Research ArticleStability and Bifurcation Analysis for a Class of GeneralizedReaction-Diffusion Neural Networks with Time Delay

Tianshi Lv Qintao Gan and Qikai Zhu

Institute of Applied Mathematics Shijiazhuang Mechanical Engineering College Shijiazhuang 050003 China

Correspondence should be addressed to Qintao Gan ganqintaosinacom

Received 5 July 2015 Revised 3 August 2015 Accepted 10 August 2015

Academic Editor Carlo Bianca

Copyright copy 2016 Tianshi Lv et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Considering the fact that results for static neural networks are much more scare than results for local field neural networks andour purpose letting the problem researched be more general in many aspects in this paper a generalized neural networks modelwhich includes reaction-diffusion local field neural networks and reaction-diffusion static neural networks is built and the stabilityand bifurcation problems for it are investigated under Neumann boundary conditions First by discussing the correspondingcharacteristic equations the local stability of the trivial uniform steady state is discussed and the existence of Hopf bifurcationsis shown By using the normal form theory and the center manifold reduction of partial function differential equations explicitformulae which determine the direction and stability of bifurcating periodic solutions are acquired Finally numerical simulationsshow the results

1 Introduction

In the past several decades the dynamics of neural networkshave been extensively investigated

The artificial neural network has been used widely invarious fields such as signal processing pattern recognitionoptimization associative memories automatic control engi-neering artificial intelligence and fault diagnosis because ithas the characteristics of self-adaption self-organization andself-learning

Most of the phenomena occurring in real-world complexsystems do not have an immediate effect but appear withsome delay for example there exist time delays in the infor-mation processing of neurons Therefore time delays havebeen inserted into mathematical models and in particularin models of the applied sciences based on ordinary differ-ential equations The delayed axonal signal transmissions inthe neural network models make the dynamical behaviorsbecome more complicated because a time delay into anordinary differential equation could change the stability of theequilibrium (stable equilibrium becomes unstable) and couldcause fluctuations and Hopf bifurcation can occur (see [1])And in [1] we can know the time delaysrsquo effects from the work

by Carlo Bianca Massimiliano Ferrara and Luca GuerriniSo the delay is an important control parameter

In addition we must consider that the activations varyin space as well as in time because the electrons move inasymmetric electromagnetic fields and there exists diffusionin neural network (see [2])

In the past the main work was to research local fieldneural networks and static neural networks were rarelystudied Considering the fact that the problem of generalizedneural network ismore general inmany aspects in this paperwe will investigate a class of generalized neural networkswhich combine local field neural networks and static neuralnetworks

In order to study the effect of time delays and diffusion onthe dynamics of a neural network model in [3] Gan and Xuconsidered the following neural network model

1205971199061

120597119905= 1198631Δ1199061minus 11988611199061(119905 119909) + 119892

1(1199062(119905 minus 120591 119909))

1205971199062

120597119905= 1198632Δ1199062minus 11988621199062(119905 119909) + 119892

2(1199061(119905 minus 120591 119909))

(1)

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2016 Article ID 4321358 9 pageshttpdxdoiorg10115520164321358

2 Discrete Dynamics in Nature and Society

Motivated by the works of Gan and Xu in this paper weare concerned with the following neural network systemwithtime delay and reaction-diffusion

120597119906

120597119905= 1198631Δ119906 minus 119886

1119906 (119905 119909) + 119887

111198921(11990811119906 (119905 119909))

+ 119887121198922(11990812V (119905 minus 120591 119909)) 119905 gt 0 119909 isin Ω

120597V120597119905= 1198632ΔV minus 119886

2V (119905 119909) + 119887

211198921(11990821119906 (119905 minus 120591 119909))

+ 119887221198922(11990822V (119905 119909)) 119905 gt 0 119909 isin Ω

(2)

with initial and boundary conditions (Neumann boundaryconditions)

120597119906

120597119897=120597V120597119897= 0 119905 gt 0 119909 isin 120597Ω

119906 (119905 119909) = 1205931(119905 119909)

V (119905 119909) = 1205932(119905 119909)

119905 isin [minus120591 0] 119909 isin Ω

(3)

where 1198861 1198862 120591 1198631 1198632ge 0 119887

11 11988712 11988721 11988722 11990811 11990812 11990821

and 11990822

are random constants where 1198861and 119886

2represent

the neuron charging time constants 120591 represents the signaltransmission time delay 119863

1and 119863

2represent the smooth

diffusion operators 11988711 11988712 11988721 and 119887

22represent connecting

weight coefficients and 11990811 11990812 11990821 and 119908

22represent the

coefficients of 119906(119905 119909) V(119905 minus 120591 119909) 119906(119905 minus 120591 119909) and V(119905 119909)respectively 119906 V and 119909 are the state variables and spacevariable respectively 119892

1and 119892

2are the action functions of

the neurons satisfying 1198921(0) = 119892

2(0) = 0 Ω is a bounded

domain in 119877119899 with smooth boundary 120597Ω where 120597120597119897 denotesthe outward normal derivative on 120597Ω

The organization of this paper is as follows In Section 2by analyzing the corresponding characteristic equations wediscuss the local stability of trivial uniform steady state andthe existence of Hopf bifurcations of (2) and (3) In Section 3by applying the normal form and the center manifold the-orem closed-form expressions are derived which allow usto determine the direction of the Hopf bifurcations and thestability of the periodic solutions in (2) and (3) (see [2]) InSection 4 numerical simulations are carried out to illustratethe main theoretical results

2 Local Stability and Hopf Bifurcation

Obviously we can easily show that system (2) always has atrivial uniform steady state 119864lowast = (0 0)

Here we use 0 = 1205831lt 1205832lt sdot sdot sdot as the eigenvalues of the

operator minusΔ on Ω with the homogeneous Neumann bound-ary conditions and 119864(120583

119894) as the eigenspace corresponding to

120583119894in 1198621(Ω) Let nabla = [1198621(Ω)]2 let 120593

119894119895 119895 = 1 dim119864(120583

119894)

be an orthonormal basis of 119864(120583119894) and let nabla

119894119895= 119888120593119894119895| 119888 isin 119877

2

Then

nabla =

infin

⨁

119894=1

nabla119894

nabla119894=

dim119864(120583119894)

⨁

119895=1

nabla119894119895

(4)

Let weierp = diag(1198631 1198632) 120577119898 = weierpΔ119898 + Z(119864lowast)119898 where

Z (119864lowast)119898

= (minus1198861+ 11988711119908111198921015840

1(0) 0

0 minus1198862+ 11988722119908221198921015840

2(0))(

119906 (119905 119909)

V (119905 119909))

+ (0 119887

12119908121198921015840

2(0)

11988721119908211198921015840

1(0) 0

)(119906 (119905 minus 120591 119909)

V (119905 minus 120591 119909))

(5)

First we linearize system (2) at 119864lowast Then 119898119905= 120577119898 nabla

119894

is invariant under the operator 120577 for each 119894 ge 1 and 120582 is aneigenvalue of 120577 if and only if it is an eigenvalue of the matrixminus120583119894weierp + Z(119864lowast) for some 119894 ge 1 in which case there is an

eigenvalue in nabla119894

The characteristic equation of minus120583119894weierp+Z(119864lowast) is of the form

1205822+ 1199011120582 + 1199012+ 1199013119890minus2120582120591

= 0 (6)

where

1199011= 1205831198941198631+ 1198861minus 11988711119908111198921015840

1(0) + 120583

1198941198632+ 1198862

minus 11988722119908221198921015840

2(0)

1199012= (1205831198941198631+ 1198861minus 11988711119908111198921015840

1(0))

sdot (1205831198941198632+ 1198862minus 11988722119908221198921015840

2(0))

1199013= minus119887121198872111990812119908211198921015840

1(0) 1198921015840

2(0)

(7)

Letting 120591 = 0 then (6) becomes

1205822+ 1199011120582 + 1199012+ 1199013= 0 (8)

Obviously

1199012+ 1199013= (1205831198941198631+ 1198861minus 11988711119908111198921015840

1(0))

sdot (1205831198941198632+ 1198862minus 11988722119908221198921015840

2(0))

minus 119887121198872111990812119908211198921015840

1(0) 1198921015840

2(0)

(9)

Obviously if the following holds(1198671)

1198861minus 11988711119908111198921015840

1(0) gt 0

1198862minus 11988722119908221198921015840

2(0) gt 0

(1198861minus 11988711119908111198921015840

1(0)) (119886

2minus 11988722119908221198921015840

2(0))

minus 119887121198872111990812119908211198921015840

1(0) 1198921015840

2(0) gt 0

(10)

Discrete Dynamics in Nature and Society 3

then 1199012+ 1199013gt 0 119901

1gt 0 Hence if (1198671) holds when 120591 = 0

the trivial uniform steady state 119864lowast of problems (2) and (3) islocally stable

Let 119894120596 (120596 gt 0) be a solution of (6) separating real andimaginary parts then we can get that

1205962minus 1199012= 1199013cos 2120596120591

1199011120596 = 119901

3sin 2120596120591

(11)

Squaring and adding the two equations of (11) we obtainthat

1205964+ (1199012

1minus 21199012) 1205962+ 1199012

2minus 1199012

3= 0 (12)

Letting 119911 = 1205962 then (12) becomes

1199112+ (1199012

1minus 21199012) 119911 + 119901

2

2minus 1199012

3= 0 (13)

Obviously it is easy to calculate that

1199012

1minus 21199012= (1205831198941198631+ 1198861minus 11988711119908111198921015840

1(0))2

+ (1205831198941198632+ 1198862minus 11988722119908221198921015840

2(0))2

gt 0

1199012

2minus 1199012

3= (1205831198941198631+ 1198861minus 11988711119908111198921015840

1(0))2

sdot (1205831198941198632+ 1198862minus 11988722119908221198921015840

2(0))2

minus 1198872

121198872

211199082

121199082

2111989210158402

1(0) 11989210158402

2(0)

(14)

Let

1199021= (1198861minus 11988711119908111198921015840

1(0)) (119886

2minus 11988722119908221198921015840

2(0))

+ 119887121198872111990812119908211198921015840

1(0) 1198921015840

2(0)

1199022= (1198861minus 11988711119908111198921015840

1(0))2

(1198862minus 11988722119908221198921015840

2(0))2

minus 1198872

121198872

211199082

121199082

2111989210158402

1(0) 11989210158402

2(0)

1199023= (1198861minus 11988711119908111198921015840

1(0))2

+ (1198862minus 11988722119908221198921015840

2(0))2

gt 0

(15)

Therefore if 1199022gt 0 (13) has no positive roots Then if

1199021gt 0 and (1198671) Holds the trivial uniform steady state 119864lowast

of system (2) is locally asymptotically stable for all 119894 ge 1 and120591 ge 0

For 119894 = 1 if 1199021lt 0 then (12) has a unique positive root

1205960 where

1205960= (

1

2(minus1199023+ radic11990223minus 41199022))

12

(16)

It means that the characteristic equation (6) admits a pairof purely imaginary roots of the form plusmn119894120596

0for 119894 = 1

Take 120596 = ((12)(minus1199023+ radic11990223minus 41199022))12 Obviously (12)

holds if and only if 119894 = 1 Now we define that

1205910119899=

1

21205960

arccos1205962

0minus 1199012

1199013

+119899120587

1205960

119899 = 0 1 (17)

Then for 119894 = 1 when 120591 = 1205910119899 (6) has a pair of purely

imaginary roots plusmn1198941205960and all roots of it have negative real

parts for 119894 ge 2 It is easy to see that if (1198671) holds the trivialuniform steady state 119864lowast is locally stable for 120591 = 0 Henceon the basis of the general theory on characteristic equationsof delay-differential equations from [3 Theorem 41] we canknow that 119864lowast remains stable when 120591 lt 120591

