bogdanov–takens bifurcation in a single inertial neuron model with delay
TRANSCRIPT
Neurocomputing 89 (2012) 193–201
Contents lists available at SciVerse ScienceDirect
Neurocomputing
0925-23
http://d
$This
China G
the Cenn Corr
E-m
journal homepage: www.elsevier.com/locate/neucom
Bogdanov–Takens bifurcation in a single inertial neuron model with delay$
Xing He a, Chuandong Li a,n, Yonglu Shu b
a College of Computer Science, Chongqing University, Chongqing 400030, PR Chinab College of Mathematics and Statistics, Chongqing University, Chongqing 401331, PR China
a r t i c l e i n f o
Article history:
Received 8 July 2011
Received in revised form
20 November 2011
Accepted 29 February 2012
Communicated by H. Jiangbifurcation, homoclinic bifurcation, heteroclinic bifurcation and double limit cycle bifurcation. Some
Available online 30 March 2012
Keywords:
Bogdanov–Takens bifurcation
Inertial neuron model
Homoclinic bifurcation
Heteroclinic bifurcation
12/$ - see front matter & 2012 Elsevier B.V. A
x.doi.org/10.1016/j.neucom.2012.02.019
research is supported by the National Na
rant No. 60974020, 11171360 and the Fund
tral Universities of China (Project No. CDJZR1
esponding author. Tel.: þ86 23 65103199.
ail address: [email protected] (C. Li).
a b s t r a c t
In this paper, we study a retarded functional differential equation modeling a single neuron with
inertial term subject to time delay. Bogdanov–Takens bifurcation is investigated by using center
manifold reduction and the normal form method for RFDE. We get the versal unfolding of the norm
forms at the B–T singularity and show that the model can exhibit saddle-node bifurcation, pitchfork
numerical simulations are given to support the analytic results.
& 2012 Elsevier B.V. All rights reserved.
1. Introduction
Since Hopfield [6] proposed a simplified neural networkmodel, there has been increasing interest in investigating thedynamical behaviors of continuous neural networks with orwithout delay due to their wide application, such as associativememory, pattern recognition, optimization and signal processing.Some important results have been reported [1,6,13].
For inertial neuron model, the inertia can be considered auseful tool, which is added to help in the generation of chaos andthere are some biological background for the inclusion of aninductance term [7,14]. More and more researchers focus on thissubject. Babcock and Westervelt [1] combined inertia and droveto explore chaos in one and two neurons system. Wheeler andSchieve [17] discussed the stability and chaos in an inertial two-neural system. Tain et al. [15,16] added inertia to neural equa-tions as a way of chaotically searching for memories in neuralnetworks. Li et al. [10] studied Hopf bifurcation and chaos in asingle inertial neuron model with delay. Liu et al. [11] illustratedthe stability of bifurcating periodic solutions for a single delayedinertial neuron model under periodic excitation. Liu et al. [12]also discussed the resonant codimension-two bifurcation in aninertial two-neuron system with time delay. However, to the best
ll rights reserved.
tural Science Foundation of
amental Research Funds for
0 18 55 01).
of the authors’ knowledge, few results for Bogdanov–Takensbifurcation in a inertial neuron model have been reported in theliterature.
In this paper, we consider that a single inertial neuron modelwith time delay [10] is described by
€x ¼�a _x�bxþcf ðx�hxðt�tÞÞ, ð1Þ
where a,b,c40,hZ0, t40 is the time delay, and f is the non-linear activation function. System (1) was analyzed in [10] fromthe point of view of Hopf bifurcation and chaos. The authors usedh as a bifurcation parameter to show that system (1) undergoesHopf bifurcation and chaotic behavior of system (1) was observedwhen adopting a non-monotonic activation function. Except thedynamic of system (1) in [10], it is interesting to further find outwhat kind of new dynamics this system has. The study carried outin the present paper may contribute to understand the codimen-sion-two Bogdanov–Takens singularity in the single inertialneuron model with time delay. We use a and b as bifurcationparameters. System (1) exhibits codimension-two singularitywhen two-parameter vary in a neighborhood of the criticalvalues. By using the normal form method for RFDE [3,4], weobtain the normal forms to study its dynamical behaviors. It isshown that different bifurcation diagrams can be constructed dueto the difference of activation function.
This paper is organized as follows. In the next section, thepreliminaries relevant to the normal forms with parameter forRFDE are presented. In Section 3, we discuss the existence ofBogdanov–Takens bifurcation. Then we analyze Bogdanov–Takens singularity in the single inertial neuron model with timedelay and get the versal unfolding of B–T bifurcation in Section 4.
