Bogdanov–Takens bifurcation in a single inertial neuron model with delay

Neurocomputing 89 (2012) 193–201

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Bogdanov–Takens bifurcation in a single inertial neuron model with delay$ Xing He a, Chuandong Li a,n, Yonglu Shu b a b

College of Computer Science, Chongqing University, Chongqing 400030, PR China College of Mathematics and Statistics, Chongqing University, Chongqing 401331, PR China

a r t i c l e i n f o

abstract

Article history: Received 8 July 2011 Received in revised form 20 November 2011 Accepted 29 February 2012 Communicated by H. Jiang Available online 30 March 2012

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 bifurcation, homoclinic bifurcation, heteroclinic bifurcation and double limit cycle bifurcation. Some numerical simulations are given to support the analytic results. & 2012 Elsevier B.V. All rights reserved.

Keywords: Bogdanov–Takens bifurcation Inertial neuron model Homoclinic bifurcation Heteroclinic bifurcation

1. Introduction Since Hopfield [6] proposed a simplified neural network model, there has been increasing interest in investigating the dynamical behaviors of continuous neural networks with or without delay due to their wide application, such as associative memory, pattern recognition, optimization and signal processing. Some important results have been reported [1,6,13]. For inertial neuron model, the inertia can be considered a useful tool, which is added to help in the generation of chaos and there are some biological background for the inclusion of an inductance term [7,14]. More and more researchers focus on this subject. Babcock and Westervelt [1] combined inertia and drove to explore chaos in one and two neurons system. Wheeler and Schieve [17] discussed the stability and chaos in an inertial twoneural system. Tain et al. [15,16] added inertia to neural equations as a way of chaotically searching for memories in neural networks. Li et al. [10] studied Hopf bifurcation and chaos in a single inertial neuron model with delay. Liu et al. [11] illustrated the stability of bifurcating periodic solutions for a single delayed inertial neuron model under periodic excitation. Liu et al. [12] also discussed the resonant codimension-two bifurcation in an inertial two-neuron system with time delay. However, to the best

$ This research is supported by the National Natural Science Foundation of China Grant No. 60974020, 11171360 and the Fundamental Research Funds for the Central Universities of China (Project No. CDJZR10 18 55 01). n Corresponding author. Tel.: þ86 23 65103199. E-mail address: [email protected] (C. Li).

0925-2312/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.neucom.2012.02.019

of the authors’ knowledge, few results for Bogdanov–Takens bifurcation in a inertial neuron model have been reported in the literature. In this paper, we consider that a single inertial neuron model with time delay [10] is described by _ x€ ¼ axbx þcf ðxhxðttÞÞ,

ð1Þ

where a,b,c 40,h Z 0, t 40 is the time delay, and f is the nonlinear activation function. System (1) was analyzed in [10] from the point of view of Hopf bifurcation and chaos. The authors used h as a bifurcation parameter to show that system (1) undergoes Hopf bifurcation and chaotic behavior of system (1) was observed when adopting a non-monotonic activation function. Except the dynamic of system (1) in [10], it is interesting to further find out what kind of new dynamics this system has. The study carried out in the present paper may contribute to understand the codimension-two Bogdanov–Takens singularity in the single inertial neuron model with time delay. We use a and b as bifurcation parameters. System (1) exhibits codimension-two singularity when two-parameter vary in a neighborhood of the critical values. By using the normal form method for RFDE [3,4], we obtain the normal forms to study its dynamical behaviors. It is shown that different bifurcation diagrams can be constructed due to the difference of activation function. This paper is organized as follows. In the next section, the preliminaries relevant to the normal forms with parameter for RFDE are presented. In Section 3, we discuss the existence of Bogdanov–Takens bifurcation. Then we analyze Bogdanov– Takens singularity in the single inertial neuron model with time delay and get the versal unfolding of B–T bifurcation in Section 4.

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X. He et al. / Neurocomputing 89 (2012) 193–201

In Section 5, these normal forms are used to predict B–T bifurcation diagrams. In Section 6, some numerical simulations are given to support the analytic results. Section 7 summarizes the main conclusions.

2. Preliminaries This section presents B–T bifurcation theory of normal form with parameters for functional differential [3,4,8]. We consider an abstract retarded functional differential equation with parameters in the phase space C ¼ Cð½t,0; Rn Þ _ ¼ LðmÞxt þ Gðxt , mÞ, xðtÞ

ð2Þ

where xt A C is defined by xt ðyÞ ¼ xðt þ yÞ, t r y r 0, the parameter m A Rp is a parameter vector in a neighborhood V of zero, LðmÞ : V-LðC,Rn Þ is C k1 , and G : C  Rp -Rn is C k ðk Z2Þ with Gð0, mÞ ¼ 0, Dx Gð0, mÞ ¼ 0. Define L ¼ Lð0Þ and Fðxt , mÞ ¼ Gðxt , mÞ þ ðLðmÞLð0ÞÞxt , then system (2) can be rewritten as _ ¼ Lxt þ Fðxt , mÞ: xðtÞ

