30), but it needs change c41 as c041 , then together with Lemma 6, we can obtain the expressions of UðhÞ and Wð0Þ as follows:

1

and c021 l1 c 0 l1 c 0 l1 c51 l1 c61 l1 ; l3 ¼ 31 ; l4 ¼ 41 ; l5 ¼ ; l6 ¼ ; l2 l3 l4 l5 l6 i P h P 3 ;i4 6¼j l1 6j¼2 c1j cj1 2i1;i2i;i1 \i l5 l5 l5 l5 D1 2 \i3 \i4  6 i1 i2 i3 i4 i; r1 ¼ P h P 3 ;i4 6¼j l4 l4 l4 l4 D2 4sl5 l2 l3 l4 l6 6j¼2 c1j cj1 2i1;i2i;i1 \i 2 \i3 \i4  6 i1 i2 i3 i4 8 0  cj1 r1 slj  l1 slj  l1 > > > ; j ¼ 2; 3; 4; > < sl2j rj ¼   > cj1 r1 slj  l1 slj  l1 > > > ; j ¼ 5; 6; : sl2j i P h P ;i2 ;i3 ;i4 6¼j  6j¼2 c1j cj1 2i1i l6 l6 l6 l6 e 1 \i2 \i3 \i4 6 i1 i2 i3 i4 j i; e1 ¼ P h P 3 ;i4 6¼j 4 l4 l4 l4 D l 20l25 l26 l22 l23 l24 s2 6j¼2 c1j cj1 i21;i2i;i1 \i 2 2 \i3 \i4  6 i1 i2 i3 i4   8 0 2 2 2 2 2 2 > c 2e s l  2l þ 2r s l þ 2r sl  2l sl  l s l > 1 1 1 1 1 1 j j j1 j j j > > >  ; > > 2s2 l3j > > > < j ¼ 2; 3; 4;   ej ¼ > 2 2 2 2 2 2 > c 2e s l  2l þ 2r s l þ 2r sl  2l sl  l s l j1 1 1 1 1 1 1 > j j j j j > >  ; > > > 2s2 l3j > > : j ¼ 5; 6; l2 ¼

Cogn Neurodyn (2016) 10:149–164

159

D1 ¼ s4 l4j þ 4s2 l2j ðslj þ 3Þ þ 24ð1 þ slj Þ;

oðg1 ; g2 ; g3 Þ oðb1 ; b2 ; b3 Þ b¼0

D2 ¼ s2 l2j ðslj þ 3Þ þ 6ð1 þ slj Þ;

¼

with

ej ¼ l5j ðl1  5r1 Þs5 þ 5l4j ðl1  4r1 Þs4 þ 20l3j ðl1  3r1 Þs3 þ 60l2j ðl1  2r1 Þs2 þ 120lj ðl1  r1 Þs þ 120l1 ; such that 0 0 J ¼ @0 0

\W; U [ ¼ I, U_ ¼ UJ, W_ ¼ JW, where 1 1 0 0 1A 0 0

The normal form of system (29) on its center takes the form 8 z_1 ¼ z2 ; > > < z_2 ¼ z3 ; ð31Þ z_ ¼ g1 z1 þ g2 z2 þ g3 z3 þ X1 z21 þ X2 z22 þ X3 z1 z2 > > : 3 þX4 z1 z3 þ h:o:t:; where c12 l1 s c1 c13 l1 sb2 c14 l1 s b3 þ þ ; l2 l3 l4 c12 ðsl2 ðr1  l1 Þ  l1 Þ c13 ðsl3 ðr1  l1 Þ  l1 Þ g2 ¼ b1 þ b2 l22 l23 c14 ðsl4 ðr1  l1 Þ  l1 Þ þ b3 ; l24 4 c ½l2 ðl þ 2e  2r Þs2 þ 2l ðl  r Þs þ 2l  X 1j j 1 1 1 1 1 j 1 bj1 ; g3 ¼ 3 2sl j j¼2 ! 6 X sc1j cj1 l1 c1j f100 ð0Þ 00 X1 ¼ þ fj ð0Þ ; lj 2lj j¼2 g1 ¼