X. He et al. / Neurocomputing 89 (2012) 193–201194
In Section 5, these normal forms are used to predict B–T bifurca-tion diagrams. In Section 6, some numerical simulations are givento support the analytic results. Section 7 summarizes the mainconclusions.
2. Preliminaries
This section presents B–T bifurcation theory of normal formwith parameters for functional differential [3,4,8]. We consider anabstract retarded functional differential equation with para-meters in the phase space C ¼ Cð½�t,0�;Rn
Þ
_xðtÞ ¼ LðmÞxtþGðxt ,mÞ, ð2Þ
where xt AC is defined by xtðyÞ ¼ xðtþyÞ, �tryr0, the para-meter mARp is a parameter vector in a neighborhood V of zero,LðmÞ : V-LðC,Rn
Þ is Ck�1, and G : C � Rp-Rn is CkðkZ2Þ with
Gð0,mÞ ¼ 0, DxGð0,mÞ ¼ 0.Define L¼ Lð0Þ and Fðxt ,mÞ ¼ Gðxt ,mÞþðLðmÞ�Lð0ÞÞxt , then sys-
tem (2) can be rewritten as
_xðtÞ ¼ LxtþFðxt ,mÞ: ð3Þ
Then the linear homogeneous retarded functional differentialequation of Eq. (3) can be written as
_xðtÞ ¼ Lxt : ð4Þ
Since L is a bounded linear operator, L can be represented by aRiemann–Stieltjes integral
Lj¼Z 0
�tdZðyÞjðyÞ, 8jAC,
by the Riesz representation theorem, where ZðyÞðyA ½�t,0�Þ is ann�n matrix function of bounded variation. Let A0 be the infini-tesimal generator for the solution semigroup defined by system(4) such that
A0j¼ _j,DðA0Þ ¼ jAC1ð½�t,0�,Rn
Þ : _jð0Þ ¼Z 0
�tdZðyÞjðyÞ
( ):
Define the bilinear form between C and C0 ¼ Cð½0,t�,RnnÞ by
/c,jS¼cð0Þjð0Þ�Z 0
�t
Z y
0cðx�yÞ dZðyÞjðxÞ dx, 8cAC0, 8jAC:
Assume that L has double zero eigenvalues and all other eigen-values have negative real parts. Let L be the set of eigenvalueswith zero real part and P be the generalized eigenvalues spaceassociated with L and Pn the space adjoint with P. Then C can bedecomposed as
C ¼ P � Q where Q ¼ fjAC : /j,cS¼ 0, 8cAPng,
with dim P¼ 2: Choose the bases F and C for P and Pn such that
/C,FS¼ I, _F ¼FJ, _C ¼�JC,
where I is the 2�2 identity matrix and
J¼0 1
0 0
� �:
Following the ideas in [3,4], we consider the enlarged phase spaceBC
BC ¼ j : ½�t,0�-Rn : j is continuous on ½�t,0Þ, ( limy-0�
jðyÞARn
� �:
Then the elements of BC can be expressed as c¼jþx0a, jAC,aARn and
x0ðyÞ ¼0, �tryo0,
I, y¼ 0,
(
where I is the identity operator on C. The space BC has the norm9jþx0a9¼ 9j9Cþ9a9Rn . The definition of the continuous projec-tion p : BC-P by
pðjþx0aÞ ¼F½ðC,jÞþCð0Þa�,
which allows us to decompose the enlarged phase spaceBC ¼ P � Ker p. Let x¼Fzþy. Then system (2) can be decomposed as
_z ¼ JzþCð0ÞFðFzþy,mÞ,_y ¼ AQ1 yþðI�pÞx0FðFzþy,mÞ, xAR2, yAQ1,
(ð5Þ
for yAQ1¼Q \ C1
� Kerp, where AQ1 is the restriction of A0 as anoperator from Q1 to the Banach space Ker p.