ð3Þ

Then the linear homogeneous retarded functional differential equation of Eq. (3) can be written as _ ¼ Lxt : xðtÞ

ð4Þ

Since L is a bounded linear operator, L can be represented by a Riemann–Stieltjes integral Z 0 dZðyÞjðyÞ, 8j A C, Lj ¼ t

by the Riesz representation theorem, where ZðyÞðy A ½t,0Þ is an n  n matrix function of bounded variation. Let A0 be the infinitesimal generator for the solution semigroup defined by system (4) such that ( ) Z A0 j ¼ j_ , DðA0 Þ ¼

j A C 1 ð½t,0,Rn Þ : j_ ð0Þ ¼

dZðyÞjðyÞ :

Define the bilinear form between C and C 0 ¼ Cð½0, t,Rnn Þ by /c, jS ¼ cð0Þjð0Þ

0

t

Z

y

for y A Q 1 ¼ Q \ C 1  Ker p, where AQ 1 is the restriction of A0 as an operator from Q1 to the Banach space Ker p. Employing Taylor’s theorem, system (5) becomes 8 < z_ ¼ Jz þ Sj Z 2 1j f 1j ðz,y, mÞ, ð6Þ : y_ ¼ A 1 yþ Sj Z 2 1 f 2j ðz,y, mÞ, j Q i

where f j ðz,y, mÞði ¼ 1; 2Þ denotes the homogeneous polynomials of degree j in variables ðz,y, mÞ. For   0 1 J¼ , 0 0 the non-resonance conditions are naturally satisfied. According to normal form theory developed in [5], system (6) can be transformed to the following normal form on the center manifold: z_ ¼ Jz þ 12g 12 ðz,0, mÞ þ h:o:t:

ð7Þ V 4j ðZÞ

the linear space of For a normed space Z, denoted by homogeneous polynomials of ðz, mÞ ¼ ðz1 ,z2 , m1 , m2 Þ with degree j and with coefficients in Z, and define Mj to be the operator in V 4j ðR2  Ker pÞ with the range in the same space by Mj ðp,hÞ ¼ ðM 1j p, M2j hÞ, where M1j p ¼ M 1j

cðxyÞ dZðyÞjðxÞ dx, 8c A C 0 , 8j AC:

p1 p2

!

0 @p

@z1 z2 p2 @p2 @z1 z2 1

¼@

1 A,

0

Assume that L has double zero eigenvalues and all other eigenvalues have negative real parts. Let L be the set of eigenvalues with zero real part and P be the generalized eigenvalues space associated with L and Pn the space adjoint with P. Then C can be decomposed as C ¼ P  Q where Q ¼ fj A C : /j, cS ¼ 0,

8c A Pn g,

M2j h ¼ M 2j hðz, mÞ ¼ Dz hðz, mÞJxAQ 1 hðz, mÞ: Using M1j , we have the following decompositions: V 4j ðR2 Þ ¼ ImðM 1j Þ  ðImðM 1j ÞÞc ,

V 4j ðR2 Þ ¼ KerðM 1j Þ  ðKerðM 1j ÞÞc :

By the above decompositions, g 12 ðz,0, mÞ can be expressed as 1

with dim P ¼ 2: Choose the bases F and C for P and Pn such that /C, FS ¼ I,

pðj þ x0 aÞ ¼ F½ðC, jÞ þ Cð0Þa, which allows us to decompose the enlarged phase space BC ¼ P  Ker p. Let x ¼ Fz þ y. Then system (2) can be decomposed as ( z_ ¼ Jz þ Cð0ÞFðFz þ y, mÞ, ð5Þ y_ ¼ AQ 1 yþ ðIpÞx0 FðFz þ y, mÞ, x A R2 , yA Q 1 ,

0

t

Z

where I is the identity operator on C. The space BC has the norm 9j þ x0 a9 ¼ 9j9C þ 9a9Rn . The definition of the continuous projection p : BC-P by

_ ¼ J C, _ ¼ FJ, C F

where I is the 2  2 identity matrix and   0 1 J¼ : 0 0 Following the ideas in [3,4], we consider the enlarged phase space BC   BC ¼ j : ½t,0-Rn : j is continuous on ½t,0Þ, ( lim jðyÞ A Rn : y-0

