global existence of periodic solutions in a special neural network model with two delays

Chaos, Solitons and Fractals 39 (2009) 2249–2257

www.elsevier.com/locate/chaos

Global existence of periodic solutions in a specialneural network model with two delays

Ying Dong a,*, Chengjun Sun b

a Department of Mathematics, Shandong University at Weihai, Weihai 264209, PR Chinab Department of Biology, McGill University, 1205, rue Docteur Penfield, Montreal, QC, Canada H3A 1B1

Accepted 26 June 2007

Abstract

A simple neural network model with two delays is considered. By analyzing the associated characteristic transcen-dental equation, it is found that Hopf bifurcation occurs when the sum of two delays passes through a sequence of crit-ical values. Using a global Hopf bifurcation theorem for FDE due to Wu [Wu J. Symmetric functional differentialequations and neural networks with memory. Trans Amer Math Soc 1998;350:4799–838], a group of sufficient condi-tions for this model to have multiple periodic solutions are obtained when the sum of delays is sufficiently large. Numer-ical simulations are presented to support the obtained theoretical results.� 2007 Elsevier Ltd. All rights reserved.

1. Introduction

Since Hopfield [1] constructed a simplified neural network model, the dynamic behavior of neural network [2–5] hasattracted much attention of many researchers. Recently, there has been increasing interest in investigating the dynamicsof neural networks. Some stability properties have been obtained for delayed neural network with two neurons (see, forexample, [5–8]). However, most work only focuses on investigating the stability. It is well known that studies on neuralnetworks not only involve a discussion of stability properties, but also involve many other dynamic behaviors such asperiodic phenomenon, bifurcation and chaos. In many applications, the properties of periodic solutions are of greatinterest, especially the global multiple periodic solutions. Based on the Hopfield neural network model, Marcus andWestervelt [9] argued that the nonlinear sigmoidal activation functions which connected to the other neurons wouldinclude discrete delays and proposed the following differential equations with delays:

0960-0doi:10.

* CoE-m

Ci _xiðt0Þ ¼ �1

Rixiðt0Þ þ

Xn

j¼1

T ijfijðxjðt0 � s0jÞÞ; i ¼ 1; 2; . . . ; n: ð1:1Þ

The variable xiðt0Þ represents the voltage on the input of the ith neuron. Each neuron is characterized by an input ofcapacitance Ci, a delay s0i, and a transfer function fij. The nonlinear transfer function fijðxÞ is sigmoidal. Assume that

779/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.1016/j.chaos.2007.06.106

rresponding author.ail address: [email protected] (Y. Dong).

mailto:[email protected]

2250 Y. Dong, C. Sun / Chaos, Solitons and Fractals 39 (2009) 2249–2257

the neurons have the same capacitance, resistance, and transfer function, that is, Ci ¼ C and Ri ¼ R. Then system (1.1)becomes

_xiðtÞ ¼ �xiðtÞ þXn

j¼1

aijfijðxjðt � sjÞÞ; i ¼ 1; 2; . . . ; n; ð1:2Þ

after recalling time, delay and T ij : t ¼ t0=RC, sj ¼ s0j=RC, aij ¼ RT ij. In the case of a single delay when sj � s, Marcusand Westervelt [9] studied the linear stability of system (1.2) when ðaijÞn�n is a symmetric matrix and discussed stabilityfor three specific network topologies: symmetric rings of neurons, symmetric random networks, and associated memorynetworks. They found out that the delay could destabilize the network as a whole and create oscillatory behavior.

Recently, Olien and Belair [10] have investigated system (1.2) with two delays for n ¼ 2, that is,

_uiðtÞ ¼ �uiðtÞ þX2

j¼1

aijfijðujðt � sjÞÞ; i ¼ 1; 2: ð1:3Þ

They discussed several cases, such as s1 ¼ s2 and a11 ¼ a22 ¼ 0 and obtained some sufficient conditions for the stabilityof the equilibrium of (1.3), which may undergo some bifurcations at certain values of parameters. A similar model rep-resenting a single pair of neurons with self-connection has been considered by Destexhe and Gaspard [11]. We refer toMagee and Roy [12], Ruan and Wei [13] and the references therein for related work on two-neuron networks with de-lays. However, the results found mainly focus on the stability of the equilibrium, the existence of Hopf bifurcation andthe properties of local periodic solutions.