Employing Taylor’s theorem, system (5) becomes
_z ¼ JzþSjZ21j f 1
j ðz,y,mÞ,
_y ¼ AQ1 yþSjZ21j f 2
j ðz,y,mÞ,
8<: ð6Þ
where f ijðz,y,mÞði¼ 1;2Þ denotes the homogeneous polynomials of
degree j in variables ðz,y,mÞ. For
J¼0 1
0 0
� �,
the non-resonance conditions are naturally satisfied. According tonormal form theory developed in [5], system (6) can be trans-formed to the following normal form on the center manifold:
_z ¼ Jzþ12g1
2ðz,0,mÞþh:o:t: ð7Þ
For a normed space Z, denoted by V4j ðZÞ the linear space of
homogeneous polynomials of ðz,mÞ ¼ ðz1,z2,m1,m2Þ with degree j
and with coefficients in Z, and define Mj to be the operator inV4
j ðR2� KerpÞ with the range in the same space by
Mjðp,hÞ ¼ ðM1j p, M2
j hÞ,
where
M1j p¼M1
j
p1
p2
!¼
@p1@z1
z2�p2
@p2@z1
z2
0@
1A,
M2j h¼M2
j hðz,mÞ ¼Dzhðz,mÞJx�AQ1 hðz,mÞ:
Using M1j , we have the following decompositions:
V4j ðR
2Þ ¼ ImðM1
j Þ � ðImðM1j ÞÞ
c , V4j ðR
2Þ ¼ KerðM1
j Þ � ðKerðM1j ÞÞ
c :
By the above decompositions, g12ðz,0,mÞ can be expressed as
g12ðz,0,mÞ ¼ Project
ðImðM12ÞÞ
c f 12ðz,0,mÞ:
The base of V42ðR
2� KerpÞ is composed by the following 20
elements:
z21
0
!,
z22
0
!,
z1z2
0
� �,
z1mi
0
� �,
z2mi
0
� �,
m2i
0
!,
m1m2
0
� �,
0
z21
!,
0
z22
!,
0
z1z2
!,
0
z1mi
!,
0
z2mi
!,
0
m2i
!,
0
m1m2
!,
i¼ 1;2,
and images of these elements under M12 are
2z1z2
0
� �,
0
0
� �,
z22
0
!,
z2mi
0
� �,�z2
1
2z1z2
!,
�z1z2
z22
!,�z2
2
0
!,�z1mi
z2mi
!,�z2mi
0
� �:
X. He et al. / Neurocomputing 89 (2012) 193–201 195
Therefore, a basis of ðImðM12ÞÞ
c can be taken as the set composedby the elements
0
z21
!,
0
z1z2
!,
0
z1mi
!,
0
z2mi
!,
0
m2i
!,
0
m1m2
!:
Then we have
1
2g1
2ðz,0,mÞ ¼0
l1z1þl2z2þZ1z21þZ2z1z2þh:o:t:
!:
If the second order normal form is degenerated, then we obtain
1
2g1
2ðz,0,mÞ ¼0
l1z1þl2z2þh:o:t:
!:
So the high-order normal must be calculated. System (6) can betransformed to the following normal form on the center manifold:
_z ¼ Jzþ12 g1
2ðz,0,mÞþ 13!g
13ðz,0,mÞþh:o:t ð8Þ
and define
U12ðzÞ ¼ ðM
12Þ�1Project
ðImðM12ÞÞ
c f 12ðz,0;0Þ, U2
2ðzÞ ¼ ðM22Þ�1f 2
2ðz,0;0Þ:
Then
g13ðz,0,mÞ ¼ Project
ðImðM13ÞÞ
c ff13ðz,0,mÞþ3
2½Dzf 12ðz,0,mÞU1
2þDyf 12ðz,0,mÞU2
2�g:
Be similar with the computation of ðImðM12ÞÞ
c. The space ðImðM13ÞÞ
c
is spanned by
0
z31
!,
0
z21z2
!,
0
z21mi
!,
0
z1z2mi
!,
0
zim2i
!,
0
z1m1m2
!,
0
z2m1m2
!,
0
m3i
!,
0
m21m2
!,
0
m1m22
!:
Then we obtain
1
3!g1
3ðz,0,mÞ ¼0
Z1z31þZ2z2
1z2þh:o:t
!:
Hence the normal form with the universal unfolding is
_z1 ¼ z2þh:o,t,
_z2 ¼ l1z1þl2z2þZ1z31þZ2z2
1z2þh:o:t:
(
3. The existence of Bogdanov–Takens bifurcation
Throughout the rest of this paper, we assume that f ð0Þ ¼ 0, andf is a nonlinear C3 function. By defining [10]
x1ðtÞ � xðtÞ�hxðt�tÞ, x2 ¼ _x1ðtÞ, tA ½�t,1Þ:
Then system (1) can be written as
_x1ðtÞ ¼ x2ðtÞ,
_x2ðtÞ ¼�ax2ðtÞ�bx1ðtÞþcf ðx1ðtÞÞ�chf ðx1ðt�tÞÞ:
(ð9Þ
It is clear that system (9) has one equilibrium ð0;0Þ.Then the linearization equation at the origin is
_x1ðtÞ ¼ x2ðtÞ,
_x2ðtÞ ¼�ax2ðtÞ�bx1ðtÞþckx1ðtÞ�chkx1ðt�tÞ
(
and the corresponding characteristic equation is
FðlÞ ¼ l2þalþb�ckþchke�lt ¼ 0, ð10Þ
where k¼ f 0ð0Þ, for t¼ 0, the two roots of Eq. (10) have negativereal parts if and only if b�ckþchk40:
Next, we focus on the case of t40. Let a0 ¼ chkt, b0 ¼ ckð1�hÞ,and we have the following result.