0 where 120591

0= 12059100

Now we claim that

119889 (Re 120582)119889120591

10038161003816100381610038161003816100381610038161003816120591=1205910

gt 0 (18)

This will mean that there exists at least one eigenvaluewith positive real part when 120591 gt 120591

0 In addition the

conditions for the existence of a Hopf bifurcation [2] arethen satisfied generating a periodic solution To this end wedifferentiate (6) about 120591 then

(2120582 + 1199011)119889120582

119889120591minus 21199013119890minus2120582120591

(120582 + 120591119889120582

119889120591) = 0 (19)

So we know that

(119889120582

119889120591)

minus1

=2120582 + 119901

1minus 2120591119901

3119890minus2120582120591

21205821199013119890minus2120582120591

=(2120582 + 119901

1) 1198902120582120591

21205821199013

minus120591

120582

(20)

Therefore

sign119889 (Re 120582)119889120591

120582=1198941205960

= signRe(119889120582119889120591)

minus1

120582=1198941205960

= signRe[(2120582 + 119901

1) 1198902120582120591

21205821199013

]

120582=1198941205960

+ Re [minus 120591120582]120582=1198941205960

= sign21205960cos 2120596

0120591 + 1199011sin 2120596

0120591

212059601199013

(21)

By (11) we can obtain that

sign119889 (Re 120582)119889120591

120582=1198941205960

= sign21205962

0+ 1199012

1minus 21199012

211990123

(22)

Because 11990121minus 21199012gt 0 so

119889 (Re 120582)119889120591

10038161003816100381610038161003816100381610038161003816120591=1205910 120596=1205960

gt 0 (23)

Hence the transversal condition holds and a Hopf bifur-cation occurs when 120596 = 120596

0and 120591 = 120591

0

Consequently we gain the following results

Theorem 1 Let 1205910= 12059100

and let 1199021be defined by (15) For

system (2) let (1198671) hold If 1199021gt 0 the trivial uniform steady

state 119864lowast of system (2) is locally asymptotically stable when 120591 ge0 if 1199021lt 0 the trivial uniform steady state119864lowast is asymptotically

stable for 0 le 120591 lt 1205910and is unstable for 120591 gt 120591

0 furthermore

system (2) undergoes a Hopf bifurcation at 119864lowast when 120591 = 1205910

4 Discrete Dynamics in Nature and Society

3 Direction and Stability of Hopf Bifurcation

In Section 2 we have demonstrated that systems (2) and (3)undergo a train of periodic solutions bifurcating from thetrivial uniform steady state 119864lowast at the critical value of 120591 Inthis section we derive explicit formulae to determine theproperties of the Hopf bifurcation at critical value 120591

0by using

the normal form theory and center manifold reduction forPFDEs In this section we also let the condition (1198671) holdand 119902

1lt 0 And the work of Bianca and Guerrini in papers

[4ndash7] is the founder of the method in this section

Set 120591 = 120572 + 1205910 We first should normalize the delay 120591 by

the time-scaling 119905 rarr 119905120591 Then (2) can be rewritten in thefixed phase space ℓlowast = 119862([minus1 0] 119883) as

(119905) = 1205910weierpΔ119898 (119905) + 120591

0Z (119864lowast) (119898 (119905))

+ 119891lowast(119898 (119905) 120572)

(24)

where 119891lowastℓlowast times 119877+ rarr 1198772 is defined by

119891lowast(120601 120572) = 120572weierpΔ120601 (0) + 120591

0Z (119864lowast) (120601) + (120591

0+ 120572)(

1

2119887121199082

1211989210158401015840

2(0) 1206012

2(minus1) +

1

3119887121199083

12119892101584010158401015840

2(0) 1206013

2(minus1) + sdot sdot sdot

1

2119887211199082

2111989210158401015840

1(0) 1206012

1(minus1) +

1

3119887211199083

21119892101584010158401015840

1(0) 1206013

1(minus1) + sdot sdot sdot

) (25)

where 120601 = (1206011 1206012)119879isin ℓlowast

By the discussion in Section 2 we can know that theorigin (0 0) is a steady state of (24) and Λ

0= minus119894120596

01205910 11989412059601205910

are a pair of simple purely imaginary eigenvalues of the linearequation

(119905) = 1205910weierpΔ119898 (119905) + 120591

0Z (119864lowast) (119898 (119905)) (26)

and the functional differential equation

(119905) = 1205910Z (119911119905) (27)

On the basis of the Riesz representation theorem thereexists a function 120578(120579 120591) of bounded variation for 120579 isin [minus1 0]such that

Z (119864lowast) (120593) =

1

1205910

int

0

minus1

119889120578 (120579 1205910) 120593 (120579) where 120593 isin C (28)

Here we choose that

120578 (120579 1205910)

= 1205910(

minus1198861+ 11988711119908111198921015840

1(0) 0

0 minus1198862+ 11988722119908221198921015840

2(0)

)120575 (120579)

minus 1205910(

0 11988712119908121198921015840

2(0)

11988721119908211198921015840

1(0) 0

)120575 (120579 + 1)

(29)

where 120575 is the Dirac delta functionLet 119860(120591

0) denote the infinitesimal generator of the semi-

group induced by the solutions of (27) and let 119860lowast be theformal adjoint of 119860(120591

0) under the bilinear pairing

⟨120595 (119904) 120593 (120579)⟩ = 120595 (0) 120593 (0)

minus int

0

minus1

int

120579

120585=0

120595 (120585 minus 120579) 119889120578 (120579) 120593 (120585) 119889120585

(30)

where 120593 isin 1198621([minus1 0] 119877

2) 120595 isin 119862

1([0 1] (119877

2)lowast) 120578(120579) =

120578(120579 1205910) Then 119860(120591

0) and 119860lowast are a pair of adjoint operators

By the discussions in Section 2 we can realize that 119860(1205910)

has a pair of simple purely imaginary eigenvalues plusmn11989411990801205910and

they are also eigenvalues of119860lowast since119860(1205910) and119860lowast are adjoint

operators Let 119875 and 119875lowast be the center spaces of 119860(1205910) and

119860lowast associated with Λ

0 respectively Hence 119875lowast is the adjoint

space of 119875 and dim119875 = dim119875lowast = 2Let

120574 =11988721119908211198921015840

1(0) 119890minus11989411990801205910

1198862+ 1198941199080minus 119887221199082211989210158402(0)

120581 =11988712119908121198921015840

2(0) 11989011989411990801205910

1198862minus 1198941199080minus 119887221199082211989210158402(0)

(31)

then

1199011(120579) = 119890

11989411990801205910120579 (1 120574)119879

1199012(120579) = 119901

1(120579)

minus 1 le 120579 le 0

(32)

is a basis of 119875 associated with Λ0and

1199021(119904) = (1 120581)

119879119890minus11989411990801205910119904

1199022(119904) = 119902

1(119904)

0 le 119904 le 1

(33)

is a basis of 119876 associated with Λ0

Let Φ = (Φ1 Φ2) where

Φ1(120579) =

1199011(120579) + 119901

2(120579)

2

Φ2(120579) =

1199011(120579) minus 119901

2(120579)

2119894

(34)

Discrete Dynamics in Nature and Society 5

for 120579 isin [minus1 0] and let Ψlowast = (Ψlowast1 Ψlowast

2)119879 where

Ψlowast

1(119904) =

1199021(119904) + 119902

2(119904)

2

Ψlowast

2(119904) =

1199021(119904) minus 119902

2(119904)

2119894

(35)

for 119904 isin [minus1 0]Now we define that (Ψlowast Φ) = (Ψlowast

119895 Φ119896) (119895 119896 = 1 2) and

construct a new basis Ψ for 119876 by

Ψ = (Ψlowast

1 Ψ2)119879

= (Ψlowast Φ)minus1

Ψlowast (36)

Hence (ΨΦ) = 1198682 which is the second-order identity

matrix Moreover we define 1198910for 1198910= (1205731

0 1205732

0) and 119888 sdot 119891

0=

11988801205731

0+ 11988821205732

0for 119888 = (119888

1 1198882)119879isin 119862 Then the center space of

linear equation (26) is given by 119875119862119873ℓlowast where

119875119862119873ℓlowast= Φ (Ψ ⟨120601 119891

0⟩) sdot 1198910 120601 isin ℓ

lowast (37)

and ℓlowast denotes the complementary subspace of119875119862119873ℓlowast where

ℓlowast= 119875119862119873ℓlowastoplus 119876 (38)

Let 1198601205910be defined by

1198601205910120601 (120579)

= 120601 (120579)

+ 1198830(120579) [weierpΔ120601 (0) + 120591

0Z (119864lowast) (120601 (120579)) minus 120601 (0)]

120601 isin ℓlowast

(39)

where1198830 [minus1 0] rarr 119861(119883119883) is given by

1198830(120579) =

0 minus1 le 120579 lt 0

119868 120579 = 0

(40)

Then we have rewritten system (24) and it can berewritten as follows

(119905) = 1198601205910119898(119905) + 119883

0119891lowast(119898 (119905) 120572) (41)

The solution of (24) on the center manifold is given by

119898lowast(119905) = Φ (119909

1 1199092)119879

sdot 1198910+119882(119909

1 1199092 120572) (42)

Letting 119911 = 1199091minus1198941199092119882 = 119882

20(11991122)+119882

11119911119911+119882

02(11991122)+

sdot sdot sdot then

= 11989411990801205910119911 + 119892 (119911 119911) (43)

where

119892 (119911 119911) = (Ψ1(0) minus 119894Ψ

2(0)) ⟨119891

lowast(119898lowast(119905) 0) 119891

0⟩

≜ 11989220

1199112

2+ 11989211119911119911 + 119892

02

1199112

2+ 11989221

1199112119911

2+ sdot sdot sdot

(44)

We can use some easy computations to show that

⟨119891lowast(119898lowast(119905) 0) 119891

0⟩ =

1205910

8(119888111199112+ 119888121199112+ 11988813119911119911

119888211199112+ 119888221199112+ 11988823119911119911

)

+1205910

16(⟨11988801 1⟩

⟨11988802 1⟩) 1199112119911 + sdot sdot sdot

(45)

where

11988811= 11988712119908121198921015840

2(0) (minus119886

1+ 11988711119908111198921015840

1(0)) 120574119890

minus11989411990801205910

11988812= 11988712119908121198921015840

2(0) (minus119886

1+ 11988711119908111198921015840

1(0)) 120574119890

11989411990801205910

11988813= 11988712119908121198921015840

2(0) (minus119886

1+ 11988711119908111198921015840

1(0))

sdot (12057411989011989411990801205910 + 120574119890

minus11989411990801205910)

11988821= 11988721119908211198921015840

1(0) (minus119886

2+ 11988722119908221198921015840

1(0)) 120574119890

minus11989411990801205910

11988822= 11988721119908211198921015840

1(0) (minus119886

2+ 11988722119908221198921015840

2(0)) 120574119890

11989411990801205910

11988823= 11988721119908211198921015840

1(0) (minus119886

2+ 11988722119908221198921015840

2(0))

sdot (12057411989011989411990801205910 + 120574119890

minus11989411990801205910)

11988801= 11988712119908121198921015840

2(0) (minus119886

1+ 11988711119908111198921015840

1(0))

sdot (120574119882(1)

20(0) 11989011989411990801205910 +119882

(2)

20(minus1)) + 2119887

12119908121198921015840

2(0)

sdot (minus1198861+ 11988711119908111198921015840

1(0))

sdot (119882(2)

11(minus1) + 120574119882

(1)

11(0) 119890minus11989411990801205910)

11988802= 11988721119908211198921015840

1(0) (minus119886

2+ 11988722119908221198921015840

2(0))

sdot (120574119882(1)

20(minus1) + 119882

(2)

20(0) 11989011989411990801205910) + 2119887

21119908211198921015840

1(0)

sdot (minus1198862+ 11988722119908221198921015840

2(0))

sdot (120574119882(1)