When there is no self-connection in the network, that is, a11 ¼ a22 ¼ 0, system (1.3) becomes

_u1ðtÞ ¼ �u1ðtÞ þ a12f12ðu2ðt � s2ÞÞ;_u2ðtÞ ¼ �u2ðtÞ þ a21f21ðu1ðt � s1ÞÞ:

ð1:4Þ

The architecture of this special case of system (1.3) is illustrated in Fig. 1.The purpose of this paper is to investigate the global existence of multiple periodic solutions for (1.4). Recently, a

great deal of research has been devoted to this topics. One method used is the ejective fixed point argument developedby [14]. Our method for showing the existence of nontrivial periodic solutions is the S1-equivariant degree (see [15,16]).More precisely, we shall use a global Hopf bifurcation result due to Wu [15] for functional differential equations, whichwas established using purely topological argument. Meanwhile, the Bendixon’s criterion for ordinary differential equa-tions will be used to rule out the existence of nontrivial periodic solutions for zero delays. To the best of our knowledge,this paper is the first time to deal with the global existence of nontrivial periodic solutions of (1.4). Our paper is orga-nized as follows: In Section 2, the general mathematical framework: a global Hopf bifurcation theory of Wu [15], isoutlined. In Section 3, the local Hopf bifurcation of system (1.4), and the global stability of system (1.4) withs ¼ s1 þ s2 ¼ 0, are investigated. Section 4 deals with the global existence of multiple periodic solutions of system(1.4). An example is analyzed and numerical simulations are presented in Section 5. This paper ends with a conclusion.

2. General framework

In this section, we briefly describe a general mathematical framework developed in [15] for proving the global exis-tence of multiple periodic solution. This general framework will be used in Section 4 to prove Theorem 4.1.

Let Y denote the Banach space of bounded continuous mappings u : R! Rn equipped with the supremum norm.For u 2 Y and t 2 R, define ut 2 Y by utðsÞ ¼ uðt þ sÞ for s 2 R.

Consider the following functional differential equation

_uðtÞ ¼ F ðut; a; T Þ; ð2:1Þ

which parameterized by two real numbers ða; T Þ 2 R� Rþ, where Rþ ¼ ð0;1Þ and F : Y � R� Rþ ! Rn is completelycontinuous. Restrict F to the subspace of constant functions u, which is identified with Rn, and then obtain a mappingbF ¼ F jRn�R�Rþ

: Rn � R� Rþ ! Rn. Assume

Fig. 1. The architecture of model (1.4).

Y. Dong, C. Sun / Chaos, Solitons and Fractals 39 (2009) 2249–2257 2251

(A1) bF is C2.Denote by u0 2 Y be the constant mapping with the value u0 2 Rn. The point ðu0; a0; T 0Þ is called a stationary solu-

tion of (2.1) if bF ðu0; a0; T 0Þ ¼ 0. Assume(A2) Du

bF ðu; a; T Þjðu0 ;a0 ;T 0Þ is an isomorphism of Rn at each stationary solution ðu0; a0; T 0Þ:Under (A1) and (A2) and by the implicit function theorem, for each stationary solution ðu0; a0; T 0Þ, there exist �0 > 0

and a C1 mapping v : B�0ða0; T 0Þ ! Rn such that bF ðvða; T Þ; a; T Þ ¼ 0, for ða; T Þ 2 B�0ða0; T 0Þ ¼ ða0 � �0; a0 þ �0Þ�ðT 0 � �0; T 0 þ �0Þ. The zeros of det Dðu0 ;a0 ;T 0ÞðkÞ ¼ 0 are the characteristic values of the stationary solution ðu0; a0; T 0Þ.Assume