Then the elements of BC can be expressed as c ¼ j þx0 a, j A C, a A Rn and ( 0, t r y o 0, x0 ðyÞ ¼ I, y ¼ 0,

g 12 ðz,0, mÞ ¼ ProjectðImðM1 ÞÞc f 2 ðz,0, mÞ: 2

The base of V 42 ðR2  Ker pÞ is composed by the following 20 elements: ! !  !        m1 m2 z1 mi z2 mi z1 z2 z21 m2i z22 , , , , , , , 0 0 0 0 0 0 0 ! ! ! ! ! ! ! 0 0 0 0 0 0 0 , , , , 2 , 2 , 2 , mi z1 z2 m1 m2 z1 mi z2 mi z1 z2 i ¼ 1; 2, and images of these elements under M 12 are ! !       z21 2z1 z2 0 z2 mi z22 , , , , , 2z1 z2 0 0 0 0 ! !  !  z1 z2 z1 mi z2 mi z22 : , , , z22 z2 mi 0 0

X. He et al. / Neurocomputing 89 (2012) 193–201

Therefore, a basis of ðImðM12 ÞÞc can be taken as the set composed by the elements ! ! ! ! ! ! 0 0 0 0 0 0 , , , , m2i , m1 m2 : z21 z1 mi z2 mi z1 z2 !

0

l1 z1 þ l2 z2 þ Z1 z21 þ Z2 z1 z2 þ h:o:t:

:

So the high-order normal must be calculated. System (6) can be transformed to the following normal form on the center manifold: 1 1 z_ ¼ Jz þ 12 g 12 ðz,0, mÞ þ 3! g 3 ðz,0, mÞ þh:o:t

ð8Þ

and define 1

2

Theorem 1. Suppose b ¼ b0 and k 4 0. Then

Proof. It is easy to calculate Fð0Þ ¼ 0 if b ¼ b0 , and

If the second order normal form is degenerated, then we obtain ! 0 1 1 g ðz,0, mÞ ¼ : l1 z1 þ l2 z2 þ h:o:t: 2 2

U 12 ðzÞ ¼ ðM12 Þ1 ProjectðImðM1 ÞÞc f 2 ðz,0; 0Þ,

Next, we focus on the case of t 40. Let a0 ¼ chkt, b0 ¼ ckð1hÞ, and we have the following result.

(i) l ¼ 0 is a single root of Eq. (10) if and only if a aa0 . (ii) l ¼ 0 is a double zero root of Eq. (10) if and only if a ¼ a0 . (iii) Eq. (10) does not have purely imaginary roots 7 oiðo 4 0Þ.

Then we have 1 1 g ðz,0, mÞ ¼ 2 2

195

F 0 ðlÞ ¼ 2l þ achktelt ,

F 00 ðlÞ ¼ 2 þ chkt2 elt :

From k 4 0, we obtain F 0 ðlÞjl ¼ 0,b ¼ b0 ,a ¼ a0 ¼ 0,

F 00 ðlÞjl ¼ 0,k ¼ k0 , t ¼ t0 ¼ 2 þ chkt2 4 0:

This completes the proof of (i) and (ii). When b ¼ b0 , substituting l ¼ io into Eq. (10) yields o2 þaoiþ b0 ck þchk0 etoi ¼ 0 and separating the real and imaginary parts, we have

2

U 22 ðzÞ ¼ ðM 22 Þ1 f 2 ðz,0; 0Þ:

o2 þchk ¼ chk cosðotÞ,

Then

resulting in 1

1

1

g 13 ðz,0, mÞ ¼ Project ðImðM1 ÞÞc ff 3 ðz,0, mÞ þ 32½Dz f 2 ðz,0, mÞU 12 þ Dy f 2 ðz,0, mÞU 22 g: 3

ðImðM 12 ÞÞc .

ðImðM 13 ÞÞc

Be similar with the computation of The space is spanned by ! ! ! ! ! ! 0 0 0 0 0 0 , , , , , , zi m2i z31 z21 z2 z21 mi z1 z2 mi z1 m1 m2 ! ! ! ! 0 0 0 0 , , , m3i m21 m2 m1 m22 : z2 m1 m2

Z1 z31 þ Z2 z21 z2 þ h:o:t :

Hence the normal form with the universal unfolding is ( z_ 1 ¼ z2 þ h:o,t, z_ 2 ¼ l1 z1 þ l2 z2 þ Z1 z31 þ Z2 z21 z2 þh:o:t:

t A ½t,1Þ:

Then system (1) can be written as ( x_ 1 ðtÞ ¼ x2 ðtÞ, x_ 2 ðtÞ ¼ ax2 ðtÞbx1 ðtÞ þcf ðx1 ðtÞÞchf ðx1 ðttÞÞ:

lt

0

¼ 0,

2

where 2

G22 ¼

3

00

000

ctðx21 hx1 ðt1ÞÞf ð0Þ ctðx31 hx1 ðt1ÞÞf ð0Þ þ : 2 6

ZðyÞ ¼ AdðyÞ þBdðy þ 1Þ, ð9Þ

where A¼

!