11(minus1) + 119882

(2)

11(0) 119890minus11989411990801205910)

(46)

Setting (1205951 1205952) = Ψ

1(0) minus 119894Ψ

2(0) by calculating we get

that

11989220=1205910

4(119888111205951+ 119888211205952)

11989202=1205910

4(119888121205951+ 119888221205952)

11989211=1205910

8(119888131205951+ 119888231205952)

11989221=1205910

8(⟨11988801 1⟩ 1205951+ ⟨11988802 1⟩ 1205952)

(47)

Because there are 11988220(120579) and 119882

11(120579) in 119892

21where 120579 isin

[minus1 0] we still need to compute themBy [4] we know that

= 1198601205910119882+119867(119911 119911) (48)

6 Discrete Dynamics in Nature and Society

where

119867(119911 119911) = 11986720

1199112

2+ 11986711119911119911 + 119867

02

1199112

2+ sdot sdot sdot

= 1198830119891lowast(119898lowast(119905) 0)

minus Φ (Ψ ⟨1198830119891lowast(119898lowast(119905) 120572) 119891

0⟩) sdot 1198910

(49)

for119867119894119895isin 119876 with 119894 + 119895 = 2 It follows from (43) (48) and (49)

that

(1198601205910minus 2119894119908

01205910)11988220(120579) = minus119867

20(120579)

119860120591011988211(120579) = minus119867

11(120579)

(50)

By (49) we have that for 120579 isin [minus1 0)

119867 (119911 119911) = minus1

2[119892201199011(120579) + 119892

021199012(120579)] 1199112sdot 1198910

minus [119892111199011(120579) + 119892

111199012(120579)] 119911119911 sdot 119891

0+ sdot sdot sdot

(51)

Comparing the coefficients with (49) we get that for 120579 isin[minus1 0)

11986720(120579) = minus [119892

201199011(120579) + 119892

021199012(120579)] sdot 119891

0 (52)

11986711(120579) = minus [119892

111199011(120579) + 119892

111199012(120579)] sdot 119891

0 (53)

By (50) (52) and the definition of 1198601205910 we get that

20(120579) = 2119894119908

0120591011988220(120579) + [119892

201199011(120579) + 119892

021199012(120579)]

sdot 1198910

(54)

Noticing that 1199011(120579) = 119901

1(0)11989011989411990801205910120579 hence

11988220(120579)

= [11989411989220

11990801205910

1199011(120579) 11989011989411990801205910120579 +

11989411989202

311990801205910

1199012(120579) 119890minus11989411990801205910120579] sdot 119891

0

+ 1198641119890211989411990801205910120579

(55)

where 1198641= (119864(1)

1 119864(2)

1) isin 1198772 which is a constant vector

In a similar way by (50) and (53) we have that

11988211(120579)

= [minus11989411989211

11990801205910

1199011(0) 11989011989411990801205910120579 +

11989411989211

11990801205910

1199012(0) 119890minus11989411990801205910120579] sdot 119891

0

+ 1198642

(56)

where 1198642= (119864(1)

2 119864(2)

2) isin 1198772 which is also a constant vector

In what follows we seek appropriate 1198641and 119864

2 From the

definition of 1198601205910and (50) we can obtain that

21198941199080120591011988220(0) minus weierpΔ119882

20(0) minus Z (119864

lowast)11988220(120579)

= 11986720(0)

(57)

minus weierpΔ11988211(0) minus Z (119864

lowast)11988211(120579) = 119867

11(0) (58)

where

11986720(0) =

1205910

4(11988811

11988821

) minus [119892201199011(0) + 119892

021199012(0)] sdot 119891

0 (59)

11986711(0) =

1205910

8(11988813

11988823

) minus [119892111199011(0) + 119892

111199012(0)] sdot 119891

0 (60)

Substituting (55) and (59) into (57) we can obtain that

1198641=1

4(

211989411990801205910+ 1198861minus 11988711119908111198921015840

1(0) minus119887

12119908121198921015840

2(0) 119890minus211989411990801205910

minus11988721119908211198921015840

1(0) 119890minus211989411990801205910 2119894119908

01205910+ 1198862minus 11988722119908221198921015840

2(0)

)

minus1

(11988811

11988821

) (61)

In a similar way substituting (56) and (60) into (58) weobtain that

1198642

=1

8(

1198861minus 11988711119908111198921015840

1(0) minus119887

12119908121198921015840

2(0)

1198862minus 11988722119908221198921015840

2(0) minus119887

21119908211198921015840

1(0)

)

minus1

(11988813

11988823

)

(62)

Therefore we can compute the following values

1198881(0) =

119894

21199080

(1198921111989220minus 2100381610038161003816100381611989211

10038161003816100381610038162

minus

10038161003816100381610038161198920210038161003816100381610038162

3) +

11989221

2

1205832= minus

Re 1198881(0)

Re 1205821015840 (1205910)

1205732= 2Re 119888

1(0)

1198792= minus

Im 1198881(0) + 120583

2Im 1205821015840 (120591

0)

1199080

(63)

which determine the quantities of bifurcating periodic solu-tions in the center manifold at the critical value 120591

0 that is

1205832determines the direction of Hopf bifurcation the Hopf

bifurcation is supercritical (subcritical) if 1205832gt 0 (120583

2lt 0)

and the bifurcating periodic solutions exist for 120591 gt 1205910(120591 lt

Discrete Dynamics in Nature and Society 7

1205910) 1205732determines the stability of the bifurcating periodic

solutions if 1205732lt 0 (120573

2gt 0) the bifurcating periodic

solutions are stable (unstable) and 1198792determines the period

of the bifurcating periodic solutions the period increases(decrease) if 119879

2gt 0 (119879

2lt 0) [8ndash11]

4 Numerical Simulations

In this section in order to illustrate the results above we willgive two examples

Example 1 In system (2) we choose that 1198631= 1198632= 1 119886

1=

11988711= 04 119908

11= 06 119886

2= 03 119887

22= 11990822= 05 119887

12= 03

11988721= 06 119908

12= 24 119908

21= 36 119892

1(119909) = minus01tan(119909) and

1198922(119909) = arctan(119909) then

120597119906

120597119905= Δ119906 minus 04119906 (119905 119909) minus 004 tan (06119906 (119905 119909))

+ 03 arctan (24V (119905 minus 120591 119909))

120597V120597119905= ΔV minus 03V (119905 119909) minus 006 tan (36119906 (119905 minus 120591 119909))

+ 05 arctan (05V (119905 119909))

(64)

in which

0 lt 119909 lt 1

119905 gt 0

(65)

with initial and Neumann boundary conditions

120597119906

120597119897=120597V120597119897= 0 119905 ge 0 119909 = 0 1

119906 (119905 119909) = 05 (1 +119905

120587) sin (120587119909)

V (119905 119909) = (1 +119905

120587) sin (120587119909)

(119905 119909) isin [minus120591 0] times [0 1]

(66)

What should be remarked is that we choose the parametervalues stochastically under the condition 119902

2lt 0 in order to

ensure the existence of Hopf bifurcation at 119864lowast when 120591 = 1205910

So 1205910= 19371 and 119908

0= 02939 Then we can know

on the basis ofTheorem 1 that the trivial uniform steady state119864lowast= (0 0) is asymptotically stable when 0 le 120591 lt 120591

0 When

120591 gt 1205910 the steady state is unstable and a Hopf bifurcation is

arising from the steady state The numerical simulations inFigures 1 and 2 illustrate the facts

When 120591 = 1205910 we get that 119888

1(0) = minus00001 + 00022119894 then

we can acquire that 1205832gt 0 and 120573

2lt 0 Hence when 120591 passes

through 1205910to the right (120591 gt 120591

0) the bifurcation turns up and

the corresponding periodic orbits are orbitally asymptoticallystable

Example 2 In system (2) we choose that 1198631= 1198632= 001

11988721= 09 119886

2= 02 119887

12= 03 119887

11= 11988722= 11990822= 05 119886

1=

11990811= 06 119908

12= 25 119908

21= 36 119892

1(119909) = minus01tan(119909) and

1198922(119909) = arctan(119909) then

120597119906

120597119905= 001Δ119906 minus 06119906 (119905 119909) minus 005 tan (06119906 (119905 119909))

+ 03 arctan (25V (119905 minus 120591 119909))

120597V120597119905= 001ΔV minus 02V (119905 119909) minus 009 tan (36119906 (119905 minus 120591 119909))

+ 05 arctan (05V (119905 119909))

(67)

in which

0 lt 119909 lt 1

119905 gt 0

(68)

with initial and Dirichlet boundary conditions

119906 (119905 0) = 119906 (119905 1) = V (119905 0) = V (119905 1) = 0 119905 ge 0

119906 (119905 119909) = 05 (1 +119905

120587) sin (120587119909)

V (119905 119909) = (1 +119905

120587) sin (120587119909)

(119905 119909) isin [minus120591 0] times [0 1]

(69)

The similar Hopf bifurcation phenomenon is illustratedby the numerical simulations in Figures 3 and 4

5 Discussion and Research Perspective

This section is devoted to a summary of discussion andresearch perspective for the generalized reaction-diffusionneural networkmodelThemodel is based on the assumptionthat the signal transmission is of a digital (McCulloch-Pitts)nature the model then described a combination of analogand digital signal processing in the network [12] Dependingon the modeling approaches neural networks can be mod-eled either as a static neural network model or as a local fieldneural network model In order to let the problem be moregeneral in many aspects we build a generalized reaction-diffusion neural network model which includes reaction-diffusion local field neural networks and reaction-diffusionstatic neural networks For a delayed neural network animportant issue is the dynamical behaviors of the network[13] Thus there has been a large body of work discussing thestability and bifurcation in delayed neural network modelsBy analyzing the characteristic equation we discussed thelocal stability of the trivial uniform of system (2) [14] Itwas shown that when the delay 120591 varies the trivial uniformsteady state exchanges its stability and Hopf bifurcationsoccur Numerical simulations illustrated the occurrence ofthe bifurcate periodic solutions when the delay 120591 passes thecritical value 120591

0

A research perspective includes the problem of deter-mining the bifurcating periodic solutions and the stabilityand directions of the Hopf bifurcation using the normal

8 Discrete Dynamics in Nature and Society

x-axis

05

1

0

minus004

minus002

0

002

004

006

008

01

u-axis

1500 50 100

t-axis(a)

x-axis

05

1

0

100 1500 50

t-axis

minus003

minus002

minus001

0

001

002

003

v-axis

(b)

Figure 1 The temporal solution found by numerical integration of systems (64) and (66) with 120591 = 185 (a) 119906(119905 119909) and (b) V(119905 119909)

minus004

minus002

0

002

004

006

008

01

u-axis

x-axis05

1

50 100 1500

t-axis(a)

minus003

minus002

minus001

0

001

002

003

v-axis

x-axis05

1

00 50

t-axis100 150

(b)

Figure 2 The temporal solution found by numerical integration of systems (64) and (66) with 120591 = 425 (a) 119906(119905 119909) and (b) V(119905 119909)

0

50t-axis

100

150

x-axis

05

1

0

minus02

0

02

04

06

u-axis

(a)0

50t-axis

100

150

x-axis05

1

0

minus05

0

05

1

v-axis

(b)

Figure 3 The temporal solution found by numerical integration of systems (67) and (69) with 120591 = 125 (a) 119906(119905 119909) and (b) V(119905 119909)

Discrete Dynamics in Nature and Society 9

0

50t-axis

100

150

x-axis

05

1

0

minus02

0

02

04

06

u-axis

(a)0

50t-axis

100

150

x-axis

05

1

0

minus05

0

05

1

v-axis

(b)

Figure 4 The temporal solution found by numerical integration of systems (67) and (69) with 120591 = 205 (a) 119906(119905 119909) and (b) V(119905 119909)

form theory and the center manifold reaction A challengingperspective is the comparison of the generalized modelintroduced in the present paper with the experimentallymea-surable quantities Indeed the mathematical models shouldreproduce both qualitatively and quantitatively empiricaldata (see [4])