(A3) F ðu; a; T Þ is differentiable with respect to u. The n� n complex matrix function Dðvða;T Þ;a;T ÞðkÞ is continuous inða; T ; kÞ 2 B�0ða0; T 0Þ � C. Define the characteristic matrix at a stationary solution ðu0; a0; T 0Þ of (2.1) as

Dðu0 ;a0 ;T 0ÞðkÞ ¼ kId� DuF ðu0; a0; T 0ÞðekIdÞ;

where DuF ðu0; a0; T 0Þ is complexification of the derivative of F ðu; a; T Þ with respect to u, evaluated at ðu0; a0; T 0Þ.Note that (A2) is equivalent to assuming that k ¼ 0 is not a characteristic value of any stationary solution of (2.1).

Definition 2.1. [15] A stationary solution ðu0; a0; T 0Þ is called a center, if it has purely imaginary characteristic values ofthe form im2p=T 0 for some positive integer m. A center ðu0; a0; T 0Þ is said to be isolated if (i) it is the only center in someneighborhood of ðu0; a0; T 0Þ and (ii) it has only finitely many purely imaginary characteristic values of the form im2p=T 0

for integer m.Let Jðu0; a0; T 0Þ be the set of all such positive integers m at an isolated center ðu0; a0; T 0Þ. Assume that there exists

m 2 Jðu0; a0; T 0Þ such that(A4) There exist �; d 2 ð0; �0Þ such that on ½a0 � d; a0 þ d� � oX�;T 0

, det Dðvða;T Þ;a;T Þðwþ im2p=T Þ ¼ 0 if and only ifa ¼ a0;w ¼ 0; T ¼ T 0, where

X�;T 0¼ fðw; T Þ : 0 < w < �; jT � T 0j < �g:

Define

H�mðu0; a0; T 0Þðw; T Þ ¼ det Dðvða0�d;T Þ;a0�d;T Þðwþ im2p=T Þ: ð2:2Þ

Then (A4) implies that H�mðu0; a0; T 0Þ–0 on oX�;T 0. Consequently, the mth crossing number cmðu0; a0; T 0Þ of ðu0; a0; T 0Þ

can be defined using topological degree of mappings H�m , as

cmðu0; a0; T 0Þ ¼ degBðH�mðu0; a0; T 0Þ;X�;T 0Þ � degBðHþmðu0; a0; T 0Þ;X�;T 0

Þ: ð2:3Þ

It is shown in Wu [15, Theworem 3.2] that cmðu0; a0; T 0Þ–0 implies the existence of a local bifurcation of periodic solu-tions with periods near T 0=m. To extend globally the local bifurcation, assume

(A5) All centers of (2.1) are isolated and (A4) holds for each center ðu0; a0; T 0Þ and each m 2 Jðu0; a0; T 0Þ.(A6) For each bounded set W # Y � R� Rþ there exists a constant L > 0 such that

jF ðu; a; T Þ � F ðw; a; T Þj 6 L sups2R

juðsÞ � wðsÞj for ðu; a; T Þ; ðw; a; T Þ 2 W :

The following is a global Hopf bifurcation result in Wu [15, Theorem 3.3].

Proposition 2.1. Assume that ðA1Þ–ðA6Þ hold. Let

RðF Þ ¼ Clfðu; a; T Þ : u is a T – periodic solution of ð2:1Þg � Y � R� Rþ;

NðF Þ ¼ fðu; a; T Þ : F ðu; a; T Þ ¼ 0g:

Denote by Cðu0; a0; T 0Þ the connected component in RðF Þ with an isolated center ðu0; a0; T 0Þ. Then either

(i) Cðu0; a0; T 0Þ is unbounded, or

(ii) Cðu0; a0; T 0Þ is bounded, Cðu0; a0; T 0ÞT

NðF Þ is finite, and
Xðu;a;T Þ2Cðu0 ;a0 ;T 0Þ\NðF Þ
cmðu; a; T Þ ¼ 0: ð2:4Þ

for all m ¼ 1; 2; . . . ; where cmðu; a; T Þ is the mth crossing number of ðu; a; T Þ if m 2 Jðu; a; T Þ, or it is zero if otherwise.