Theorem 1. Suppose b¼ b0 and k40. Then
(i)
l¼ 0 is a single root of Eq. (10) if and only if aaa0. (ii) l¼ 0 is a double zero root of Eq. (10) if and only if a¼ a0.(iii)
Eq. (10) does not have purely imaginary roots 7oiðo40Þ.Proof. It is easy to calculate Fð0Þ ¼ 0 if b¼ b0, and
F 0ðlÞ ¼ 2lþa�chkte�lt, F 00ðlÞ ¼ 2þchkt2e�lt:
From k40, we obtain
F 0ðlÞjl ¼ 0,b ¼ b0 ,a ¼ a0¼ 0, F 00ðlÞjl ¼ 0,k ¼ k0 ,t ¼ t0
¼ 2þchkt240:
This completes the proof of (i) and (ii).
When b¼ b0, substituting l¼ io into Eq. (10) yields
�o2þaoiþb0�ckþchk0e�toi ¼ 0
and separating the real and imaginary parts, we have
o2þchk¼ chk cosðotÞ,
resulting in
cos ðotÞ ¼ 1þo2
chk41,
which is meaningless. This proves claim (iii). &
4. Bogdanov–Takens bifurcation
In this section, we investigate B–T bifurcation by using themethod in Section 2. From Theorem 1, we know that, at the origin,the characteristic equation of system (9) has a double zero root ifa0 ¼ chkt, b0 ¼ ckð1�hÞ and k40. So we treat (a,b) as bifurcationparameters near ða0,b0Þ.
Rescaling the time by t-t=t to normalize the delay, system (9)can be written as
_x1ðtÞ ¼ tx2ðtÞ,
_x2ðtÞ ¼�atx2ðtÞ�btx1ðtÞþctf ðx1ðtÞÞ�chtf ðx1ðt�1ÞÞ:
(ð11Þ
Let a¼ a0þm1, b¼ b0þm2 and expand the function f. Then system(11) becomes
_x1ðtÞ ¼ tx2ðtÞ,
_x2ðtÞ ¼�ða0þm1Þtx2ðtÞ�ðb0þm2Þtx1ðtÞþctkx1ðtÞ,
�chtkx1ðt�1ÞþG22þh:o:t:,
8><>: ð12Þ
where
G22 ¼
ctðx21�hx2
1ðt�1ÞÞf 00ð0Þ
2þ
ctðx31�hx3
1ðt�1ÞÞf 000ð0Þ
6:
From Section 2, let
ZðyÞ ¼ AdðyÞþBdðyþ1Þ,
where
A¼0 t
ð�b0þckÞt �a0t
!, B¼
0 0
�chtk 0
� �
and define
L0j¼Z 0
�1dZðyÞjðyÞ, 8jAC:
The infinitesimal generator
A0j¼_j, �1ryo0,R 0�1 dZðyÞjðyÞ, y¼ 0:
(
X. He et al. / Neurocomputing 89 (2012) 193–201196
Rewrite system (12) as
_xt ¼ LðmÞxtþGðxt ,mÞþh:o:t¼ ðL0þL1ðmÞÞxtþGðxt ,mÞþh:o:t,
where
L0xt ¼tx2ð0Þ
�a0tx2ð0Þ�b0tx1ð0Þþctkðx1ð0Þ�hx1ð�1ÞÞ
!,
Gðxt ,mÞ ¼0
G22
!,
L1ðmÞxt ¼0
�m2tx1ð0Þ�m1tx2ð0Þ
!:
Then system (12) can be transformed into
_xt ¼ L0xtþFðxt ,mÞþh:o:t,
where Fðxt ,mÞ ¼ L1ðmÞxtþGðxt ,mÞ, and the bilinear form on Cn� C is
/c,jS¼cð0Þjð0ÞþZ 0
�1cðxþ1ÞBjðxÞ dx,
where FðyÞ ¼ ðj1ðyÞ,j2ðyÞÞAC,
CðsÞ ¼c1ðsÞ
c2ðsÞ
!ACn
are, respectively, the bases for the center space P and its dualspace Pn.
Next we will find the FðyÞ and CðsÞ based on the techniquesdeveloped by [18].