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (no 61305076) and the ScientificResearch Foundation for the Returned Overseas ChineseScholars State Education Ministry

References

[1] C Bianca M Ferrara and L Guerrini ldquoThe time delaysrsquo effectson the qualitative behavior of an economic growth modelrdquoAbstract and Applied Analysis vol 2013 Article ID 901014 10pages 2013

[2] LV Ballestra LGuerrini andG Pacelli ldquoStability switches andbifurcation analysis of a time delay model for the diffusion of anew technologyrdquo International Journal of Bifurcation amp Chaosvol 24 no 9 Article ID 1450113 2014

[3] Q Gan and R Xu ldquoStability and Hopf bifurcation of a delayedreaction-diffusion neural networkrdquo Mathematical Methods inthe Applied Sciences vol 34 no 12 pp 1450ndash1459 2011

[4] C Bianca and LGuerrini ldquoOn theDalgaard-Strulikmodel withlogistic population growth rate and delayed-carrying capacityrdquoActa Applicandae Mathematicae vol 128 pp 39ndash48 2013

[5] O G Jepps C Bianca and L Rondoni ldquoOnset of diffusivebehavior in confined transport systemsrdquo Chaos vol 18 no 1Article ID 013127 2008

[6] L Gori L Guerrini and M Sodini ldquoEquilibrium and dis-equilibrium dynamics in cobweb models with time delaysrdquo

International Journal of Bifurcation and Chaos vol 25 no 6Article ID 1550088 2015

[7] C Bianca L Guerrini and A Lemarchand ldquoExistence of solu-tions of a partial integrodifferential equation with thermostatand time delayrdquoAbstract and Applied Analysis vol 2014 ArticleID 463409 7 pages 2014

[8] Y Kuang ldquoDelay differential equations with applications inpopulation dynamicsrdquo Discrete amp Continuous Dynamical Sys-tems vol 33 no 4 pp 1633ndash1644 2013

[9] C Huang Y He L Huang and Y Zhaohui ldquoHopf bifurcationanalysis of two neurons with three delaysrdquo Nonlinear AnalysisReal World Applications vol 8 no 3 pp 903ndash921 2007

[10] C Huang L Huang J Feng M Nai and Y He ldquoHopfbifurcation analysis for a two-neuron networkwith four delaysrdquoChaos Solitons amp Fractals vol 34 no 3 pp 795ndash812 2007

[11] Y Song M Han and J Wei ldquoStability and Hopf bifurcationanalysis on a simplified BAM neural network with delaysrdquoPhysica D Nonlinear Phenomena vol 200 no 3-4 pp 185ndash2042005

[12] X-P Yan ldquoHopf bifurcation and stability for a delayed tri-neuron network modelrdquo Journal of Computational and AppliedMathematics vol 196 no 2 pp 579ndash595 2006

[13] H Zhao and L Wang ldquoHopf bifurcation in Cohen-Grossbergneural network with distributed delaysrdquo Nonlinear AnalysisReal World Applications vol 8 no 1 pp 73ndash89 2007

[14] B Zheng Y Zhang and C Zhang ldquoStability and bifurcationof a discrete BAM neural network model with delaysrdquo ChaosSolitons amp Fractals vol 36 no 3 pp 612ndash616 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Discrete Dynamics in Nature and Society

Motivated by the works of Gan and Xu in this paper weare concerned with the following neural network systemwithtime delay and reaction-diffusion

120597119906

120597119905= 1198631Δ119906 minus 119886

1119906 (119905 119909) + 119887

111198921(11990811119906 (119905 119909))

+ 119887121198922(11990812V (119905 minus 120591 119909)) 119905 gt 0 119909 isin Ω

120597V120597119905= 1198632ΔV minus 119886

2V (119905 119909) + 119887

211198921(11990821119906 (119905 minus 120591 119909))

+ 119887221198922(11990822V (119905 119909)) 119905 gt 0 119909 isin Ω

(2)

with initial and boundary conditions (Neumann boundaryconditions)

120597119906

120597119897=120597V120597119897= 0 119905 gt 0 119909 isin 120597Ω

119906 (119905 119909) = 1205931(119905 119909)

V (119905 119909) = 1205932(119905 119909)

119905 isin [minus120591 0] 119909 isin Ω

(3)

where 1198861 1198862 120591 1198631 1198632ge 0 119887

11 11988712 11988721 11988722 11990811 11990812 11990821

and 11990822

are random constants where 1198861and 119886

2represent

the neuron charging time constants 120591 represents the signaltransmission time delay 119863

1and 119863

2represent the smooth

diffusion operators 11988711 11988712 11988721 and 119887

22represent connecting

weight coefficients and 11990811 11990812 11990821 and 119908

22represent the

coefficients of 119906(119905 119909) V(119905 minus 120591 119909) 119906(119905 minus 120591 119909) and V(119905 119909)respectively 119906 V and 119909 are the state variables and spacevariable respectively 119892

1and 119892

2are the action functions of

the neurons satisfying 1198921(0) = 119892

2(0) = 0 Ω is a bounded

domain in 119877119899 with smooth boundary 120597Ω where 120597120597119897 denotesthe outward normal derivative on 120597Ω

The organization of this paper is as follows In Section 2by analyzing the corresponding characteristic equations wediscuss the local stability of trivial uniform steady state andthe existence of Hopf bifurcations of (2) and (3) In Section 3by applying the normal form and the center manifold the-orem closed-form expressions are derived which allow usto determine the direction of the Hopf bifurcations and thestability of the periodic solutions in (2) and (3) (see [2]) InSection 4 numerical simulations are carried out to illustratethe main theoretical results

2 Local Stability and Hopf Bifurcation

Obviously we can easily show that system (2) always has atrivial uniform steady state 119864lowast = (0 0)

Here we use 0 = 1205831lt 1205832lt sdot sdot sdot as the eigenvalues of the

operator minusΔ on Ω with the homogeneous Neumann bound-ary conditions and 119864(120583

119894) as the eigenspace corresponding to

120583119894in 1198621(Ω) Let nabla = [1198621(Ω)]2 let 120593

119894119895 119895 = 1 dim119864(120583

119894)

be an orthonormal basis of 119864(120583119894) and let nabla

119894119895= 119888120593119894119895| 119888 isin 119877

2

Then

nabla =

infin

⨁

119894=1

nabla119894

nabla119894=

dim119864(120583119894)

⨁

119895=1

nabla119894119895

(4)

Let weierp = diag(1198631 1198632) 120577119898 = weierpΔ119898 + Z(119864lowast)119898 where

Z (119864lowast)119898

= (minus1198861+ 11988711119908111198921015840

1(0) 0

0 minus1198862+ 11988722119908221198921015840

2(0))(

119906 (119905 119909)

V (119905 119909))

+ (0 119887

12119908121198921015840

2(0)

11988721119908211198921015840

1(0) 0

)(119906 (119905 minus 120591 119909)

V (119905 minus 120591 119909))

(5)

First we linearize system (2) at 119864lowast Then 119898119905= 120577119898 nabla

119894

is invariant under the operator 120577 for each 119894 ge 1 and 120582 is aneigenvalue of 120577 if and only if it is an eigenvalue of the matrixminus120583119894weierp + Z(119864lowast) for some 119894 ge 1 in which case there is an

eigenvalue in nabla119894

The characteristic equation of minus120583119894weierp+Z(119864lowast) is of the form

1205822+ 1199011120582 + 1199012+ 1199013119890minus2120582120591

= 0 (6)

where

1199011= 1205831198941198631+ 1198861minus 11988711119908111198921015840

1(0) + 120583

1198941198632+ 1198862

minus 11988722119908221198921015840

2(0)

1199012= (1205831198941198631+ 1198861minus 11988711119908111198921015840

1(0))

sdot (1205831198941198632+ 1198862minus 11988722119908221198921015840

2(0))

1199013= minus119887121198872111990812119908211198921015840

1(0) 1198921015840

2(0)

(7)

Letting 120591 = 0 then (6) becomes

1205822+ 1199011120582 + 1199012+ 1199013= 0 (8)

Obviously

1199012+ 1199013= (1205831198941198631+ 1198861minus 11988711119908111198921015840

1(0))

sdot (1205831198941198632+ 1198862minus 11988722119908221198921015840

2(0))

minus 119887121198872111990812119908211198921015840

1(0) 1198921015840

2(0)

(9)

Obviously if the following holds(1198671)

1198861minus 11988711119908111198921015840

1(0) gt 0

1198862minus 11988722119908221198921015840

2(0) gt 0

(1198861minus 11988711119908111198921015840

1(0)) (119886

2minus 11988722119908221198921015840

2(0))

minus 119887121198872111990812119908211198921015840

1(0) 1198921015840

2(0) gt 0

(10)

Discrete Dynamics in Nature and Society 3

then 1199012+ 1199013gt 0 119901

1gt 0 Hence if (1198671) holds when 120591 = 0

the trivial uniform steady state 119864lowast of problems (2) and (3) islocally stable

Let 119894120596 (120596 gt 0) be a solution of (6) separating real andimaginary parts then we can get that

1205962minus 1199012= 1199013cos 2120596120591

1199011120596 = 119901

3sin 2120596120591

(11)

Squaring and adding the two equations of (11) we obtainthat

1205964+ (1199012

1minus 21199012) 1205962+ 1199012

2minus 1199012

3= 0 (12)

Letting 119911 = 1205962 then (12) becomes

1199112+ (1199012

1minus 21199012) 119911 + 119901

2

2minus 1199012

3= 0 (13)

Obviously it is easy to calculate that

1199012

1minus 21199012= (1205831198941198631+ 1198861minus 11988711119908111198921015840

1(0))2

+ (1205831198941198632+ 1198862minus 11988722119908221198921015840

2(0))2

gt 0

1199012

2minus 1199012

3= (1205831198941198631+ 1198861minus 11988711119908111198921015840

1(0))2

sdot (1205831198941198632+ 1198862minus 11988722119908221198921015840

2(0))2

minus 1198872

121198872

211199082

121199082

2111989210158402

1(0) 11989210158402

2(0)

(14)

Let

1199021= (1198861minus 11988711119908111198921015840

1(0)) (119886

2minus 11988722119908221198921015840

2(0))

+ 119887121198872111990812119908211198921015840

1(0) 1198921015840

2(0)

1199022= (1198861minus 11988711119908111198921015840

1(0))2

(1198862minus 11988722119908221198921015840

2(0))2

minus 1198872

121198872

211199082

121199082

2111989210158402

1(0) 11989210158402

2(0)

1199023= (1198861minus 11988711119908111198921015840

1(0))2

+ (1198862minus 11988722119908221198921015840

2(0))2

gt 0

(15)

Therefore if 1199022gt 0 (13) has no positive roots Then if

1199021gt 0 and (1198671) Holds the trivial uniform steady state 119864lowast

of system (2) is locally asymptotically stable for all 119894 ge 1 and120591 ge 0

For 119894 = 1 if 1199021lt 0 then (12) has a unique positive root

1205960 where

1205960= (

1

2(minus1199023+ radic11990223minus 41199022))

12

(16)

It means that the characteristic equation (6) admits a pairof purely imaginary roots of the form plusmn119894120596

0for 119894 = 1

Take 120596 = ((12)(minus1199023+ radic11990223minus 41199022))12 Obviously (12)

holds if and only if 119894 = 1 Now we define that

1205910119899=

1

21205960

arccos1205962

0minus 1199012

1199013

+119899120587

1205960

119899 = 0 1 (17)

Then for 119894 = 1 when 120591 = 1205910119899 (6) has a pair of purely

imaginary roots plusmn1198941205960and all roots of it have negative real

parts for 119894 ge 2 It is easy to see that if (1198671) holds the trivialuniform steady state 119864lowast is locally stable for 120591 = 0 Henceon the basis of the general theory on characteristic equationsof delay-differential equations from [3 Theorem 41] we canknow that 119864lowast remains stable when 120591 lt 120591