By Proposition 2.1, to show Cðu0; a0; T 0Þ is unbounded, one can show that the sum in (2.4) is not equal to zero, for aparticular integer m. This will be done for system Eq. (1.4) in Section 4.


3. Local Hopf bifurcation and global stability

In this section, we present some important results, which will be used in the subsequent section to establishglobal existence of nontrivial periodic solutions. Let �u1ðtÞ ¼ u1ðt � s1Þ, �u2ðtÞ ¼ u2ðtÞ, f1 ¼ f12, f2 ¼ f21 ands ¼ s1 þ s2. Dropping the bars for simplification of notations, we can then rewrite (1.4) as the following equivalentsystem

_u1ðtÞ ¼ �u1ðtÞ þ a12f1ðu2ðt � sÞÞ;_u2ðtÞ ¼ �u2ðtÞ þ a21f2ðu1ðtÞÞ:

ð3:1Þ

We make the following assumptions for (3.1).(H1) fi 2 C2, fið0Þ ¼ 0 ði ¼ 1; 2Þ, and there exists L > 0 such that jfiðuÞj 6 L for all u 2 R. The origin (0,0) is the

unique equilibrium of (3.1).The linearization of (3.1) at (0,0) takes the form

_u1ðtÞ ¼ �u1ðtÞ þ a12u2ðt � sÞ;_u2ðtÞ ¼ �u2ðtÞ þ a21u1ðtÞ:

ð3:2Þ

where a12 ¼ a12f 01ð0Þ, a21 ¼ a21f 02ð0Þ, and f 0i ðujÞ denotes the derivative of fi with respect to uj ðj ¼ 1; 2Þ. The associatedcharacteristic equation of (3.2) goes as follows

k2 þ 2k� a12a21e�ks þ 1 ¼ 0: ð3:3Þ

(H2) Suppose that a12a21 < �1.Under the assumption (H2), we have the following results.

Lemma 3.1.

(i) When s ¼ sj ¼def 1x0fsin�1 � 2x0

a12a21

� �þ 2jpg, j ¼ 0; 1; 2; . . ., Eq. (3.3) has a simple pair of purely imaginary roots �ix0,

where x0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffija12a21j � 1

p.

(ii) For s 2 ½0; s0Þ, all roots of Eq. (3.3) have strictly negative real parts.

(iii) When s ¼ s0, Eq. (3.3) has a pair of imaginary roots �ix0 and all other roots have strictly negative real

parts.

Proof . ix0ðx0 > 0Þ is a purely imaginary root of (3.3) if and only if x0 satisfies

�x20 þ i2x0 � a12a21 cos x0sþ ia12a21 sin x0sþ 1 ¼ 0: ð3:4Þ

Separating the real and imaginary parts, we have

x20 � 1 ¼ �a12a21 cos x0s;

2x0 ¼ �a12a21 sin x0s:

�ð3:5Þ

It follows from Eq. (3.5) that

x40 þ 2x2

0 þ 1 ¼ a212a

221;

hence, x20 ¼ �1� ja12a21j, i.e., x0 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffija12a21j � 1

p. It is clear that x0 is well defined if a12a21 < �1. Denote

x0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffija12a21j � 1

p;

sj ¼def 1x0fsin�1ð� 2x0

a12a21Þ þ 2jpg; j ¼ 0; 1; 2; . . . :

ð3:6Þ

From (3.5) we know that (3.3) has a simple pair of purely imaginary roots �ix0 at sj (j = 0,1,2, . . .).