X2 ¼

6 X c21j cj1 sf100 ð0Þ j¼2

l2j

e1 þ

þ 2lj ðl1  r1 Þs þ 2l1  þ

sl4j

2

6 X c1j cj1 ½slj ðr1  l1 Þ  l1 

l3j

j¼2

X4 ¼

2sl3j

6 X c21j cj1 sf100 ð0Þ j¼2

l2j

e1 þ

 þ 2lj ðl1  r1 Þs þ 2l1  þ

6 f 00 ð0Þc2 c X 1 1j j1 j¼2

2sl4j

One can see that

;

ðf100 ð0Þc1j þ fj00 ð0Þlj Þ;

fj00 ð0Þc1j cj1 2sl3j

where d ¼ l45 l46 ½l2 s3 l1 l3 l4 þ 3ðl2 l1 l3 þ 3l2 l1 l4 þ l3 l1 l4 þ l2 l3 l4 Þs2 þ 6ðl1 l3 þ l1 l2 þ l2 l4 þ 6l1 l4 þ l2 l3 þ 6l3 l4 Þs þ 6ðl3 þ 6l2 þ 6l1 þ 6l4 Þ þ 6c51 c15 l46 ðl4  l5 Þ  ðl3  l5 Þðl2  l5 Þ þ 6c61 c16 l45 ðl4  l6 Þðl3  l6 Þðl2  l6 Þ: Theorem 3 Let ðH1Þ, ðH3Þ and ðH5Þ hold. If fi00 ð0Þ 6¼ 0, i ¼ 1; . . .; 6, then on the center manifold, system (29) is equivalent to the normal form (31). Following Campbell and Yuan (2008) the bifurcation diagrams of system (31) at the origin are as follows: (1)

system (31) undergoes a transcritical bifurcation when T ¼ fðb1 ; b2 ; b3 Þ : g1 ¼ 0g;

(2)

system (31) undergoes a Hopf bifurcation when

~1 ¼ ðb1 ; b2 ; b3 Þ : g3 ¼  g1 ; g2 \0 ; H g2

(3)

system (31) undergoes a Hopf bifurcation at the nontrivial equilibrium point ð gh11 ; 0; 0Þ when

 X1 ~2 ¼ ðb1 ; b2 ; b3 Þ : g3 ¼ X4  g1 ; H X1 X3 g1  X1 g2

X1 [0 ; X3 g1  X1 g2

(4)

system (31) undergoes a B–T bifurcation when

½l2j ðl1 þ 2e1  2r1 Þs2

6 c2 c f 00 ð0Þ X ð1 þ slj Þðl1  2slj r1 Þ 1j j1 1 j¼2

X3 ¼



fj00 ð0Þc1j cj1

12 216c12 s6 l12 6 l5 c13 c14 ðl3  l4 Þðl2  l4 Þðl2  l3 Þ 6¼ 0; d3

B~ ¼ fðb1 ; b2 ; b3 Þ : g1 ¼ 0; g2 ¼ 0g; (5)

system (31) undergoes a zero–Hopf bifurcation when ~3 ¼ fðb1 ; b2 ; b3 Þ : g1 ¼ 0; g3 ¼ 0; g2 \0g: H

½l2j ðl1 þ 2e1  2r1 Þs2

½l2j ðl1  2r1 Þs2 þ 2lj ðl1  r1 Þs þ 2l1 :

Numerical examples and simulations In this section, first several numerical examples and bifurcation curves corresponding to our results are given. Second, some numerical simulations of an example is given to verify our results.

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160

Example 1 Take l1 ¼ 0:2, l2 ¼ 0:6, l3 ¼ 0:2, l4 ¼ 0:4, l5 ¼ 0:5, l6 ¼ 0:8, c41 ¼ 0:5, c51 ¼ 0:6, c61 ¼ 0:8, c12 ¼ c13 ¼ c15 ¼ 1, c14 ¼ 3, c16 ¼ 2, fi ðxÞ ¼ tanhðxÞþ 0:2x2 ði ¼ 1; 2; . . .; 6Þ, s ¼ 2, then fi0 ð0Þ ¼ 1, fi00 ð0Þ ¼ 0:4, by (5)-(8), we can obtain c021 ¼ 3:2535, c031 ¼ 0:2655 c41 6¼ c041 ¼ 0:7344. Eq. (13) has only one positive root w1 0:2957156781, then s0 5:341141624 which implies 0\s\s0 , one can obtain that (17) is satisfied. Hence, the conditions Theorem 2 are all satisfied. The bifurcation curves of Theorem 2 were shown in Fig. 1.