0

t

ðb0 þckÞt

a0 t

and define Z 0 dZðyÞjðyÞ, L0 j ¼

,

B¼



0

0

chtk

0

8j A C:

1

and the corresponding characteristic equation is 2

&

From Section 2, let

It is clear that system (9) has one equilibrium ð0; 0Þ. Then the linearization equation at the origin is ( x_ 1 ðtÞ ¼ x2 ðtÞ, x_ 2 ðtÞ ¼ ax2 ðtÞbx1 ðtÞ þckx1 ðtÞchkx1 ðttÞ

FðlÞ ¼ l þ al þ bck þ chke

which is meaningless. This proves claim (iii).

1

Throughout the rest of this paper, we assume that f ð0Þ ¼ 0, and f is a nonlinear C3 function. By defining [10] x2 ¼ x_ 1 ðtÞ,

4 1,

Let a ¼ a0 þ m1 , b ¼ b0 þ m2 and expand the function f. Then system (11) becomes 8_ x ðtÞ ¼ tx2 ðtÞ, > < 1 x_ 2 ðtÞ ¼ ða0 þ m1 Þtx2 ðtÞðb0 þ m2 Þtx1 ðtÞ þ ctkx1 ðtÞ, ð12Þ > : chtkx ðt1Þ þ G2 þh:o:t:,

3. The existence of Bogdanov–Takens bifurcation

x1 ðtÞ  xðtÞhxðttÞ,

chk

In this section, we investigate B–T bifurcation by using the method in Section 2. From Theorem 1, we know that, at the origin, the characteristic equation of system (9) has a double zero root if a0 ¼ chkt, b0 ¼ ckð1hÞ and k 4 0. So we treat (a,b) as bifurcation parameters near ða0 ,b0 Þ. Rescaling the time by t-t=t to normalize the delay, system (9) can be written as ( x_ 1 ðtÞ ¼ tx2 ðtÞ, ð11Þ x_ 2 ðtÞ ¼ atx2 ðtÞbtx1 ðtÞ þ ctf ðx1 ðtÞÞchtf ðx1 ðt1ÞÞ:

!

0

o2

4. Bogdanov–Takens bifurcation

Then we obtain 1 1 g ðz,0, mÞ ¼ 3! 3

cos ðotÞ ¼ 1 þ

ð10Þ

where k ¼ f ð0Þ, for t ¼ 0, the two roots of Eq. (10) have negative real parts if and only if bck þchk 4 0:

The infinitesimal generator ( j_ , 1 r y o 0, A0 j ¼ R 0 y ¼ 0: 1 dZðyÞjðyÞ,



196

X. He et al. / Neurocomputing 89 (2012) 193–201

Rewrite system (12) as

x2 ð0Þ ¼

x_t ¼ LðmÞxt þGðxt , mÞ þ h:o:t ¼ ðL0 þ L1 ðmÞÞxt þ Gðxt , mÞ þ h:o:t,

t

z2 þ y2 ð0Þ:

From Section 2, system (12) can be decomposed as

where !

tx2 ð0Þ , a0 tx2 ð0Þb0 tx1 ð0Þ þ ctkðx1 ð0Þhx1 ð1ÞÞ

L0 xt ¼

1

Gðxt , mÞ ¼

L1 ðmÞxt ¼

0

!

z_ ¼ Jz þ Cð0ÞFðFz þ y, mÞ þ h:o:t,

0

00

where Fðxt , mÞ ¼ L1 ðmÞxt þ Gðxt , mÞ, and the bilinear form on C n  C is Z 0 /c, jS ¼ cð0Þjð0Þ þ cðx þ 1ÞBjðxÞ dx, 1

where FðyÞ ¼ ðj1 ðyÞ, j2 ðyÞÞ A C, ! c1 ðsÞ A Cn CðsÞ ¼ c2 ðsÞ

00

ctf ð0Þ 2 cthf ð0Þ z1  ðz1 z2 Þ2 2 2 000 000 ctf ð0Þ 3 chtf ð0Þ z1  ðz1 z2 Þ3 : þ 6 6

Following the computation of the normal form for functional differential equations introduced by Section 2, we get the normal form with versal unfolding on the center manifold ( z_1 ¼ z2 , ð15Þ z_2 ¼ l1 z1 þ l2 z2 þ Z1 z21 þ Z2 z1 z2 þ h:o:t, where

are, respectively, the bases for the center space P and its dual space Pn. Next we will find the FðyÞ and CðsÞ based on the techniques developed by [18].

l1 ¼ n1 tm2 , l2 ¼ n1 m1 m2 tm2 ,

Lemma 1 (see Xu and Huang [18]). The bases of P and its dual space Pn have the following representations

If the following condition is satisfied   @ðl1 , l2 Þ  a 0,  @ðm1 , m2 Þ m ¼ 0

FðyÞ ¼ ðj1 ðyÞ, j2 ðyÞÞ, 1r y r0,

where j1 ðyÞ ¼ j01 A Rn \f0g, j2 ðyÞ ¼ j02 þ j01 y, j02 A Rn , and c2 ðsÞ ¼ c02 A Rnn \f0g, c1 ðsÞ ¼ c01 sc02 , c01 A Rnn , which satisfy