When s ¼ 0, Eq. (3.3) deduces to

k2 þ 2kþ 1� a12a21 ¼ 0: ð3:7Þ

It is obvious that all roots of (3.7) have negative real parts. Applying the lemma in Cooke and Grossman [17], we obtainconclusions (ii) and (iii), completing the proof. h

Let kðsÞ ¼ ajðsÞ þ ixjðsÞ be the root of (3.3) satisfying ajðsjÞ ¼ 0, xjðsjÞ ¼ x0. Then we have the followingtransversality condition.


Lemma 3.2. If (H2) is satisfied, then

dajðsÞds

��s¼sj

> 0

Proof . Differentiating the characteristic equation (3.3) with respect to s leads to

ð2kþ 2þ sa12a21e�ksÞ dkds¼ �ka12a21e�ks:

Thus,

dkds

� ��1

¼ �ð2kþ 2Þeks

ka12a21

� sk:

When s ¼ sj, ix0 is a purely imaginary root of (3.3). We then easily get

dajðsÞds

� ��1��s¼sj

¼ Re �ð2kþ 2Þeks

ka12a21

� s¼sj

¼ � 2x0 cos x0sþ 2 sin x0sx0a12a21

¼ 2ðx20 þ 1Þ

a212a

221

> 0:

We know that signdajðsÞ

ds js¼sj

n o¼ sign ðdajðsÞ

ds Þ�1js¼sj

n o, i.e., dajðsÞ=dsjs¼sj > 0, completing the proof. h

Remark . The Hopf bifurcation takes place if a pair of imaginary eigenvalues cross transversally the imaginary axis onGauss plane. To prove this fact we can check that

(1) There are pairs of conjugated complex solutions of (3.3) taking the form kðsÞ ¼ aðsÞ � ixðsÞ and there is only onepair for which real part of eigenvalues Rek ¼ 0;

(2) The transversality condition is fulfilled, i.e., dds RekðsÞ–0.

Theorem 3.1. Suppose (H2) is satisfied. Then the equilibrium (0,0) of (3.1) is asymptotically stable when s 2 ½0; s0Þ, and

unstable when s > s0. Moreover, at s ¼ sj; j ¼ 0; 1; 2; . . . ;�ix0 is a simple pair of purely imaginary roots of (3.3), and (3.1)

undergoes Hopf bifurcation near (0,0).

Theorem 3.1 can be illustrated by Fig. 2, where the shadowed region is the stable domain, Hopf bifurcations occur atthe line s1 þ s2 ¼ sj ðj ¼ 0; 1; 2; . . .Þ.

To extend the local Hopf branches described in Theorem 3.1 for large delay values, we apply a global Hopf bifur-cation result due to Wu [15], which has been outlined in Section 2. When s ¼ 0, (3.1) reduces to

_u1ðtÞ ¼ �u1ðtÞ þ a12f1ðu2ðtÞÞ;_u2ðtÞ ¼ �u2ðtÞ þ a21f2ðu1ðtÞÞ:

ð3:8Þ

Theorem 3.2. If assumption (H1) holds, system (3.8) has no nontrivial periodic solutions. Furthermore, the unique

equilibrium (0,0) is globally asymptotically stable in R2.

Proof . We first prove that solutions of (3.8) are bounded. Let

V ðu1; u2Þ ¼1

2ðu2

1 þ u22Þ:

Fig. 2. The stable diagram in the (s1; s2) plane.


Differentiating V along a solution of (3.8) gives

dVdt

��ð3:8Þ¼ �u2

1 � u22 þ a12u1f1ðu2Þ þ a21u2f2ðu1Þ:

Using (H1) we have

dVdt

��ð3:8Þ6 �u2

1 � u22 þ Lja12jju1j þ Lja21jju2j: ð3:9Þ

There must exist M > 1 such thatffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2

1 þ u22

pP M implies

dVdt

��ð3:8Þ

< 0:

This shows that solutions of (3.8) are stable and uniformly ultimately bounded. Let

ðP ðu1; u2Þ;Qðu1; u2ÞÞ ¼ ð�u1ðtÞ þ a12f1ðu2ðtÞÞ;�u2ðtÞ þ a21f2ðu1ðtÞÞÞ

denote the vector field of (3.8). Then

oPou1

þ oQou2

¼ �2 < 0

for all ðu1; u2Þ. The classical Bendixson’s negative criterion implies that (3.8) has no nontrivial periodic solutions. Thiscompletes the proof. h

Theorem 3.3. System (3.1) has no nontrivial periodic solutions of period s.