Cogn Neurodyn (2016) 10:149–164

c15 ¼ 1; c14 ¼ 3; c16 ¼ 2; fi ðxÞ ¼ tanhðxÞ þ 0:2x2 ði ¼ 1; 2; . . .; 6Þ, s ¼ 2; then fi0 ð0Þ ¼ 1; fi00 ð0Þ ¼ 0:4: By (5)–(8), one can obtain c021 ¼ 0:9126; c031 ¼ 0:1974; c041 ¼ 0:7344; c51 6¼ c051 59:52708333. Eq. (13) has only 1 0:8509803928, then s0

one positive root w 2:040926698 which implies 0\s\ s0 , one can obtain that (17) is satisfied. Hence, the conditions of Theorem 3 are all satisfied.

Example 4 Take l1 ¼ 0:2; l2 ¼ 0:6; l3 ¼ 0:2; l4 ¼ 0:4; l5 ¼ 0:5; l6 ¼ 0:8; c51 ¼ 0:6; c61 ¼ 0:8; c12 ¼ c13 ¼

Next, for the Example 3 we carry out some numerical simulations to demonstrate the corresponding dynamical behaviors when parameters ða1 ; a2 Þ are chosen in appropriate regions in Fig. 3. First we take parameters ða1 ; a2 Þ=ð0:003; 0:003Þ, then under initial conditions (0.02, 0.000004, 0.0002, 0.0004, 0.0002, 0.0004) and (-0.02, -0.000004, -0.0002, -0.0004, -0.0002, 0.0004), a time evolutions of system (20) is shown in Fig. 4a. As expected, two trajectories asymptotically reach the trivial equilibrium which, however, loses its stability, and bifurcates two asymptotically stable nontrivial equilibrium when the parameters ða1 ; a2 Þ cross the Pitchfork bifurcation curve S to the side of v1 [ 0 (see Fig. 4b). Within the region surrounded by the curves S and H0 , system (20) has a stable trivial equilibrium which may undergo a nondegenerate Hopf bifurcation on the curve H0 as shown in Fig. 5. One can see that a stable limit cycle is bifurcated when ða1 ; a2 Þ ¼ ð0:002; 0:002Þ. When parameters ða1 ; a2 Þ cross the Pitchfork bifurcation curve Sþ from the bottom up, two unstable nontrivial equilibria are branched from the trivial equilibrium. If choose ða1 ; a2 Þ ¼ ð0:006; 0:0014Þ, then with different

Fig. 1 The bifurcation curves of Theorem 2 corresponding to the Example 1

Fig. 2 The bifurcation curves for s ¼ 1 when f ð0Þ ¼ 0, fi0 ð0Þ ¼ 1, fi00 ð0Þ ¼ 0 and fi000 ð0Þ 6¼ 0

Example 2 For the case of fi00 ð0Þ ¼ 0, i ¼ 1; . . .; 6, we take fi ðxÞ ¼ tanhðxÞ þ 2x3 =3, i ¼ 1; . . .; 6. Other parameters share the same values as in Example 1. It is easy to verify that fi0 ð0Þ ¼ 1; fi00 ð0Þ ¼ 0; fi000 ð0Þ ¼ 2, a3