Lemma 1 (see Xu and Huang [18]). The bases of P and its dual space
Pn have the following representations
P¼ spanF, FðyÞ ¼ ðj1ðyÞ,j2ðyÞÞ, �1ryr0,
Pn¼ spanC, CðsÞ ¼ colðc1ðsÞ, c2ðsÞÞ, 0rsr1,
where j1ðyÞ ¼j01ARn
\f0g, j2ðyÞ ¼j02þj0
1y, j02ARn, and c2ðsÞ ¼
c02ARnn
\f0g, c1ðsÞ ¼c01�sc0
2, c01ARnn, which satisfy
ð1Þ ðAþBÞj01 ¼ 0, ð2Þ ðAþBÞj0
2 ¼ ðBþ IÞj01, ð3Þ c0
2ðAþBÞ ¼ 0,
ð4Þ c01ðAþBÞ ¼c0
2ðBþ IÞ, ð5Þ c02j
02�
12c
02Bj0
1þc02Bj0
2 ¼ 1,
ð6Þ c01j
02�
12 c
01Bj0
1þc01Bj0
2þ16 c
02Bj0
1�12c
02Bj0
2 ¼ 0:
So it is not difficult to verify that
FðyÞ ¼1 y0 1
t
!, CðsÞ ¼
m1�n1as m2�n1s
n1a n1
!,
such that _F ¼FJ, _C ¼�JC, /C,FS¼ I, where
J¼0 1
0 0
� �,
and
n1 ¼2a0ð1�hÞ
a20ð1�hÞþb0h
, m1 ¼n1a2
0t3ta0þ6
þ2b0h
a20ð1�hÞþb0h
,
m2 ¼n1a0t
3ta0þ6:
Let x¼Fzþy, namely
x1ðyÞ ¼ z1þyz2þy1ðyÞ, x2ðyÞ ¼1
t z2þy2ðyÞ:
Then
x1ð0Þ ¼ z1þy1ð0Þ, x1ð�1Þ ¼ z1�z2þy1ð�1Þ,
x2ð0Þ ¼1
t z2þy2ð0Þ:
From Section 2, system (12) can be decomposed as
_z ¼ JzþCð0ÞFðFzþy,mÞþh:o:t,
_y ¼ AQ1 yþðI�pÞx0FðFzþy,mÞþh:o:t, zAR2, yAQ1:
(ð13Þ
On the center manifold, system (13) can be written as
_z1 ¼ z2þm1F12þm2F2
2þh:o:t,
_z2 ¼ naF12þnF2
2þh:o:t,
(ð14Þ
where F12 ¼ 0,
F22 ¼�m1z2�m2tz1þ
ctf 00ð0Þ
2z2
1�cthf 00ð0Þ
2ðz1�z2Þ
2
þctf 000ð0Þ
6z3
1�chtf 000ð0Þ
6ðz1�z2Þ
3:
Following the computation of the normal form for functionaldifferential equations introduced by Section 2, we get the normalform with versal unfolding on the center manifold
_z1 ¼ z2,
_z2 ¼ l1z1þl2z2þZ1z21þZ2z1z2þh:o:t,
(ð15Þ
where
l1 ¼�n1tm2, l2 ¼�n1m1�m2tm2,
Z1 ¼n1ctð1�hÞf 00ð0Þ
2, Z2 ¼m2ctð1�hÞf 00ð0Þþn1chtf 00ð0Þ:
If the following condition is satisfied
@ðl1,l2Þ
@ðm1,m2Þ
� ����m ¼ 0
a0, ð16Þ
the map ðm1,m2Þ/ðl1,l2Þ is regular and
n1ctð1�hÞ40, m2ctð1�hÞ40, n1cht40:
Thus we can get the following result:
Theorem 2. Under condition (16), if f 00ð0Þa0, then, on the center
manifold, system (12) is equivalent to the normal form (15), where
Z1 � Z240.
If f 00ð0Þ ¼ 0, then Z1 ¼ 0, Z2 ¼ 0, hence the system (15) is degen-erate, we need to calculate the high order normal form. Followingthe computation of the normal form for functional differentialequations introduced by Section 2, we can get high-order normalform:
_z1 ¼ z2,
_z2 ¼ l1z1þl2z2þZ3z31þZ4z2
1z2þh:o:t,
(ð17Þ
where Z3 ¼ n1ctð1�hÞf 000ð0Þ=6,
Z4 ¼m2ctð1�hÞf 000ð0Þ
2þ
n1chtf 000ð0Þ
2:
So we obtain the following result
Theorem 3. Under condition (16), if f 00ð0Þ ¼ 0, f 000ð0Þa0, then, on
the center manifold, system (12) is equivalent to the normal form
(17), where Z3 � Z440.
5. Bifurcation diagrams
In this section, we discuss the bifurcation diagrams of system(9), first we consider the truncated normal form of system (15):
_z1 ¼ z2,
_z2 ¼ l1z1þl2z2þZ1z21þZ2z1z2:
(ð18Þ
Fig. 1. Bifurcation curve of Theorem 4.