Proof . Note that if uðtÞ ¼ ðu1ðtÞ; u2ðtÞÞT is a s-periodic solution of (3.1), then uðtÞ is a s-periodic solution of the ordin-ary differential equation (3.8). Applying Theorem 3.2 we know that, under (H1), system (3.1) has no nontrivial s-peri-odic solutions, completing the proof. h

4. Global existence of periodic solutions

In this section, we study the global continuation of local Hopf branch bifurcating from the point ð0; sjÞðj ¼ 0; 1; 2; . . .Þ by using a global Hopf bifurcation theorem due to Wu [15].

For convenience of the reader, we copy system (3.1) in the following.

_u1ðtÞ ¼ �u1ðtÞ þ a12f1ðu2ðt � sÞÞ;_u2ðtÞ ¼ �u2ðtÞ þ a21f2ðu1ðtÞÞ:

ð4:1Þ

Following the work of Wu [15], we define

U ¼ Cð½�T ; 0�;R2Þ;RðF Þ ¼ Clfðu; s; T Þ : u is a T-periodic solution ofð4:1Þg � Y � R� Rþ;

NðF Þ ¼ fðu; s; T Þ : F ðu; s; T Þ ¼ 0g:

Let Cð0; sj; 2p=x0Þ denote the connected component of ð0; sj; 2p=x0Þ in R, where sj and x0 are defined in Eqs. (3.4) and(3.5), respectively. RðF Þ stands for the Fuller space, where F is differentiable with respect to u. NðF Þ is called the sta-tionary solution set for Eq. (4.1).

Lemma 4.1. Cð0; sj; 2p=x0Þ is unbounded for each center ð0; sj; 2p=x0Þ.

Proof . Our key approach is to verify assumptions (A1)–(A6) of Proposition 2.1 for (4.1). Smoothness conditions in (A1)and (A6) are easy ensured by (H1). Also by (H1), we know that (0,0) is the only equilibrium of (4.1), and thus all sta-tionary solutions of (4.1) are of the form ð0; s; T Þ. By Theorem 3.1, k ¼ 0 is not a characteristic root of the equilibrium(0,0), and thus (A2) is satisfied. The corresponding characteristic function takes the form:

k2 þ 2k� a12a21e�ks þ 1 ¼ 0;

where a12 and a21 are denoted as in (3.3).


Obviously, Dð0;s;T ÞðkÞ is continuous in ðs; T ; kÞ 2 Rþ � Rþ � C. This verifies (A3). A stationary solution ð0; sj; 2p=x0Þis a center. In fact, the set of centers is countable and given by

fð0; sj; 2p=x0Þ; j ¼ 0; 1; 2; . . .g;

where sj is determined by (3.6). Thus ð0; sj; 2p=x0Þ is an isolated point. This verifies (A5).By Lemma 3.2 and Theorem 3.1, for fixed j, there exist � > 0; d > 0 and a smooth curve k : ðsj � d; sj þ dÞ ! C such

that Dð0;s;T ÞðkðsÞÞ ¼ 0, jkðsÞ � ix0j < � for all s 2 ðsj � d; sj þ dÞ and

kðsjÞ ¼ ix0;d

dsRekðsÞ

��s¼sj

> 0:

Denote T j ¼ 2p=x0 and let

X� ¼ fðv; T Þ : 0 < v < �; jT � T jj < �g:

Clearly, if js� sjj 6 d and ðv; T Þ 2 oX� such that Dð0;s;T Þðvþ i2p=T Þ ¼ 0, then s ¼ sj; v ¼ 0 and T ¼ T j. This verifiesassumption (A4) for m ¼ 1. Moreover, if we put