0

ð6Þ

c01

0

j

0 1 2 2

c01 B

j

0

ð2Þ ðA þBÞj02 ¼ ðB þ IÞj01 , 0

ð3Þ c2 ðA þBÞ ¼ 0,

0

0

ð5Þ c2 j02 12c2 Bj01 þ c2 Bj02 ¼ 1,

0 0 1 þ c1 B

j

0 1 2 þ6

c02 B

j

0 1 0 1 2c2 B

So it is not difficult to verify that ! 1 y m1 n1 as FðyÞ ¼ , CðsÞ ¼ 1 0 t n1 a

0 2

j ¼ 0:

m2 n1 s n1

00

Z1 ¼

n1 ctð1hÞf ð0Þ , 2

Z2 ¼ m2 ctð1hÞf 00 ð0Þ þ n1 chtf 00 ð0Þ:

ð16Þ

the map ðm1 , m2 Þ/ðl1 , l2 Þ is regular and

CðsÞ ¼ colðc1 ðsÞ, c2 ðsÞÞ, 0 rs r 1,

ð4Þ c1 ðA þBÞ ¼ c2 ðB þIÞ,

ð14Þ

F 22 ¼ m1 z2 m2 tz1 þ

x_t ¼ L0 xt þ Fðxt , mÞ þ h:o:t,

ð1Þ ðA þ BÞj01 ¼ 0,

ð13Þ

where F 12 ¼ 0,

:

Then system (12) can be transformed into

P n ¼ spanC,

yA Q 1 :

z_2 ¼ naF 12 þ nF 22 þ h:o:t,

!

m2 tx1 ð0Þm1 tx2 ð0Þ

P ¼ spanF,

z A R2 ,

y_ ¼ AQ 1 yþ ðIpÞx0 FðFz þ y, mÞ þh:o:t,

On the center manifold, system (13) can be written as ( z_1 ¼ z2 þm1 F 12 þm2 F 22 þ h:o:t,

,

G22

(

! ,

_ ¼ JC, /C, FS ¼ I, where _ ¼ FJ, C such that F   0 1 J¼ , 0 0

n1 ctð1hÞ 40,

m2 ctð1hÞ 4 0,

n1 cht 4 0:

Thus we can get the following result: 00

Theorem 2. Under condition (16), if f ð0Þ a 0, then, on the center manifold, system (12) is equivalent to the normal form (15), where Z1  Z2 4 0. 00

If f ð0Þ ¼ 0, then Z1 ¼ 0, Z2 ¼ 0, hence the system (15) is degenerate, we need to calculate the high order normal form. Following the computation of the normal form for functional differential equations introduced by Section 2, we can get high-order normal form: ( z_1 ¼ z2 , ð17Þ z_2 ¼ l1 z1 þ l2 z2 þ Z3 z31 þ Z4 z21 z2 þ h:o:t, 000

where Z3 ¼ n1 ctð1hÞf ð0Þ=6, 000

Z4 ¼

000

m2 ctð1hÞf ð0Þ n1 chtf ð0Þ þ : 2 2

So we obtain the following result

and

00

2a0 ð1hÞ , n1 ¼ 2 a0 ð1hÞ þ b0 h n1 a0 t : m2 ¼ 3ta0 þ 6

n1 a20 t 2b0 h þ m1 ¼ , 3ta0 þ 6 a20 ð1hÞ þ b0 h

000

Theorem 3. Under condition (16), if f ð0Þ ¼ 0, f ð0Þ a0, then, on the center manifold, system (12) is equivalent to the normal form (17), where Z3  Z4 4 0. 5. Bifurcation diagrams

Let x ¼ Fz þ y, namely x1 ðyÞ ¼ z1 þ yz2 þy1 ðyÞ,

x2 ðyÞ ¼

1

t

z2 þy2 ðyÞ:

Then x1 ð0Þ ¼ z1 þy1 ð0Þ,

x1 ð1Þ ¼ z1 z2 þ y1 ð1Þ,

In this section, we discuss the bifurcation diagrams of system (9), first we consider the truncated normal form of system (15): ( z_1 ¼ z2, ð18Þ z_2 ¼ l1 z1 þ l2 z2 þ Z1 z21 þ Z2 z1 z2 :

X. He et al. / Neurocomputing 89 (2012) 193–201

197

By introducing the change of variables and rescaling of time   Z2 Z l1 Z z1 ¼ 12 x1  , z2 ¼ 13 x2 , t ¼ 2 t,

Z2

Z2

Z1

Z2

system (18) becomes (still using z1 , z2 for simplicity) ( z_1 ¼ z2, z_2 ¼ u1 þ u2 z2 þ z21 þz1 z2 ,

ð19Þ

where u1 ¼

Z22 Z21

   Z  2 k1 þk2 m2 þ k3 m1 ,

Z1

u2 ¼

Z2 Z1







Z2 k 2k2 m2 2k3 m1 , Z1 1

k1 ¼ n1 t, k2 ¼ m2 t, k3 ¼ n1 : The complete bifurcation diagrams of system (19) can be found in [2,5,9]. Here we just briefly list some results. (i) System (19) undergoes a saddle-node bifurcation on the curve   u2 S ¼ ðu1 ,u2 Þ : u1 ¼ 2 , 4 (ii) system (19) undergoes an unstable Hopf bifurcation on the curve

Fig. 1. Bifurcation curve of Theorem 4.