X. He et al. / Neurocomputing 89 (2012) 193–201 197
By introducing the change of variables and rescaling of time
z1 ¼Z1
Z22
x1�l1
Z2
� �, z2 ¼
Z21
Z32
x2, t¼Z2
Z1
t,
system (18) becomes (still using z1, z2 for simplicity)
_z1 ¼ z2,
_z2 ¼ u1þu2z2þz21þz1z2,
(ð19Þ
where
u1 ¼Z2
2
Z21
�Z2
Z1
k1þk2
� �m2þk3m1
� �, u2 ¼
Z2
Z1
Z2
Z1
k1�2k2
� �m2�2k3m1
� �,
k1 ¼�n1t, k2 ¼�m2t, k3 ¼�n1:
The complete bifurcation diagrams of system (19) can befound in [2,5,9]. Here we just briefly list some results.
(i) System (19) undergoes a saddle-node bifurcation on thecurve
S¼ ðu1,u2Þ : u1 ¼u2
2
4
� �,
(ii) system (19) undergoes an unstable Hopf bifurcation on thecurve
H¼ fðu1,u2Þ : u1 ¼ 0, u2o0g,
(iii) system (19) undergoes a saddle homoclinic bifurcation on thecurve
T ¼ ðu1,u2Þ : u1 ¼�6
25u2
2, u2o0
� �:
Applying the above results and using the expressions of u1, u2, weobtain the following result.
Theorem 4. Under condition (16), if f 00ð0Þa0, for sufficiently small
m1, m2,
(i) system (9) undergoes a saddle-node bifurcation on the curve:
S¼ ðm1,m2Þ :Z2
Z1
k1�2k2
� �m1�2k3m2
� �2
¼ 4 �Z2
Z1
k1þk2
� �m1þk3m2
� �( ),
(ii) system (9) undergoes an unstable Hopf bifurcation at the non-
trivial equilibrium on the curve:
H¼ ðm1,m2Þ : �Z2
Z1
k1þk2
� �m1þk3m2 ¼ 0,
Z2
Z1
k1�2k2
� �m1o2k3m2
� �,
(iii) system (9) undergoes a saddle homoclinic bifurcation on the
curve:
T ¼ ðm1,m2Þ : �Z2
Z1
k1þk2
� �m1þk3m2 ¼�
6
25
Z2
Z1
k1�2k2
� �m1�2k3m2
� �2
,
(
Z2
Z1
k1�2k2
� �m1o2k3m2
�:
The bifurcation curve of Theorem 4 is in Fig. 1. Next, we considerthe truncated normal form of system (17)
_z1 ¼ z2,
_z2 ¼ l1z1þl2z2þZ3z31þZ4z2
1z2:
(ð20Þ
In order to put system (20) in an appropriate form, we make thetransformation
x1 ¼Z4ffiffiffiffiffiffiffiffiffi9Z39
q z1, x2 ¼�Z2
4
9Z39ffiffiffiffiffiffiffiffiffi9Z39
q z2, t¼�9Z39Z4
t,
bring system (20) into (still using z1, z2 for simplicity)
_z1 ¼ z2,
_z2 ¼ v1z1þv2z2þsz31�z2
1z2,
(ð21Þ
where s¼ sgnðZ3Þ, v1 ¼ ðZ4=Z3Þ2l1 ¼ ðZ4=Z3Þ
2k1m2, v2 ¼�ðZ4=
9Z39Þ l2 ¼�ðZ4=9Z39Þðk2m2þk3m1Þ:
The complete bifurcation diagrams of system (21) can befound in [2,5,9]. Here we just briefly list some results:
when s¼1,(i) system (21) undergoes a pitchfork bifurcation on the curve:
S¼ fðv1,v2Þ : v1 ¼ 0, v2ARg,
(ii) system (21) undergoes a Hopf bifurcation at the trivialequilibrium on the curve:
H¼ fðv1,v2Þ : v2 ¼ 0, v1o0g,
(iii) system (21) undergoes a heteroclinic bifurcation on thecurve:
T ¼ fðv1,v2Þ : v2 ¼�15v1þOðv2
1Þ, v1o0g:
Applying the above results and using the expressions of v1, v2, weobtain the following results.
Theorem 5. Under condition (16), if f 00ð0Þ ¼ 0 and f 000ð0Þ40, for
sufficiently small m1,m2,
(i)
system (9) undergoes a pitchfork bifurcation on the curve:S¼ fðm1,m2Þ : m2 ¼ 0, m1ARg,
(ii)
system (9) undergoes a stable Hopf bifurcation at the trivialequilibrium on the curve:
H¼ ðm1,m2Þ : m2 ¼�k3
k2m1, m240
� �,
(iii)
system (9) undergoes a heteroclinic bifurcation on the curve:T ¼ ðm1,m2Þ : m2 ¼5Z3k3
Z4k1�5Z3k2m1þOðm2
2Þ, m240
� �:
Fig. 3. Bifurcation curve of Theorem 6.