0 where 120591

0= 12059100

Now we claim that

119889 (Re 120582)119889120591

10038161003816100381610038161003816100381610038161003816120591=1205910

gt 0 (18)

This will mean that there exists at least one eigenvaluewith positive real part when 120591 gt 120591

0 In addition the

conditions for the existence of a Hopf bifurcation [2] arethen satisfied generating a periodic solution To this end wedifferentiate (6) about 120591 then

(2120582 + 1199011)119889120582

119889120591minus 21199013119890minus2120582120591

(120582 + 120591119889120582

119889120591) = 0 (19)

So we know that

(119889120582

119889120591)

minus1

=2120582 + 119901

1minus 2120591119901

3119890minus2120582120591

21205821199013119890minus2120582120591

=(2120582 + 119901

1) 1198902120582120591

21205821199013

minus120591

120582

(20)

Therefore

sign119889 (Re 120582)119889120591

120582=1198941205960

= signRe(119889120582119889120591)

minus1

120582=1198941205960

= signRe[(2120582 + 119901

1) 1198902120582120591

21205821199013

]

120582=1198941205960

+ Re [minus 120591120582]120582=1198941205960

= sign21205960cos 2120596

0120591 + 1199011sin 2120596

0120591

212059601199013

(21)

By (11) we can obtain that

sign119889 (Re 120582)119889120591

120582=1198941205960

= sign21205962

0+ 1199012

1minus 21199012

211990123

(22)

Because 11990121minus 21199012gt 0 so

119889 (Re 120582)119889120591

10038161003816100381610038161003816100381610038161003816120591=1205910 120596=1205960

gt 0 (23)

Hence the transversal condition holds and a Hopf bifur-cation occurs when 120596 = 120596

0and 120591 = 120591

0

Consequently we gain the following results

Theorem 1 Let 1205910= 12059100

and let 1199021be defined by (15) For

system (2) let (1198671) hold If 1199021gt 0 the trivial uniform steady

state 119864lowast of system (2) is locally asymptotically stable when 120591 ge0 if 1199021lt 0 the trivial uniform steady state119864lowast is asymptotically

stable for 0 le 120591 lt 1205910and is unstable for 120591 gt 120591

0 furthermore

system (2) undergoes a Hopf bifurcation at 119864lowast when 120591 = 1205910

4 Discrete Dynamics in Nature and Society

3 Direction and Stability of Hopf Bifurcation

In Section 2 we have demonstrated that systems (2) and (3)undergo a train of periodic solutions bifurcating from thetrivial uniform steady state 119864lowast at the critical value of 120591 Inthis section we derive explicit formulae to determine theproperties of the Hopf bifurcation at critical value 120591

0by using

the normal form theory and center manifold reduction forPFDEs In this section we also let the condition (1198671) holdand 119902

1lt 0 And the work of Bianca and Guerrini in papers

[4ndash7] is the founder of the method in this section

Set 120591 = 120572 + 1205910 We first should normalize the delay 120591 by

the time-scaling 119905 rarr 119905120591 Then (2) can be rewritten in thefixed phase space ℓlowast = 119862([minus1 0] 119883) as

(119905) = 1205910weierpΔ119898 (119905) + 120591

0Z (119864lowast) (119898 (119905))

+ 119891lowast(119898 (119905) 120572)

(24)

where 119891lowastℓlowast times 119877+ rarr 1198772 is defined by

119891lowast(120601 120572) = 120572weierpΔ120601 (0) + 120591

0Z (119864lowast) (120601) + (120591

0+ 120572)(

1

2119887121199082

1211989210158401015840

2(0) 1206012

2(minus1) +

1

3119887121199083

12119892101584010158401015840

2(0) 1206013

2(minus1) + sdot sdot sdot

1

2119887211199082

2111989210158401015840

1(0) 1206012

1(minus1) +

1

3119887211199083

21119892101584010158401015840

1(0) 1206013

1(minus1) + sdot sdot sdot

) (25)

where 120601 = (1206011 1206012)119879isin ℓlowast

By the discussion in Section 2 we can know that theorigin (0 0) is a steady state of (24) and Λ

0= minus119894120596

01205910 11989412059601205910

are a pair of simple purely imaginary eigenvalues of the linearequation

(119905) = 1205910weierpΔ119898 (119905) + 120591

0Z (119864lowast) (119898 (119905)) (26)

and the functional differential equation

(119905) = 1205910Z (119911119905) (27)

On the basis of the Riesz representation theorem thereexists a function 120578(120579 120591) of bounded variation for 120579 isin [minus1 0]such that

Z (119864lowast) (120593) =

1

1205910

int

0

minus1

119889120578 (120579 1205910) 120593 (120579) where 120593 isin C (28)

Here we choose that

120578 (120579 1205910)

= 1205910(

minus1198861+ 11988711119908111198921015840

1(0) 0

0 minus1198862+ 11988722119908221198921015840

2(0)

)120575 (120579)

minus 1205910(

0 11988712119908121198921015840

2(0)

11988721119908211198921015840

1(0) 0

)120575 (120579 + 1)

(29)

where 120575 is the Dirac delta functionLet 119860(120591

0) denote the infinitesimal generator of the semi-

group induced by the solutions of (27) and let 119860lowast be theformal adjoint of 119860(120591

0) under the bilinear pairing

⟨120595 (119904) 120593 (120579)⟩ = 120595 (0) 120593 (0)

minus int

0

minus1

int

120579

120585=0

120595 (120585 minus 120579) 119889120578 (120579) 120593 (120585) 119889120585

(30)

where 120593 isin 1198621([minus1 0] 119877

2) 120595 isin 119862

1([0 1] (119877

2)lowast) 120578(120579) =

120578(120579 1205910) Then 119860(120591

0) and 119860lowast are a pair of adjoint operators

By the discussions in Section 2 we can realize that 119860(1205910)

has a pair of simple purely imaginary eigenvalues plusmn11989411990801205910and

they are also eigenvalues of119860lowast since119860(1205910) and119860lowast are adjoint

operators Let 119875 and 119875lowast be the center spaces of 119860(1205910) and

119860lowast associated with Λ

0 respectively Hence 119875lowast is the adjoint

space of 119875 and dim119875 = dim119875lowast = 2Let

120574 =11988721119908211198921015840

1(0) 119890minus11989411990801205910

1198862+ 1198941199080minus 119887221199082211989210158402(0)

120581 =11988712119908121198921015840

2(0) 11989011989411990801205910

1198862minus 1198941199080minus 119887221199082211989210158402(0)

(31)

then

1199011(120579) = 119890

11989411990801205910120579 (1 120574)119879

1199012(120579) = 119901

1(120579)

minus 1 le 120579 le 0

(32)

is a basis of 119875 associated with Λ0and

1199021(119904) = (1 120581)

119879119890minus11989411990801205910119904

1199022(119904) = 119902

1(119904)

0 le 119904 le 1

(33)

is a basis of 119876 associated with Λ0

Let Φ = (Φ1 Φ2) where

Φ1(120579) =

1199011(120579) + 119901

2(120579)

2

Φ2(120579) =

1199011(120579) minus 119901

2(120579)

2119894

(34)

Discrete Dynamics in Nature and Society 5

for 120579 isin [minus1 0] and let Ψlowast = (Ψlowast1 Ψlowast

2)119879 where

Ψlowast

1(119904) =

1199021(119904) + 119902

2(119904)

2

Ψlowast

2(119904) =

1199021(119904) minus 119902

2(119904)

2119894

(35)

for 119904 isin [minus1 0]Now we define that (Ψlowast Φ) = (Ψlowast

119895 Φ119896) (119895 119896 = 1 2) and

construct a new basis Ψ for 119876 by

Ψ = (Ψlowast

1 Ψ2)119879

= (Ψlowast Φ)minus1

Ψlowast (36)

Hence (ΨΦ) = 1198682 which is the second-order identity

matrix Moreover we define 1198910for 1198910= (1205731

0 1205732

0) and 119888 sdot 119891

0=

11988801205731

0+ 11988821205732

0for 119888 = (119888

1 1198882)119879isin 119862 Then the center space of

linear equation (26) is given by 119875119862119873ℓlowast where

119875119862119873ℓlowast= Φ (Ψ ⟨120601 119891

0⟩) sdot 1198910 120601 isin ℓ

lowast (37)

and ℓlowast denotes the complementary subspace of119875119862119873ℓlowast where

ℓlowast= 119875119862119873ℓlowastoplus 119876 (38)

Let 1198601205910be defined by

1198601205910120601 (120579)

= 120601 (120579)

+ 1198830(120579) [weierpΔ120601 (0) + 120591

0Z (119864lowast) (120601 (120579)) minus 120601 (0)]

120601 isin ℓlowast

(39)

where1198830 [minus1 0] rarr 119861(119883119883) is given by

1198830(120579) =

0 minus1 le 120579 lt 0

119868 120579 = 0

(40)

Then we have rewritten system (24) and it can berewritten as follows

(119905) = 1198601205910119898(119905) + 119883

0119891lowast(119898 (119905) 120572) (41)

The solution of (24) on the center manifold is given by

119898lowast(119905) = Φ (119909

1 1199092)119879

sdot 1198910+119882(119909

1 1199092 120572) (42)

Letting 119911 = 1199091minus1198941199092119882 = 119882

20(11991122)+119882

11119911119911+119882

02(11991122)+

sdot sdot sdot then

= 11989411990801205910119911 + 119892 (119911 119911) (43)

where

119892 (119911 119911) = (Ψ1(0) minus 119894Ψ

2(0)) ⟨119891

lowast(119898lowast(119905) 0) 119891

0⟩

≜ 11989220

1199112

2+ 11989211119911119911 + 119892

02

1199112

2+ 11989221

1199112119911

2+ sdot sdot sdot

(44)

We can use some easy computations to show that

⟨119891lowast(119898lowast(119905) 0) 119891

0⟩ =

1205910

8(119888111199112+ 119888121199112+ 11988813119911119911

119888211199112+ 119888221199112+ 11988823119911119911

)

+1205910

16(⟨11988801 1⟩

⟨11988802 1⟩) 1199112119911 + sdot sdot sdot

(45)

where

11988811= 11988712119908121198921015840

2(0) (minus119886

1+ 11988711119908111198921015840

1(0)) 120574119890

minus11989411990801205910

11988812= 11988712119908121198921015840

2(0) (minus119886

1+ 11988711119908111198921015840

1(0)) 120574119890

11989411990801205910

11988813= 11988712119908121198921015840

2(0) (minus119886

1+ 11988711119908111198921015840

1(0))

sdot (12057411989011989411990801205910 + 120574119890

minus11989411990801205910)

11988821= 11988721119908211198921015840

1(0) (minus119886

2+ 11988722119908221198921015840

1(0)) 120574119890

minus11989411990801205910

11988822= 11988721119908211198921015840

1(0) (minus119886

2+ 11988722119908221198921015840

2(0)) 120574119890

11989411990801205910

11988823= 11988721119908211198921015840

1(0) (minus119886

2+ 11988722119908221198921015840

2(0))

sdot (12057411989011989411990801205910 + 120574119890

minus11989411990801205910)

11988801= 11988712119908121198921015840

2(0) (minus119886

1+ 11988711119908111198921015840

1(0))

sdot (120574119882(1)

20(0) 11989011989411990801205910 +119882

(2)

20(minus1)) + 2119887

12119908121198921015840

2(0)

sdot (minus1198861+ 11988711119908111198921015840

1(0))

sdot (119882(2)

11(minus1) + 120574119882

(1)

11(0) 119890minus11989411990801205910)

11988802= 11988721119908211198921015840

1(0) (minus119886

2+ 11988722119908221198921015840

2(0))

sdot (120574119882(1)