H ¼ fðu1 ,u2 Þ : u1 ¼ 0, u2 o 0g, (iii) system (19) undergoes a saddle homoclinic bifurcation on the curve   6 2 u2 , u2 o 0 : T ¼ ðu1 ,u2 Þ : u1 ¼  25 Applying the above results and using the expressions of u1 , u2 , we obtain the following result. 00

Theorem 4. Under condition (16), if f ð0Þ a 0, for sufficiently small m1 , m2 , (i) system (9) undergoes a saddle-node bifurcation on the curve: ( ðm1 , m2 Þ :

S¼





Z2 k 2k2 m1 2k3 m2 Z1 1

2

¼4







Z2 k þk m þk m Z1 1 2 1 3 2

)

bring system (20) into (still using z1 , z2 for simplicity) ( z_1 ¼ z2 , z_2 ¼ v1 z1 þv2 z2 þ sz31 z21 z2 ,

ð21Þ

where s ¼ sgnðZ3 Þ, v1 ¼ ðZ4 =Z3 Þ2 l1 ¼ ðZ4 =Z3 Þ2 k1 m2 , v2 ¼ ðZ4 = 9Z3 9Þ l2 ¼ ðZ4 =9Z3 9Þðk2 m2 þ k3 m1 Þ: The complete bifurcation diagrams of system (21) can be found in [2,5,9]. Here we just briefly list some results: when s¼1, (i) system (21) undergoes a pitchfork bifurcation on the curve: S ¼ fðv1 ,v2 Þ : v1 ¼ 0, v2 A Rg,

,

(ii) system (21) undergoes a Hopf bifurcation at the trivial equilibrium on the curve:

(ii) system (9) undergoes an unstable Hopf bifurcation at the nontrivial equilibrium on the curve:

H ¼ fðv1 ,v2 Þ : v2 ¼ 0, v1 o0g,

      Z Z H ¼ ðm1 , m2 Þ :  2 k1 þ k2 m1 þ k3 m2 ¼ 0, 2 k1 2k2 m1 o 2k3 m2 ,

(iii) system (21) undergoes a heteroclinic bifurcation on the curve:

Z1

Z1

(iii) system (9) undergoes a saddle homoclinic bifurcation on the curve: ( ðm1 , m2 Þ :

T¼









Z2 6 k þk m þk m ¼  Z1 1 2 1 3 2 25









2 Z2 k 2k2 m1 2k3 m2 , Z1 1



Z2 k 2k2 m1 o2k3 m2 : Z1 1

T ¼ fðv1 ,v2 Þ : v2 ¼ 15v1 þOðv21 Þ, v1 o 0g: Applying the above results and using the expressions of v1 , v2 , we obtain the following results. 00

000

Theorem 5. Under condition (16), if f ð0Þ ¼ 0 and f ð0Þ 4 0, for sufficiently small m1 , m2 , (i) system (9) undergoes a pitchfork bifurcation on the curve: S ¼ fðm1 , m2 Þ : m2 ¼ 0, m1 A Rg,

The bifurcation curve of Theorem 4 is in Fig. 1. Next, we consider the truncated normal form of system (17) ( z_1 ¼ z2 , ð20Þ z_2 ¼ l1 z1 þ l2 z2 þ Z3 z31 þ Z4 z21 z2 :

(ii) system (9) undergoes a stable Hopf bifurcation at the trivial equilibrium on the curve:   k3 H ¼ ðm1 , m2 Þ : m2 ¼  m1 , m2 4 0 , k2

In order to put system (20) in an appropriate form, we make the transformation

(iii) system (9) undergoes a heteroclinic bifurcation on the curve:

Z

4 z1 , x2 ¼  x1 ¼ qffiffiffiffiffiffiffiffiffi

9Z3 9

9Z 9 qffiffiffiffiffiffiffiffiffi z2 , t ¼  3 t, Z4 9Z3 9 9Z3 9

Z24

 T ¼ ðm1 , m2 Þ : m2 ¼

 5Z3 k3 m1 þ Oðm22 Þ, m2 4 0 : Z4 k1 5Z3 k2

198

X. He et al. / Neurocomputing 89 (2012) 193–201

Fig. 3. Bifurcation curve of Theorem 6.