5
6
7
x1x2
X. He et al. / Neurocomputing 89 (2012) 193–201198
The bifurcation curve of Theorem 5 is in Fig. 2. For system (21),when s¼�1,
Fig. 2. Bifurcation curve of Theorem 5.
4
(i)2
3x
system (21) undergoes a pitchfork bifurcation on the curve
S¼ fðv1,v2Þ : v1 ¼ 0, v2ARg,
(ii)
0
1
system (21) undergoes an unstable Hopf bifurcation at thenon-trivial equilibrium on the curveH¼ fðv1,v2Þ : v2 ¼ v1, v140g,
−1
(iii)0 50 100 150 200 250t
system (21) undergoes homoclinic bifurcation on the curve
T ¼ fðv1,v2Þ : v2 ¼45v1þOðv2
1Þ, v140g,
Fig. 4. Waveform plot of the variable x of system (9) for m1 ¼�0:213,m2 ¼ 0:05.
(iv) system (21) undergoes a double cycle bifurcation on thecurveHd ¼ fðv1,v2Þ : v2 ¼ dv1þOðv21Þ, v140, d 0:752g:
Applying the above results and using the expressions ofv1, v2, we obtain the following results.
Theorem 6. Under condition (16), if f 00ð0Þ ¼ 0 and f 000ð0Þo0, for
sufficiently small m1, m2,
(i)
system (9) undergoes a pitchfork bifurcation on the curve:S¼ fðm1,m2Þ : m2 ¼ 0, m1ARg,
(ii)
system (9) undergoes an unstable Hopf bifurcation at the non-trivial equilibrium on the curve:
H¼ ðm1,m2Þ : m1 ¼k1Z4�k2Z3
k3Z3
m2, m2o0
� �,
(iii)
system (9) undergoes homoclinic bifurcation on the curve:T ¼ ðm1,m2Þ : m1 ¼4k1Z4�5k2Z3
5k3Z3
m2þOðm22Þ, m2o0
� �,
(iv)
system (9) undergoes a double cycle bifurcation on the curve:Hd ¼ ðm1,m2Þ : m1 ¼dk1Z4�k2Z3
k3Z3
m2þOðm22Þ, m2o0, d 0:752
� �:
The bifurcation curve of Theorem 6 is in Fig. 3.
6. Numerical simulations
In this section, we give some examples to verify the theoreticalresults. For system (9), we fix h¼0.5, c¼2, t¼ 1. Then wecalculate n1 ¼ 1, m2 ¼
19, k1 ¼�1, k2 ¼�
19, k3 ¼�1, Z2=Z1 ¼
209 ,
Z4=Z3 ¼103 .
−1 0 1 2 3 4 5 6 7−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
x1
x 2
Fig. 5. Phase portraits of system (9) for m1 ¼�0:213,m2 ¼ 0:05.
0 500 1000 1500 2000 2500−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
3
t
x
x1x2
Fig. 6. Waveform plot of the variable x of system (9) for m1 ¼�0:02111,m2 ¼�0:01.
−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
x1
x 2
Fig. 7. Phase portraits of system (9) for m1 ¼�0:02111,m2 ¼�0:01.
X. He et al. / Neurocomputing 89 (2012) 193–201 199
Example 1. This example supports the result of Theorem 4. Forsystem (9), we fix f ðsÞ ¼ tanhðsÞþ0:01s2, and we get a0 ¼ 1, b0 ¼ 1.By the condition (i) of Theorem 4, system (9) undergoes a saddle-node bifurcation on the curve S¼ fðm1,m2Þ : ðm1�m2Þ
2¼ 19
9 m1�m2g,and there should exist a non-trivial node and the origin is a saddlepoint. If we set m1 ¼�0:213, m2 ¼ 0:05, Figs. 4 and 5 verify thisresult. By the condition (ii) of Theorem 4, system (9) undergoes anunstable Hopf bifurcation on the curve H¼ fðm1,m2Þ : m1 ¼
2:1111m2, m24m1g. We set m1 ¼�0:02111, m2 ¼�0:01. and thesystem (9) has two equilibrium points E1 ¼ ð0;0Þ andE2 ¼ ð�0:09,0Þ, and an unstable limit cycle exists through theHopf bifurcation near E2. Figs. 6 and 7 verify this result.