H�mð0; sj; 2p=x0Þðv; T Þ ¼ Dð0;sj�d;T Þðvþ i2pm=T Þ;

then, at m ¼ 1, we have the crossing number of isolated center ð0; sj; 2p=x0Þ as follows:

c1ð0; sj; 2p=x0Þ ¼ degBðH�1 ð0; sj; 2p=x0Þ;X�Þ � degBðHþ1 ð0; sj; 2p=x0Þ;X�Þ ¼ �1:

By Proposition 2.1, we conclude that Cð0; sj; 2p=x0Þ is nonempty and unbounded, completing the proof. h

Lemma 4.2. Periodic solutions of (4.1) are uniformly bounded.

Proof . Let rðtÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2

1ðtÞ þ u22ðtÞ

p, M P f1; Lðja12j þ ja21jÞg. Differentiating rðtÞ along a solution of (4.1) we have

_rðtÞ ¼ 1

rðtÞ ½u1ðtÞ _u1ðtÞ þ u2ðtÞ _u2ðtÞ� ¼1

rðtÞ f�u21 � u2

2 þ a12u1f1ðu2Þ þ a21u2f2ðu1Þg

61

rðtÞ f�½u21ðtÞ þ u2

2ðtÞ� þ L½ja12jju1ðtÞj þ ja21jju2ðtÞj�g:

If there exists t0 > 0 such that rðt0Þ ¼ A P M , then we have

_rðt0Þ 61

A½�A2 þ LAðja12j þ ja21jÞ� ¼ �Aþ Lðja12j þ ja21jÞ < 0:

It follows that if uðtÞ ¼ ðu1ðtÞ; u2ðtÞÞT is a periodic solution of (4.1), then r < M for all t. This shows that periodic solu-tions of (4.1) are uniformly bounded. h

Lemma 4.3. The periods of periodic solutions of (4.1) are uniformly bounded.

Proof . By the definition of sj in Eq. (3.6), we know that x0sj > 2p ðj ¼ 1; 2; . . .Þ and hence 2p=x0 < sj ðj ¼ 1; 2; . . .Þ:From Theorem 3.1, we know that s0 > 0. Hence for s > sj, there exists an integer m such that s=m < 2p=x0 < s. Sincesystem (4.1) has no s-periodic solution, it has no s=n-periodic solution for any integer n. This implies that the period T

of a periodic solution on the connected component Cð0; sj; 2p=x0Þ satisfies s=m < T < s. Hence, we know that the peri-ods of periodic solutions of (4.1) on Cð0; sj; 2p=x0Þ are uniformly bounded. h

Theorem 4.1. Suppose that assumptions (H1) and (H2) are satisfied. Then system (4.1) has at least j nontrivial periodic

solutions when s > sj; j P 1, where sj is defined by (3.6).

Proof . It is sufficient to verify that the projection of Cð0; sj; 2p=x0Þ onto s-space is ½~s;1Þ, where 0 < ~s 6 sj; j ¼ 1; 2; . . ..Lemma 4.2 implies that the projection of Cð0; sj; 2p=x0Þ onto the u-space is bounded. Lemma 4.3 implies that the pro-jection of Cð0; sj; 2p=x0Þ onto the T-space is bounded. Theorem 3.2 implies that, when s ¼ 0, system (4.1) has no non-trivial periodic solutions. Therefore, the projection of Cð0; sj; 2p=x0Þ onto the s-space is bounded below. This shows thatin order for Cð0; sj; 2p=x0Þ to be unbounded, its projection onto the s-space must be unbounded. Consequently, the pro-jection of Cð0; sj; 2p=x0Þ onto the s-space includes ½sj;1Þ. This shows that, for each s > sj, system (4.1) has at least j

nontrivial periodic solutions, completing the proof of Theorem 4.1. h

0 1 2x 104

—0.4

—0.2

0

0.2

(a) tu1 plane

u1(t)