20(minus1) + 119882

(2)

20(0) 11989011989411990801205910) + 2119887

21119908211198921015840

1(0)

sdot (minus1198862+ 11988722119908221198921015840

2(0))

sdot (120574119882(1)

11(minus1) + 119882

(2)

11(0) 119890minus11989411990801205910)

(46)

Setting (1205951 1205952) = Ψ

1(0) minus 119894Ψ

2(0) by calculating we get

that

11989220=1205910

4(119888111205951+ 119888211205952)

11989202=1205910

4(119888121205951+ 119888221205952)

11989211=1205910

8(119888131205951+ 119888231205952)

11989221=1205910

8(⟨11988801 1⟩ 1205951+ ⟨11988802 1⟩ 1205952)

(47)

Because there are 11988220(120579) and 119882

11(120579) in 119892

21where 120579 isin

[minus1 0] we still need to compute themBy [4] we know that

= 1198601205910119882+119867(119911 119911) (48)

6 Discrete Dynamics in Nature and Society

where

119867(119911 119911) = 11986720

1199112

2+ 11986711119911119911 + 119867

02

1199112

2+ sdot sdot sdot

= 1198830119891lowast(119898lowast(119905) 0)

minus Φ (Ψ ⟨1198830119891lowast(119898lowast(119905) 120572) 119891

0⟩) sdot 1198910

(49)

for119867119894119895isin 119876 with 119894 + 119895 = 2 It follows from (43) (48) and (49)

that

(1198601205910minus 2119894119908

01205910)11988220(120579) = minus119867

20(120579)

119860120591011988211(120579) = minus119867

11(120579)

(50)

By (49) we have that for 120579 isin [minus1 0)

119867 (119911 119911) = minus1

2[119892201199011(120579) + 119892

021199012(120579)] 1199112sdot 1198910

minus [119892111199011(120579) + 119892

111199012(120579)] 119911119911 sdot 119891

0+ sdot sdot sdot

(51)

Comparing the coefficients with (49) we get that for 120579 isin[minus1 0)

11986720(120579) = minus [119892

201199011(120579) + 119892

021199012(120579)] sdot 119891

0 (52)

11986711(120579) = minus [119892

111199011(120579) + 119892

111199012(120579)] sdot 119891

0 (53)

By (50) (52) and the definition of 1198601205910 we get that

20(120579) = 2119894119908

0120591011988220(120579) + [119892

201199011(120579) + 119892

021199012(120579)]

sdot 1198910

(54)

Noticing that 1199011(120579) = 119901

1(0)11989011989411990801205910120579 hence

11988220(120579)

= [11989411989220

11990801205910

1199011(120579) 11989011989411990801205910120579 +

11989411989202

311990801205910

1199012(120579) 119890minus11989411990801205910120579] sdot 119891

0

+ 1198641119890211989411990801205910120579

(55)

where 1198641= (119864(1)

1 119864(2)

1) isin 1198772 which is a constant vector

In a similar way by (50) and (53) we have that

11988211(120579)

= [minus11989411989211

11990801205910

1199011(0) 11989011989411990801205910120579 +

11989411989211

11990801205910

1199012(0) 119890minus11989411990801205910120579] sdot 119891

0

+ 1198642

(56)

where 1198642= (119864(1)

2 119864(2)

2) isin 1198772 which is also a constant vector

In what follows we seek appropriate 1198641and 119864

2 From the

definition of 1198601205910and (50) we can obtain that

21198941199080120591011988220(0) minus weierpΔ119882

20(0) minus Z (119864

lowast)11988220(120579)

= 11986720(0)

(57)

minus weierpΔ11988211(0) minus Z (119864

lowast)11988211(120579) = 119867

11(0) (58)

where

11986720(0) =

1205910

4(11988811

11988821

) minus [119892201199011(0) + 119892

021199012(0)] sdot 119891

0 (59)

11986711(0) =

1205910

8(11988813

11988823

) minus [119892111199011(0) + 119892

111199012(0)] sdot 119891

0 (60)

Substituting (55) and (59) into (57) we can obtain that

1198641=1

4(

211989411990801205910+ 1198861minus 11988711119908111198921015840

1(0) minus119887

12119908121198921015840

2(0) 119890minus211989411990801205910

minus11988721119908211198921015840

1(0) 119890minus211989411990801205910 2119894119908

01205910+ 1198862minus 11988722119908221198921015840

2(0)

)

minus1

(11988811

11988821

) (61)

In a similar way substituting (56) and (60) into (58) weobtain that

1198642

=1

8(

1198861minus 11988711119908111198921015840

1(0) minus119887

12119908121198921015840

2(0)

1198862minus 11988722119908221198921015840

2(0) minus119887

21119908211198921015840

1(0)

)

minus1

(11988813

11988823

)

(62)

Therefore we can compute the following values

1198881(0) =

119894

21199080

(1198921111989220minus 2100381610038161003816100381611989211

10038161003816100381610038162

minus

10038161003816100381610038161198920210038161003816100381610038162

3) +

11989221

2

1205832= minus

Re 1198881(0)

Re 1205821015840 (1205910)

1205732= 2Re 119888

1(0)

1198792= minus

Im 1198881(0) + 120583

2Im 1205821015840 (120591

0)

1199080

(63)

which determine the quantities of bifurcating periodic solu-tions in the center manifold at the critical value 120591

0 that is

1205832determines the direction of Hopf bifurcation the Hopf

bifurcation is supercritical (subcritical) if 1205832gt 0 (120583

2lt 0)

and the bifurcating periodic solutions exist for 120591 gt 1205910(120591 lt

Discrete Dynamics in Nature and Society 7

1205910) 1205732determines the stability of the bifurcating periodic

solutions if 1205732lt 0 (120573

2gt 0) the bifurcating periodic

solutions are stable (unstable) and 1198792determines the period

of the bifurcating periodic solutions the period increases(decrease) if 119879

2gt 0 (119879

2lt 0) [8ndash11]

4 Numerical Simulations

In this section in order to illustrate the results above we willgive two examples

Example 1 In system (2) we choose that 1198631= 1198632= 1 119886

1=

11988711= 04 119908

11= 06 119886

2= 03 119887

22= 11990822= 05 119887

12= 03

11988721= 06 119908

12= 24 119908

21= 36 119892

1(119909) = minus01tan(119909) and

1198922(119909) = arctan(119909) then

120597119906

120597119905= Δ119906 minus 04119906 (119905 119909) minus 004 tan (06119906 (119905 119909))

+ 03 arctan (24V (119905 minus 120591 119909))

120597V120597119905= ΔV minus 03V (119905 119909) minus 006 tan (36119906 (119905 minus 120591 119909))

+ 05 arctan (05V (119905 119909))

(64)

in which

0 lt 119909 lt 1

119905 gt 0

(65)

with initial and Neumann boundary conditions

120597119906

120597119897=120597V120597119897= 0 119905 ge 0 119909 = 0 1

119906 (119905 119909) = 05 (1 +119905

120587) sin (120587119909)

V (119905 119909) = (1 +119905

120587) sin (120587119909)

(119905 119909) isin [minus120591 0] times [0 1]

(66)

What should be remarked is that we choose the parametervalues stochastically under the condition 119902

2lt 0 in order to

ensure the existence of Hopf bifurcation at 119864lowast when 120591 = 1205910

So 1205910= 19371 and 119908

0= 02939 Then we can know

on the basis ofTheorem 1 that the trivial uniform steady state119864lowast= (0 0) is asymptotically stable when 0 le 120591 lt 120591

0 When

120591 gt 1205910 the steady state is unstable and a Hopf bifurcation is

arising from the steady state The numerical simulations inFigures 1 and 2 illustrate the facts

When 120591 = 1205910 we get that 119888

1(0) = minus00001 + 00022119894 then

we can acquire that 1205832gt 0 and 120573

2lt 0 Hence when 120591 passes

through 1205910to the right (120591 gt 120591

0) the bifurcation turns up and

the corresponding periodic orbits are orbitally asymptoticallystable

Example 2 In system (2) we choose that 1198631= 1198632= 001

11988721= 09 119886

2= 02 119887

12= 03 119887

11= 11988722= 11990822= 05 119886

1=

11990811= 06 119908

12= 25 119908

21= 36 119892

1(119909) = minus01tan(119909) and

1198922(119909) = arctan(119909) then

120597119906

120597119905= 001Δ119906 minus 06119906 (119905 119909) minus 005 tan (06119906 (119905 119909))

+ 03 arctan (25V (119905 minus 120591 119909))

120597V120597119905= 001ΔV minus 02V (119905 119909) minus 009 tan (36119906 (119905 minus 120591 119909))

+ 05 arctan (05V (119905 119909))

(67)

in which

0 lt 119909 lt 1

119905 gt 0

(68)

with initial and Dirichlet boundary conditions

119906 (119905 0) = 119906 (119905 1) = V (119905 0) = V (119905 1) = 0 119905 ge 0

119906 (119905 119909) = 05 (1 +119905

120587) sin (120587119909)

V (119905 119909) = (1 +119905

120587) sin (120587119909)

(119905 119909) isin [minus120591 0] times [0 1]

(69)

The similar Hopf bifurcation phenomenon is illustratedby the numerical simulations in Figures 3 and 4

5 Discussion and Research Perspective

This section is devoted to a summary of discussion andresearch perspective for the generalized reaction-diffusionneural networkmodelThemodel is based on the assumptionthat the signal transmission is of a digital (McCulloch-Pitts)nature the model then described a combination of analogand digital signal processing in the network [12] Dependingon the modeling approaches neural networks can be mod-eled either as a static neural network model or as a local fieldneural network model In order to let the problem be moregeneral in many aspects we build a generalized reaction-diffusion neural network model which includes reaction-diffusion local field neural networks and reaction-diffusionstatic neural networks For a delayed neural network animportant issue is the dynamical behaviors of the network[13] Thus there has been a large body of work discussing thestability and bifurcation in delayed neural network modelsBy analyzing the characteristic equation we discussed thelocal stability of the trivial uniform of system (2) [14] Itwas shown that when the delay 120591 varies the trivial uniformsteady state exchanges its stability and Hopf bifurcationsoccur Numerical simulations illustrated the occurrence ofthe bifurcate periodic solutions when the delay 120591 passes thecritical value 120591

0

A research perspective includes the problem of deter-mining the bifurcating periodic solutions and the stabilityand directions of the Hopf bifurcation using the normal

8 Discrete Dynamics in Nature and Society

x-axis

05

1

0

minus004

minus002

0

002

004

006

008

01

u-axis

1500 50 100

t-axis(a)

x-axis

05

1

0

100 1500 50

t-axis

minus003

minus002

minus001

0

001

002

003

v-axis

(b)

Figure 1 The temporal solution found by numerical integration of systems (64) and (66) with 120591 = 185 (a) 119906(119905 119909) and (b) V(119905 119909)

minus004

minus002

0

002

004

006

008

01

u-axis

x-axis05

1

50 100 1500

t-axis(a)

minus003

minus002

minus001

0

001

002

003

v-axis

x-axis05

1

00 50

t-axis100 150

(b)

Figure 2 The temporal solution found by numerical integration of systems (64) and (66) with 120591 = 425 (a) 119906(119905 119909) and (b) V(119905 119909)

0

50t-axis

100

150

x-axis

05

1

0

minus02

0

02

04

06

u-axis

(a)0

50t-axis

100

150

x-axis05

1

0

minus05

0

05

1

v-axis

(b)

Figure 3 The temporal solution found by numerical integration of systems (67) and (69) with 120591 = 125 (a) 119906(119905 119909) and (b) V(119905 119909)

Discrete Dynamics in Nature and Society 9

0

50t-axis

100

150

x-axis

05

1

0

minus02

0

02

04

06

u-axis

(a)0

50t-axis

100

150

x-axis

05

1

0

minus05

0

05

1

v-axis

(b)