6:228127027; b3 37:84176265, thus s ¼ 1. The corresponding bifurcation curves are shown in Fig. 2. Example 3 To get the case of s ¼ 1 when fi ð0Þ ¼ 0, fi0 ð0Þ ¼ 1, fi00 ð0Þ ¼ 0 and fi000 ð0Þ 6¼ 0, we choose fi ðxÞ ¼ tanhðxÞ, ði ¼ 1; 2; . . .; 6Þ, l1 ¼ 0:2, l2 ¼ 0:6, l3 ¼ 0:2, l4 ¼ 0:4, l5 ¼ 0:5, l6 ¼ 0:8, c41 ¼ 0:5, c51 ¼ 0:6, c61 ¼ 0:8, c12 ¼ 1, c13 ¼ 1, c14 ¼ 3, c15 ¼ 0:1, c16 ¼ 2 and s ¼ 1:5. To satisfy the conditions (H1) and (H2), we take c021 ¼ 2:6883 and c031 ¼ 0:2379. Some simple computations show s ¼ 1, fi ð0Þ ¼ 0, fi0 ð0Þ ¼ 1, fi00 ð0Þ ¼ 0 and fi000 ð0Þ ¼ 2, (i ¼ 1; . . .; 6), thus, by using the results obtained in ‘‘B–T bifurcation’’ section, we get a full picture of bifurcation diagram of system (20) in the parameter space ða1 ; a2 Þ which is shown in Fig. 3.

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Cogn Neurodyn (2016) 10:149–164

161

initial conditions, all trajectories approach the stable periodic solution (see Fig. 6). Moving parameters toward the Hopf bifurcation curve H1 , the big stable limit cycle is still present. Two nontrivial eqilibria become stable, and two unstable small limit cycles surrounding two nontrivial equilibria are yielded when ða1 ; a2 Þ pass H1 . Moreover, the two cycles are located inside the big limit cycle. From Fig. 7, we can see that if the initial condition is chosen close the outer of one of small cycle, then the solution asymptotically approaches the small limit cycle. However, if the initial condition is far away from two nontrivial equilibria, then the solution reach the big cycle. As parameters sequentially cross the homoclinic curve T, two small limit cycles disappear via a homoclinic loop bifurcation as shown in Fig. 8a where two nontrivial equilibria are still asymptotically stable and surrounded by the limit cycle. Finally, we choose ða1 ; a2 Þ in the interior made up by the cures Hd and S where the big limit cycle also disappears, only the nontrivial equilibria are stable.

Fig. 3 The bifurcation curves for s ¼ 1 when f ð0Þ ¼ 0, fi0 ð0Þ ¼ 1, fi00 ð0Þ ¼ 0 and fi000 ð0Þ ¼ 2

(a)

(b)

0.03

0.04

0.02

0.02

0

−0.01

−0.02

−0.02 −0.03

0

u1

u1

0.01

0

2000

4000

−0.04

6000

0

1000

time t

3000

4000

(the black curve) and ð0:02; 0:000004; 0:0002; 0:0004; 0:0002; 0:0004Þ (the blue curve). b System (20) has two locally asymptotically stable nontrivial equilibria when ða1 ; a2 Þ = ð0:003; 0:003Þ with the same initial conditions as a. (Color figure online)

Fig. 4 System (20) undergoes a Pitchfork bifurcation when parameters ða1 ; a2 Þ pass the curve S in Fig. 2. a The origin is locally asymptotically stable when ða1 ; a2 Þ ¼ ð0:003; 0:003Þ under initial conditions ð0:02; 0:000004; 0:0002; 0:0004; 0:0002; 0:0004Þ

(a)

2000

time t

(b)

0.03 0.02

0.05

0

u3

u1

0.01 0

−0.01 −0.05

−0.02 −0.03

0

2000 4000 6000 8000 10000 12000

time t

Fig. 5 System (20) undergoes a Hopf bifurcation at the trivial equilibrium when ða1 ; a2 Þ ¼ ð0:002; 0:002Þ with initial conditions ð0:02; 0:000004; 0:0002; 0:0004; 0:0002; 0:0004Þ (the blue curve), ð0:02; 0:000004; 0:0002; 0:0004; 0:0002; 0:0004Þ (the black

−0.02

−0.01

0

u

0.01

0.02

1

curve) and ð0:001; 0:0; 0:0; 0:0; 0:0; 0:0Þ (the red curve), respectively. a The time series in the plane ðt; u1 Þ. b The phase portrait in plane ðu1 ; u3 Þ. (Color figure online)

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Cogn Neurodyn (2016) 10:149–164

(a)

(b)