0 1 2x 104

—0.1

0

0.1

0.2

(b) tu2 plane

u2(t)

0 1 2x 104

—0.1

—0.05

0

0.05

0.1

(c) u1(t)

u2(t)

Fig. 3. Matlab simulations of multiple periodic solutions to system (5.1) with s1 ¼ 3:0816; s2 ¼ 7:0908. The total delays ¼ s1 þ s2 ¼ 10:1724 is between the two Hopf bifurcation values ~s1 ¼ 7:8540 and ~s2 ¼ 14:1372.


5. Numerical example

In this section, we carry out numerical simulations on system (1.4) to demonstrate the global Hopf bifurcationresults in Theorem 4.1 by using the Matlab software.

As a model, we consider the following no self-connection network model with two neurons:

_x1ðtÞ ¼ �x1ðtÞ þ tanhðx2ðt � s2ÞÞ;_x2ðtÞ ¼ �x2ðtÞ � 2 tanhðx1ðt � s1ÞÞ:

ð5:1Þ

We make some transformation just as in Section 3, and system is equivalent to (5.2).

_u1ðtÞ ¼ �u1ðtÞ þ tanhðu2ðt � sÞÞ;_u2ðtÞ ¼ �u2ðtÞ � 2 tanhðu1ðtÞÞ;

ð5:2Þ

which has a unique equilibrium (0,0). Applying the results of previous sections, we establish the global existence of peri-odic solutions for the system (5.1).

Notice that fi 2 C2, fið0Þ ¼ 0 and jfiðuÞj 6 1 ði ¼ 1; 2Þ for all u 2 R. This verifies assumption (H1). a12 ¼ 1; a21 ¼ �2and a12a21 < �1. This verifies assumption (H2). There exists ~sj ¼ p=2þ 2jp ðj ¼ 0; 1; 2; . . .Þ such that at ~sj;�i (i.e.x0 ¼ 1) is a pair of purely imaginary roots of (5.2). System (5.2) undergoes Hopf bifurcation near (0,0). In Fig. 3,we show the existence of periodic solutions for values of s ¼ s1 þ s2 far away from ~sj. For j ¼ 0; 1; 2; 3; . . .,~sj ¼ 1:5708; 7:8540; 14:1372; 20:4203; . . .. The delays are chosen as s1 ¼ 3:0816; s2 ¼ 7:0908 such thats ¼ s1 þ s2 ¼ 10:1724 is between the two Hopf bifurcation values ~s1 ¼ 7:8540 and ~s2 ¼ 14:1372. Periodic solutionsare shown in Fig. 3a–c.

6. Conclusion

Recently, a few applicable sufficient criteria have been established for the stability of neural network models withdelays (see [5,18–20] and the references cited therein). Bifurcation in neural network models with a single delay havealso been observed by many researchers [10,21]. However, there are few papers on the bifurcations of the neural net-work models with multiple delays, especially the global existence of multiple periodic solutions.

In this paper, we consider a two-neuron network model with two delays and no self-connection. We have found thatwhen the sum of two delays, s ¼ s1 þ s2, varies, the zero solution loses its stability and a Hopf bifurcation occurs, that is,a family of periodic solutions bifurcate the zero solution when s passes a critical value, say s0. Applying a global Hopfbifurcation theory due to Wu [15], multiple periodic solutions have been proved to exist when s ¼ s1 þ s2 is much larger.

Neural network with delays has very rich dynamics. From the point of view of nonlinear dynamics their analysis areuseful in solving problems of both theoretical and practical importance. The two-neuron network with two delays dis-cussed above are quite simple, but it potentially useful since the complexity found in this simple case might be carriedout to larger networks with multiple delays.

Acknowledgement

My work was supported by an NSF grant to Oswald Schmitz, Yale University, and an NSERC Discovery grant toMichel Loreau, McGill University.


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global existence of periodic solutions in a special neural network model with two delays

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