7 Fig. 2. Bifurcation curve of Theorem 5.

6

The bifurcation curve of Theorem 5 is in Fig. 2. For system (21), when s ¼ 1,

5

x1 x2

4 x

(i) system (21) undergoes a pitchfork bifurcation on the curve S ¼ fðv1 ,v2 Þ : v1 ¼ 0, v2 A Rg, (ii) system (21) undergoes an unstable Hopf bifurcation at the non-trivial equilibrium on the curve H ¼ fðv1 ,v2 Þ : v2 ¼ v1 , v1 40g,

3 2 1 0

(iii) system (21) undergoes homoclinic bifurcation on the curve T ¼ fðv1 ,v2 Þ : v2 ¼ 45v1 þ Oðv21 Þ, v1 40g,

Hd ¼ fðv1 ,v2 Þ : v2 ¼ dv1 þOðv21 Þ, v1 4 0, d 0:752g: Applying the above results and using the expressions of v1 , v2 , we obtain the following results. 000

Theorem 6. Under condition (16), if f ð0Þ ¼ 0 and f ð0Þ o 0, for sufficiently small m1 , m2 , (i) system (9) undergoes a pitchfork bifurcation on the curve: S ¼ fðm1 , m2 Þ : m2 ¼ 0, m1 A Rg, (ii) system (9) undergoes an unstable Hopf bifurcation at the nontrivial equilibrium on the curve:   k1 Z4 k2 Z3 H ¼ ðm1 , m2 Þ : m1 ¼ m2 , m2 o0 , k3 Z3

0

50

100

150

200

250

t

(iv) system (21) undergoes a double cycle bifurcation on the curve

00

−1

Fig. 4. Waveform plot of the variable x of system (9) for m1 ¼ 0:213, m2 ¼ 0:05.

(iii) system (9) undergoes homoclinic bifurcation on the curve:   4k1 Z4 5k2 Z3 T ¼ ðm1 , m2 Þ : m1 ¼ m2 þ Oðm22 Þ, m2 o 0 , 5k3 Z3 (iv) system (9) undergoes a double cycle bifurcation on the curve:   dk1 Z4 k2 Z3 Hd ¼ ðm1 , m2 Þ : m1 ¼ m2 þ Oðm22 Þ, m2 o 0, d 0:752 : k3 Z3

The bifurcation curve of Theorem 6 is in Fig. 3.

6. Numerical simulations In this section, we give some examples to verify the theoretical results. For system (9), we fix h¼0.5, c ¼2, t ¼ 1. Then we calculate n1 ¼ 1, m2 ¼ 19, k1 ¼ 1, k2 ¼ 19, k3 ¼ 1, Z2 =Z1 ¼ 20 9, Z4 =Z3 ¼ 10 3.

X. He et al. / Neurocomputing 89 (2012) 193–201

0.6

199

Example 1. This example supports the result of Theorem 4. For system (9), we fix f ðsÞ ¼ tanhðsÞ þ 0:01s2 , and we get a0 ¼ 1, b0 ¼ 1. By the condition (i) of Theorem 4, system (9) undergoes a saddlenode bifurcation on the curve S ¼ fðm1 , m2 Þ : ðm1 m2 Þ2 ¼ 19 9 m1 m2 g, and there should exist a non-trivial node and the origin is a saddle point. If we set m1 ¼ 0:213, m2 ¼ 0:05, Figs. 4 and 5 verify this result. By the condition (ii) of Theorem 4, system (9) undergoes an unstable Hopf bifurcation on the curve H ¼ fðm1 , m2 Þ : m1 ¼ 2:1111m2 , m2 4 m1 g. We set m1 ¼ 0:02111, m2 ¼ 0:01. and the system (9) has two equilibrium points E1 ¼ ð0; 0Þ and E2 ¼ ð0:09,0Þ, and an unstable limit cycle exists through the Hopf bifurcation near E2. Figs. 6 and 7 verify this result.

0.5 0.4

x2

0.3 0.2 0.1 0 −0.1 −1

0

1

2

3 x1

4

5

6

7

Example 2. This example demonstrates the result of Theorem 5. For system (9), we set f ðsÞ ¼ tanhðsÞ þ 23s3 . By the condition (i) of Theorem 5, system (9) undergoes a pitchfork bifurcation on the curve S ¼ fðm1 , m2 Þ : m1 ¼ 0, m2 A Rg. If we set m1 ¼ 0:01, m2 ¼ 0:001, there should exist two stable non-trivial equilibria and the origin is unstable. Fig. 8 shows the good agreement with the

Fig. 5. Phase portraits of system (9) for m1 ¼ 0:213, m2 ¼ 0:05.