0.01

0.05

0

0

3

0.1

−0.01

−0.05

−0.02

−0.1

−0.03

0

2000

time t

4000

6000

−0.03 −0.02 −0.01

Fig. 6 System (20) has two unstable nodes when ða1 ; a2 Þ ¼ ð0:006; 0:0014Þ with initial conditions ð0:00006; 0:0; 0:0; 0:0; 0:0; 0:0Þ (the blue curve), ð0:00006; 0:0; 0:0; 0:0; 0:0; 0:0Þ

(b)

0.03

0.01

0.05

0

0

3

0.1

−0.01

−0.05

−0.02

−0.1 0

2000

4000

time t

6000

8000

0

0

1

0.01

u

u1

0.01

−0.01

−0.01

−0.02

−0.02

time t

0

u

0.01

0.02

0.03

0.03 0.02

5000

0.03

black curve), respectively. b The phase portrait in plane ðu1 ; u3 Þ when ða1 ; a2 Þ ¼ ð0:006; 0:00124Þ with initial conditions ð0:00004; 0:0; 0:0; 0:0; 0:0; 0:0Þ (the red curve) and ð0:02; 0:0; 0:0; 0:0; 0:0; 0:0Þ (the black curve). (Color figure online)

0.02

10000

Fig. 8 The dynamics of system (20) when ða1 ; a2 Þ across the lines Hd from the bottom up. a System (20) has two locally asymptotically stable nontrivial equilibria and a stable limit cycle when ða1 ; a2 Þ ¼ ð0:006; 0:00112Þ with initial conditions ð0:00005; 0:0; 0:0;

123

−0.03 −0.02 −0.01

(b)

0.03

0

0.02

1

Fig. 7 The dynamics of system (20) when ða1 ; a2 Þ are located between the lines H1 and Hd . a System (20) has a unstable limit cycle bifurcated from a nontrivial equilibrium when ða1 ; a2 Þ ¼ ð0:006; 0:00134Þ with initial conditions ð0:00003; 0:0; 0:0; 0:0; 0:0; 0:0Þ (the red curve) and ð0:02; 0:0; 0:0; 0:0; 0:0; 0:0Þ (the

−0.03

0.01

0.15

0.02

−0.03

0

u1

(the red curve) and ð0:02; 0:0; 0:0; 0:0; 0:0; 0:0Þ (the black curve), respectively. a The time series in the plane ðt; u1 Þ. b The phase portrait in plane ðu1 ; u3 Þ. (Color figure online)

u

u1

(a)

(a)

0.15

0.02

u

u1

0.03

−0.03

0

600

1200

time t

1800

2400

0:0; 0:0; 0:0Þ (the red curve), ð0:00005; 0:0; 0:0; 0:0; 0:0; 0:0Þ (the blue curve) and ð0:02; 0:0; 0:0; 0:0; 0:0; 0:0Þ (the black curve), respectively. b The case of ða1 ; a2 Þ ¼ ð0:006; 0Þ where initial conditions are same as a. (Color figure online)

Cogn Neurodyn (2016) 10:149–164

Thus, in the region, all trajectories are attracted to one of the nontrivial equilibria (see Fig. 8b).

Discussion In this work, we have devoted to derive the sufficient conditions for the existence of B–T and triple zero bifurcations, and the normal forms in a delayed BAM network with six neurons. Moreover, the bifurcation analysis near the B–T and triple zero critical points are given respectively, showing that the system may exhibit Pithfork bifurcation, heteroclinic bifurcation, transcritical bifurcation, Hopf bifurcation, zero–Hopf bifurcation in the neighborhood of the degenerate equilibrium. These give an important guiding significance in improving the design of BAM and extending associated application in the more fields. However, for (1), it is believed that there are still interesting and complex dynamical behaviors to be completely exploited. In the future, we will focus on the B–T and triple zero bifurcations analysis for neural networks with multiple neurons and multiple delays. Acknowledgments The authors would like to thank the anonymous reviewers for their valuable comments and suggestions. This research is supported by the National Natural Science Foundation of China (No. 11171206).

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High codimensional bifurcation analysis to a six-neuron BAM neural network.

In this article, the high codimension bifurcations of a six-neuron BAM neural network system with multiple delays are addressed. We first deduce the e...
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