Figure 4 The temporal solution found by numerical integration of systems (67) and (69) with 120591 = 205 (a) 119906(119905 119909) and (b) V(119905 119909)

form theory and the center manifold reaction A challengingperspective is the comparison of the generalized modelintroduced in the present paper with the experimentallymea-surable quantities Indeed the mathematical models shouldreproduce both qualitatively and quantitatively empiricaldata (see [4])

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (no 61305076) and the ScientificResearch Foundation for the Returned Overseas ChineseScholars State Education Ministry

References

[1] C Bianca M Ferrara and L Guerrini ldquoThe time delaysrsquo effectson the qualitative behavior of an economic growth modelrdquoAbstract and Applied Analysis vol 2013 Article ID 901014 10pages 2013

[2] LV Ballestra LGuerrini andG Pacelli ldquoStability switches andbifurcation analysis of a time delay model for the diffusion of anew technologyrdquo International Journal of Bifurcation amp Chaosvol 24 no 9 Article ID 1450113 2014

[3] Q Gan and R Xu ldquoStability and Hopf bifurcation of a delayedreaction-diffusion neural networkrdquo Mathematical Methods inthe Applied Sciences vol 34 no 12 pp 1450ndash1459 2011

[4] C Bianca and LGuerrini ldquoOn theDalgaard-Strulikmodel withlogistic population growth rate and delayed-carrying capacityrdquoActa Applicandae Mathematicae vol 128 pp 39ndash48 2013

[5] O G Jepps C Bianca and L Rondoni ldquoOnset of diffusivebehavior in confined transport systemsrdquo Chaos vol 18 no 1Article ID 013127 2008

[6] L Gori L Guerrini and M Sodini ldquoEquilibrium and dis-equilibrium dynamics in cobweb models with time delaysrdquo

International Journal of Bifurcation and Chaos vol 25 no 6Article ID 1550088 2015

[7] C Bianca L Guerrini and A Lemarchand ldquoExistence of solu-tions of a partial integrodifferential equation with thermostatand time delayrdquoAbstract and Applied Analysis vol 2014 ArticleID 463409 7 pages 2014

[8] Y Kuang ldquoDelay differential equations with applications inpopulation dynamicsrdquo Discrete amp Continuous Dynamical Sys-tems vol 33 no 4 pp 1633ndash1644 2013

[9] C Huang Y He L Huang and Y Zhaohui ldquoHopf bifurcationanalysis of two neurons with three delaysrdquo Nonlinear AnalysisReal World Applications vol 8 no 3 pp 903ndash921 2007

[10] C Huang L Huang J Feng M Nai and Y He ldquoHopfbifurcation analysis for a two-neuron networkwith four delaysrdquoChaos Solitons amp Fractals vol 34 no 3 pp 795ndash812 2007

[11] Y Song M Han and J Wei ldquoStability and Hopf bifurcationanalysis on a simplified BAM neural network with delaysrdquoPhysica D Nonlinear Phenomena vol 200 no 3-4 pp 185ndash2042005

[12] X-P Yan ldquoHopf bifurcation and stability for a delayed tri-neuron network modelrdquo Journal of Computational and AppliedMathematics vol 196 no 2 pp 579ndash595 2006

[13] H Zhao and L Wang ldquoHopf bifurcation in Cohen-Grossbergneural network with distributed delaysrdquo Nonlinear AnalysisReal World Applications vol 8 no 1 pp 73ndash89 2007

[14] B Zheng Y Zhang and C Zhang ldquoStability and bifurcationof a discrete BAM neural network model with delaysrdquo ChaosSolitons amp Fractals vol 36 no 3 pp 612ndash616 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 3

then 1199012+ 1199013gt 0 119901

1gt 0 Hence if (1198671) holds when 120591 = 0

the trivial uniform steady state 119864lowast of problems (2) and (3) islocally stable

Let 119894120596 (120596 gt 0) be a solution of (6) separating real andimaginary parts then we can get that

1205962minus 1199012= 1199013cos 2120596120591

1199011120596 = 119901

3sin 2120596120591

(11)

Squaring and adding the two equations of (11) we obtainthat

1205964+ (1199012

1minus 21199012) 1205962+ 1199012

2minus 1199012

3= 0 (12)

Letting 119911 = 1205962 then (12) becomes

1199112+ (1199012

1minus 21199012) 119911 + 119901

2

2minus 1199012

3= 0 (13)

Obviously it is easy to calculate that

1199012

1minus 21199012= (1205831198941198631+ 1198861minus 11988711119908111198921015840

1(0))2

+ (1205831198941198632+ 1198862minus 11988722119908221198921015840

2(0))2

gt 0

1199012

2minus 1199012

3= (1205831198941198631+ 1198861minus 11988711119908111198921015840

1(0))2

sdot (1205831198941198632+ 1198862minus 11988722119908221198921015840

2(0))2

minus 1198872

121198872

211199082

121199082

2111989210158402

1(0) 11989210158402

2(0)

(14)

Let

1199021= (1198861minus 11988711119908111198921015840

1(0)) (119886

2minus 11988722119908221198921015840

2(0))

+ 119887121198872111990812119908211198921015840

1(0) 1198921015840

2(0)

1199022= (1198861minus 11988711119908111198921015840

1(0))2

(1198862minus 11988722119908221198921015840

2(0))2

minus 1198872

121198872

211199082

121199082

2111989210158402

1(0) 11989210158402

2(0)

1199023= (1198861minus 11988711119908111198921015840

1(0))2

+ (1198862minus 11988722119908221198921015840

2(0))2

gt 0

(15)

Therefore if 1199022gt 0 (13) has no positive roots Then if

1199021gt 0 and (1198671) Holds the trivial uniform steady state 119864lowast

of system (2) is locally asymptotically stable for all 119894 ge 1 and120591 ge 0

For 119894 = 1 if 1199021lt 0 then (12) has a unique positive root

1205960 where

1205960= (

1

2(minus1199023+ radic11990223minus 41199022))

12

(16)

It means that the characteristic equation (6) admits a pairof purely imaginary roots of the form plusmn119894120596

0for 119894 = 1

Take 120596 = ((12)(minus1199023+ radic11990223minus 41199022))12 Obviously (12)

holds if and only if 119894 = 1 Now we define that

1205910119899=

1

21205960

arccos1205962

0minus 1199012

1199013

+119899120587

1205960

119899 = 0 1 (17)

Then for 119894 = 1 when 120591 = 1205910119899 (6) has a pair of purely

imaginary roots plusmn1198941205960and all roots of it have negative real

parts for 119894 ge 2 It is easy to see that if (1198671) holds the trivialuniform steady state 119864lowast is locally stable for 120591 = 0 Henceon the basis of the general theory on characteristic equationsof delay-differential equations from [3 Theorem 41] we canknow that 119864lowast remains stable when 120591 lt 120591

0 where 120591

0= 12059100

Now we claim that

119889 (Re 120582)119889120591

10038161003816100381610038161003816100381610038161003816120591=1205910

gt 0 (18)

This will mean that there exists at least one eigenvaluewith positive real part when 120591 gt 120591

0 In addition the

conditions for the existence of a Hopf bifurcation [2] arethen satisfied generating a periodic solution To this end wedifferentiate (6) about 120591 then

(2120582 + 1199011)119889120582

119889120591minus 21199013119890minus2120582120591

(120582 + 120591119889120582

119889120591) = 0 (19)

So we know that

(119889120582

119889120591)

minus1

=2120582 + 119901

1minus 2120591119901

3119890minus2120582120591

21205821199013119890minus2120582120591

=(2120582 + 119901

1) 1198902120582120591

21205821199013

minus120591

120582

(20)

Therefore

sign119889 (Re 120582)119889120591

120582=1198941205960

= signRe(119889120582119889120591)

minus1

120582=1198941205960

= signRe[(2120582 + 119901

1) 1198902120582120591

21205821199013

]

120582=1198941205960

+ Re [minus 120591120582]120582=1198941205960

= sign21205960cos 2120596

0120591 + 1199011sin 2120596

0120591

212059601199013

(21)

By (11) we can obtain that

sign119889 (Re 120582)119889120591

120582=1198941205960

= sign21205962

0+ 1199012

1minus 21199012

211990123

(22)

Because 11990121minus 21199012gt 0 so

119889 (Re 120582)119889120591

10038161003816100381610038161003816100381610038161003816120591=1205910 120596=1205960

gt 0 (23)

Hence the transversal condition holds and a Hopf bifur-cation occurs when 120596 = 120596

0and 120591 = 120591

0

Consequently we gain the following results

Theorem 1 Let 1205910= 12059100

and let 1199021be defined by (15) For

system (2) let (1198671) hold If 1199021gt 0 the trivial uniform steady

state 119864lowast of system (2) is locally asymptotically stable when 120591 ge0 if 1199021lt 0 the trivial uniform steady state119864lowast is asymptotically

stable for 0 le 120591 lt 1205910and is unstable for 120591 gt 120591

0 furthermore

system (2) undergoes a Hopf bifurcation at 119864lowast when 120591 = 1205910

4 Discrete Dynamics in Nature and Society

3 Direction and Stability of Hopf Bifurcation

In Section 2 we have demonstrated that systems (2) and (3)undergo a train of periodic solutions bifurcating from thetrivial uniform steady state 119864lowast at the critical value of 120591 Inthis section we derive explicit formulae to determine theproperties of the Hopf bifurcation at critical value 120591

0by using

the normal form theory and center manifold reduction forPFDEs In this section we also let the condition (1198671) holdand 119902

1lt 0 And the work of Bianca and Guerrini in papers

[4ndash7] is the founder of the method in this section

Set 120591 = 120572 + 1205910 We first should normalize the delay 120591 by

the time-scaling 119905 rarr 119905120591 Then (2) can be rewritten in thefixed phase space ℓlowast = 119862([minus1 0] 119883) as

(119905) = 1205910weierpΔ119898 (119905) + 120591

0Z (119864lowast) (119898 (119905))

+ 119891lowast(119898 (119905) 120572)

(24)

where 119891lowastℓlowast times 119877+ rarr 1198772 is defined by

119891lowast(120601 120572) = 120572weierpΔ120601 (0) + 120591

0Z (119864lowast) (120601) + (120591

0+ 120572)(

1

2119887121199082

1211989210158401015840

2(0) 1206012

2(minus1) +

1

3119887121199083

12119892101584010158401015840

2(0) 1206013

2(minus1) + sdot sdot sdot

1

2119887211199082

2111989210158401015840

1(0) 1206012

1(minus1) +

1

3119887211199083

21119892101584010158401015840

1(0) 1206013

1(minus1) + sdot sdot sdot

) (25)

where 120601 = (1206011 1206012)119879isin ℓlowast

By the discussion in Section 2 we can know that theorigin (0 0) is a steady state of (24) and Λ

0= minus119894120596

01205910 11989412059601205910

are a pair of simple purely imaginary eigenvalues of the linearequation

(119905) = 1205910weierpΔ119898 (119905) + 120591

0Z (119864lowast) (119898 (119905)) (26)

and the functional differential equation

(119905) = 1205910Z (119911119905) (27)

On the basis of the Riesz representation theorem thereexists a function 120578(120579 120591) of bounded variation for 120579 isin [minus1 0]such that

Z (119864lowast) (120593) =

1

1205910

int

0

minus1

119889120578 (120579 1205910) 120593 (120579) where 120593 isin C (28)

Here we choose that

120578 (120579 1205910)

= 1205910(

minus1198861+ 11988711119908111198921015840

1(0) 0

0 minus1198862+ 11988722119908221198921015840

2(0)

)120575 (120579)

minus 1205910(

0 11988712119908121198921015840

2(0)

11988721119908211198921015840

1(0) 0

)120575 (120579 + 1)

(29)

where 120575 is the Dirac delta functionLet 119860(120591

0) denote the infinitesimal generator of the semi-

group induced by the solutions of (27) and let 119860lowast be theformal adjoint of 119860(120591