3

0.5

2.5

0.4

x1 x2

2

0.3

1.5

0.2

1 x

0.1 x

0.5

0

0

−0.1

−0.5

−0.2

−1 −1.5

x1 x2

−0.3 0

500

1000

1500

2000

2500

−0.4

t

0

1000

2000

3000 t

4000

5000

6000

Fig. 6. Waveform plot of the variable x of system (9) for m1 ¼ 0:02111, m2 ¼ 0:01. Fig. 8. Waveform plot of the variable x of system (9) for m1 ¼ 0:01, m2 ¼ 0:001.

0.45

0.4

0.4 0.35

0.2

0.3

0.1

0.2

x

x2

0.25

0

0.15 0.1

−0.1

0.05

−0.2

0 −0.05 −1.5

x1 x2

0.3

−1

−0.5

0

0.5

x1

1

1.5

2

2.5

Fig. 7. Phase portraits of system (9) for m1 ¼ 0:02111, m2 ¼ 0:01.

3

−0.3

0

1000

2000

3000 t

4000

5000

6000

Fig. 9. Waveform plot of the variable x of system (9) for m1 ¼ 0:0001, m2 ¼ 0:0009.

200

X. He et al. / Neurocomputing 89 (2012) 193–201

0.03

theoretical analysis. By the condition (ii) of Theorem 5, system (9) undergoes a stable Hopf bifurcation at the origin on the curve H ¼ fðm1 , m2 Þ : m2 ¼ 9m1 , m1 o0g, If we set m1 ¼ 0:0001 and m2 ¼ 0:0009, a stable limit cycle exists through the Hopf bifurcation near the origin. Figs. 9 and 10 verify this result.

0.02

x2

0.01

Example 3. This example supports the result of Theorem 6. For system (9), we fix f ðsÞ ¼ tanhðsÞ. From Theorem 6, system (9) undergoes homoclinic bifurcation on the curve H ¼ fðm1 , m2 Þ : m1 ¼ 2:5556m2 þ Oðm22 Þ, m2 o 0g, If we set m1 ¼ 0:025556 and m2 ¼ 0:01, a closed orbit exists through the homoclinic bifurcation and three equilibria of system (9) are in the closed orbit, and two homoclinic orbits emerge from the origin in Fig. 11. System (9) also undergoes a double limit cycle bifurcation on the curve Hd ¼ fðm1 , m2 Þ : m1 ¼ 2:3956m2 þ Oðm22 Þ, m2 o 0g. Fix m1 ¼ 0:00043 and m2 ¼ 0:00018. Fig. 12 verifies this result.

0 −0.01 −0.02 −0.03 −0.3

−0.2

−0.1

0

x1

0.1

0.2

0.3

0.4

Fig. 10. Phase portraits of system (9) for m1 ¼ 0:0001, m2 ¼ 0:0009.

7. Conclusion A single delayed neuron model with inertial term has been studied in the neighborhood of Bogdanov–Takens codimensiontwo bifurcation point where the linear part of the system has double zero eigenvalues. By applying normal form theory and center manifold reduction, we are able to predict their corresponding bifurcation diagrams such as saddle-node bifurcation, pitchfork bifurcation, homoclinic bifurcation, heteroclinic bifurcation and double limit cycle bifurcation.

0.02 0.015 0.01

x2

0.005 0

References

−0.005 −0.01 −0.015 −0.02 −0.4

−0.3

−0.2

−0.1

0

0.1

0.2

x1 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.

0.3

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X. He et al. / Neurocomputing 89 (2012) 193–201 Xing He received the B.Sc. degree in mathematics and applied mathematics from the Department of Mathematics, Guizhou University, Guiyang, China, in 2009. Currently, he is working towards the Ph.D degree with the college of computer science, Chongqing University, Chongqing, China. His research interests include neural network, bifurcation theory, and nonlinear dynamical system.

Chuandong Li received the B.S. degree in Applied Mathematics from Sichuan University, Chengdu, China in 1992, and the M.S. degree in operational research and control theory and Ph.D. degree in Computer Software and Theory from Chongqing University, Chongqing, China, in 2001 and in 2005, respectively. He has been a Professor at the College of Computer Science, Chongqing University, Chongqing 400030, China, since 2007, and been the IEEE Senior member since 2010. From November 2006 to November 2008, he serves as a research fellow in the Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Hong Kong, China. His current research interest covers iterative learning control of time-delay systems, neural networks, chaos control and synchronization, and impulsive dynamical systems.

201 Yonglu Shu received the B.S. degree in mathematics and the M.S. degree in applied mathematics from Sichuan University, Chengdu, China, in 1985 and 1989, respectively, and Ph.D. degree in electrical engineering from Chongqing University, Chongqing, China, in 2004. He is currently a Professor with the School of Mathematics and Statistics, Chongqing University, Chongqing, China. He is the author or coauthor of about 30 referred international journal. His research interests include nonlinear dynamical systems; analysis, control and synchronization of chaotic dynamical system; dynamics of linear operators.