Example 2. This example demonstrates the result of Theorem 5.For system (9), we set f ðsÞ ¼ tanhðsÞþ2
3s3. By the condition (i) ofTheorem 5, system (9) undergoes a pitchfork bifurcation on thecurve S¼ fðm1,m2Þ : m1 ¼ 0, m2ARg. If we set m1 ¼ 0:01, m2 ¼ 0:001,there should exist two stable non-trivial equilibria and theorigin is unstable. Fig. 8 shows the good agreement with the
0 1000 2000 3000 4000 5000 6000−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
t
xx1x2
Fig. 8. Waveform plot of the variable x of system (9) for m1 ¼ 0:01,m2 ¼ 0:001.
0 1000 2000 3000 4000 5000 6000−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
t
x
x1x2
Fig. 9. Waveform plot of the variable x of system (9) for m1 ¼�0:0001,m2 ¼ 0:0009.
−0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4−0.03
−0.02
−0.01
0
0.01
0.02
0.03
x1
x 2
Fig. 10. Phase portraits of system (9) for m1 ¼�0:0001,m2 ¼ 0:0009.
−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3−0.02
−0.015
−0.01
−0.005
0
0.005
0.01
0.015
0.02
x1
x 2
Fig. 11. Phase portraits of system (9) for m1 ¼�0:025556,m2 ¼�0:01.
Fig. 12. Phase portraits of system (9) for m1 ¼�0:00043,m2 ¼�0:00018.
X. He et al. / Neurocomputing 89 (2012) 193–201200
theoretical analysis. By the condition (ii) of Theorem 5, system (9)undergoes a stable Hopf bifurcation at the origin on the curveH¼ fðm1,m2Þ : m2 ¼�9m1, m1o0g, If we set m1 ¼�0:0001 andm2 ¼ 0:0009, a stable limit cycle exists through the Hopf bifurca-tion near the origin. Figs. 9 and 10 verify this result.
Example 3. This example supports the result of Theorem 6. Forsystem (9), we fix f ðsÞ ¼ tanhðsÞ. From Theorem 6, system (9) under-goes homoclinic bifurcation on the curve H¼ fðm1,m2Þ : m1 ¼
2:5556m2þOðm22Þ, m2o0g, If we set m1 ¼�0:025556 and m2 ¼
�0:01, a closed orbit exists through the homoclinic bifurcation andthree equilibria of system (9) are in the closed orbit, and twohomoclinic orbits emerge from the origin in Fig. 11. System (9)also undergoes a double limit cycle bifurcation on the curveHd ¼ fðm1,m2Þ : m1 ¼ 2:3956m2þOðm2
2Þ,m2o0g. Fix m1 ¼�0:00043and m2 ¼�0:00018. Fig. 12 verifies this result.
7. Conclusion
A single delayed neuron model with inertial term has beenstudied in the neighborhood of Bogdanov–Takens codimension-two bifurcation point where the linear part of the system hasdouble zero eigenvalues. By applying normal form theory andcenter manifold reduction, we are able to predict their corre-sponding bifurcation diagrams such as saddle-node bifurcation,pitchfork bifurcation, homoclinic bifurcation, heteroclinic bifurca-tion and double limit cycle bifurcation.
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X. He et al. / Neurocomputing 89 (2012) 193–201 201
Xing He received the B.Sc. degree in mathematics andapplied mathematics from the Department of Mathe-matics, Guizhou University, Guiyang, China, in 2009.Currently, he is working towards the Ph.D degree withthe college of computer science, Chongqing University,Chongqing, China. His research interests includeneural network, bifurcation theory, and nonlineardynamical system.
Chuandong Li received the B.S. degree in AppliedMathematics from Sichuan University, Chengdu, Chinain 1992, and the M.S. degree in operational researchand control theory and Ph.D. degree in ComputerSoftware and Theory from Chongqing University,Chongqing, China, in 2001 and in 2005, respectively.
He has been a Professor at the College of ComputerScience, Chongqing University, Chongqing 400030,China, since 2007, and been the IEEE Senior membersince 2010. From November 2006 to November 2008,he serves as a research fellow in the Department ofManufacturing Engineering and Engineering Manage-
ment, City University of Hong Kong, Hong Kong, China.His current research interest covers iterative learning control of time-delaysystems, neural networks, chaos control and synchronization, and impulsivedynamical systems.
Yonglu Shu received the B.S. degree in mathematicsand the M.S. degree in applied mathematics fromSichuan University, Chengdu, China, in 1985 and1989, respectively, and Ph.D. degree in electrical engi-neering from Chongqing University, Chongqing, China,in 2004. He is currently a Professor with the School ofMathematics and Statistics, Chongqing University,Chongqing, China. He is the author or coauthor ofabout 30 referred international journal. His researchinterests include nonlinear dynamical systems; analy-sis, control and synchronization of chaotic dynamicalsystem; dynamics of linear operators.