0) under the bilinear pairing

⟨120595 (119904) 120593 (120579)⟩ = 120595 (0) 120593 (0)

minus int

0

minus1

int

120579

120585=0

120595 (120585 minus 120579) 119889120578 (120579) 120593 (120585) 119889120585

(30)

where 120593 isin 1198621([minus1 0] 119877

2) 120595 isin 119862

1([0 1] (119877

2)lowast) 120578(120579) =

120578(120579 1205910) Then 119860(120591

0) and 119860lowast are a pair of adjoint operators

By the discussions in Section 2 we can realize that 119860(1205910)

has a pair of simple purely imaginary eigenvalues plusmn11989411990801205910and

they are also eigenvalues of119860lowast since119860(1205910) and119860lowast are adjoint

operators Let 119875 and 119875lowast be the center spaces of 119860(1205910) and

119860lowast associated with Λ

0 respectively Hence 119875lowast is the adjoint

space of 119875 and dim119875 = dim119875lowast = 2Let

120574 =11988721119908211198921015840

1(0) 119890minus11989411990801205910

1198862+ 1198941199080minus 119887221199082211989210158402(0)

120581 =11988712119908121198921015840

2(0) 11989011989411990801205910

1198862minus 1198941199080minus 119887221199082211989210158402(0)

(31)

then

1199011(120579) = 119890

11989411990801205910120579 (1 120574)119879

1199012(120579) = 119901

1(120579)

minus 1 le 120579 le 0

(32)

is a basis of 119875 associated with Λ0and

1199021(119904) = (1 120581)

119879119890minus11989411990801205910119904

1199022(119904) = 119902

1(119904)

0 le 119904 le 1

(33)

is a basis of 119876 associated with Λ0

Let Φ = (Φ1 Φ2) where

Φ1(120579) =

1199011(120579) + 119901

2(120579)

2

Φ2(120579) =

1199011(120579) minus 119901

2(120579)

2119894

(34)

Discrete Dynamics in Nature and Society 5

for 120579 isin [minus1 0] and let Ψlowast = (Ψlowast1 Ψlowast

2)119879 where

Ψlowast

1(119904) =

1199021(119904) + 119902

2(119904)

2

Ψlowast

2(119904) =

1199021(119904) minus 119902

2(119904)

2119894

(35)

for 119904 isin [minus1 0]Now we define that (Ψlowast Φ) = (Ψlowast

119895 Φ119896) (119895 119896 = 1 2) and

construct a new basis Ψ for 119876 by

Ψ = (Ψlowast

1 Ψ2)119879

= (Ψlowast Φ)minus1

Ψlowast (36)

Hence (ΨΦ) = 1198682 which is the second-order identity

matrix Moreover we define 1198910for 1198910= (1205731

0 1205732

0) and 119888 sdot 119891

0=

11988801205731

0+ 11988821205732

0for 119888 = (119888

1 1198882)119879isin 119862 Then the center space of

linear equation (26) is given by 119875119862119873ℓlowast where

119875119862119873ℓlowast= Φ (Ψ ⟨120601 119891

0⟩) sdot 1198910 120601 isin ℓ

lowast (37)

and ℓlowast denotes the complementary subspace of119875119862119873ℓlowast where

ℓlowast= 119875119862119873ℓlowastoplus 119876 (38)

Let 1198601205910be defined by

1198601205910120601 (120579)

= 120601 (120579)

+ 1198830(120579) [weierpΔ120601 (0) + 120591

0Z (119864lowast) (120601 (120579)) minus 120601 (0)]

120601 isin ℓlowast

(39)

where1198830 [minus1 0] rarr 119861(119883119883) is given by

1198830(120579) =

0 minus1 le 120579 lt 0

119868 120579 = 0

(40)

Then we have rewritten system (24) and it can berewritten as follows

(119905) = 1198601205910119898(119905) + 119883

0119891lowast(119898 (119905) 120572) (41)

The solution of (24) on the center manifold is given by

119898lowast(119905) = Φ (119909

1 1199092)119879

sdot 1198910+119882(119909

1 1199092 120572) (42)

Letting 119911 = 1199091minus1198941199092119882 = 119882

20(11991122)+119882

11119911119911+119882

02(11991122)+

sdot sdot sdot then

= 11989411990801205910119911 + 119892 (119911 119911) (43)

where

119892 (119911 119911) = (Ψ1(0) minus 119894Ψ

2(0)) ⟨119891

lowast(119898lowast(119905) 0) 119891

0⟩

≜ 11989220

1199112

2+ 11989211119911119911 + 119892

02

1199112

2+ 11989221

1199112119911

2+ sdot sdot sdot

(44)

We can use some easy computations to show that

⟨119891lowast(119898lowast(119905) 0) 119891

0⟩ =

1205910

8(119888111199112+ 119888121199112+ 11988813119911119911

119888211199112+ 119888221199112+ 11988823119911119911

)

+1205910

16(⟨11988801 1⟩

⟨11988802 1⟩) 1199112119911 + sdot sdot sdot

(45)

where

11988811= 11988712119908121198921015840

2(0) (minus119886

1+ 11988711119908111198921015840

1(0)) 120574119890

minus11989411990801205910

11988812= 11988712119908121198921015840

2(0) (minus119886

1+ 11988711119908111198921015840

1(0)) 120574119890

11989411990801205910

11988813= 11988712119908121198921015840

2(0) (minus119886

1+ 11988711119908111198921015840

1(0))

sdot (12057411989011989411990801205910 + 120574119890

minus11989411990801205910)

11988821= 11988721119908211198921015840

1(0) (minus119886

2+ 11988722119908221198921015840

1(0)) 120574119890

minus11989411990801205910

11988822= 11988721119908211198921015840

1(0) (minus119886

2+ 11988722119908221198921015840

2(0)) 120574119890

11989411990801205910

11988823= 11988721119908211198921015840

1(0) (minus119886

2+ 11988722119908221198921015840

2(0))

sdot (12057411989011989411990801205910 + 120574119890

minus11989411990801205910)

11988801= 11988712119908121198921015840

2(0) (minus119886

1+ 11988711119908111198921015840

1(0))

sdot (120574119882(1)

20(0) 11989011989411990801205910 +119882

(2)

20(minus1)) + 2119887

12119908121198921015840

2(0)

sdot (minus1198861+ 11988711119908111198921015840

1(0))

sdot (119882(2)

11(minus1) + 120574119882

(1)

11(0) 119890minus11989411990801205910)

11988802= 11988721119908211198921015840

1(0) (minus119886

2+ 11988722119908221198921015840

2(0))

sdot (120574119882(1)

20(minus1) + 119882

(2)

20(0) 11989011989411990801205910) + 2119887

21119908211198921015840

1(0)

sdot (minus1198862+ 11988722119908221198921015840

2(0))

sdot (120574119882(1)

11(minus1) + 119882

(2)

11(0) 119890minus11989411990801205910)

(46)

Setting (1205951 1205952) = Ψ

1(0) minus 119894Ψ

2(0) by calculating we get

that

11989220=1205910

4(119888111205951+ 119888211205952)

11989202=1205910

4(119888121205951+ 119888221205952)

11989211=1205910

8(119888131205951+ 119888231205952)

11989221=1205910

8(⟨11988801 1⟩ 1205951+ ⟨11988802 1⟩ 1205952)

(47)

Because there are 11988220(120579) and 119882

11(120579) in 119892

21where 120579 isin

[minus1 0] we still need to compute themBy [4] we know that

= 1198601205910119882+119867(119911 119911) (48)

6 Discrete Dynamics in Nature and Society

where

119867(119911 119911) = 11986720

1199112

2+ 11986711119911119911 + 119867

02

1199112

2+ sdot sdot sdot

= 1198830119891lowast(119898lowast(119905) 0)

minus Φ (Ψ ⟨1198830119891lowast(119898lowast(119905) 120572) 119891

0⟩) sdot 1198910

(49)

for119867119894119895isin 119876 with 119894 + 119895 = 2 It follows from (43) (48) and (49)

that

(1198601205910minus 2119894119908

01205910)11988220(120579) = minus119867

20(120579)

119860120591011988211(120579) = minus119867

11(120579)

(50)

By (49) we have that for 120579 isin [minus1 0)

119867 (119911 119911) = minus1

2[119892201199011(120579) + 119892

021199012(120579)] 1199112sdot 1198910

minus [119892111199011(120579) + 119892

111199012(120579)] 119911119911 sdot 119891

0+ sdot sdot sdot

(51)

Comparing the coefficients with (49) we get that for 120579 isin[minus1 0)

11986720(120579) = minus [119892

201199011(120579) + 119892

021199012(120579)] sdot 119891

0 (52)

11986711(120579) = minus [119892

111199011(120579) + 119892

111199012(120579)] sdot 119891

0 (53)

By (50) (52) and the definition of 1198601205910 we get that

20(120579) = 2119894119908

0120591011988220(120579) + [119892

201199011(120579) + 119892

021199012(120579)]

sdot 1198910

(54)

Noticing that 1199011(120579) = 119901

1(0)11989011989411990801205910120579 hence

11988220(120579)

= [11989411989220

11990801205910

1199011(120579) 11989011989411990801205910120579 +

11989411989202

311990801205910

1199012(120579) 119890minus11989411990801205910120579] sdot 119891

0

+ 1198641119890211989411990801205910120579

(55)

where 1198641= (119864(1)

1 119864(2)

1) isin 1198772 which is a constant vector

In a similar way by (50) and (53) we have that

11988211(120579)

= [minus11989411989211

11990801205910

1199011(0) 11989011989411990801205910120579 +

11989411989211

11990801205910

1199012(0) 119890minus11989411990801205910120579] sdot 119891

0

+ 1198642

(56)

where 1198642= (119864(1)

2 119864(2)

2) isin 1198772 which is also a constant vector

In what follows we seek appropriate 1198641and 119864

2 From the

definition of 1198601205910and (50) we can obtain that

21198941199080120591011988220(0) minus weierpΔ119882

20(0) minus Z (119864

lowast)11988220(120579)

= 11986720(0)

(57)

minus weierpΔ11988211(0) minus Z (119864

lowast)11988211(120579) = 119867

11(0) (58)

where

11986720(0) =

1205910

4(11988811

11988821

) minus [119892201199011(0) + 119892

021199012(0)] sdot 119891

0 (59)

11986711(0) =

1205910

8(11988813

11988823

) minus [119892111199011(0) + 119892

111199012(0)] sdot 119891

0 (60)

Substituting (55) and (59) into (57) we can obtain that

1198641=1

4(

211989411990801205910+ 1198861minus 11988711119908111198921015840

1(0) minus119887

12119908121198921015840

2(0) 119890minus211989411990801205910

minus11988721119908211198921015840

1(0) 119890minus211989411990801205910 2119894119908

01205910+ 1198862minus 11988722119908221198921015840

2(0)

)

minus1

(11988811

11988821

) (61)

In a similar way substituting (56) and (60) into (58) weobtain that

1198642

=1

8(

1198861minus 11988711119908111198921015840

1(0) minus119887

12119908121198921015840

2(0)

1198862minus 11988722119908221198921015840

2(0) minus119887

21119908211198921015840

1(0)

)

minus1

(11988813

11988823

)

(62)

Therefore we can compute the following values

1198881(0) =

119894

21199080

(1198921111989220minus 2100381610038161003816100381611989211

10038161003816100381610038162

minus

10038161003816100381610038161198920210038161003816100381610038162

3) +

11989221

2

1205832= minus

Re 1198881(0)

Re 1205821015840 (1205910)

1205732= 2Re 119888

1(0)

1198792= minus

Im 1198881(0) + 120583

2Im 1205821015840 (120591

0)

1199080

(63)

which determine the quantities of bifurcating periodic solu-tions in the center manifold at the critical value 120591

0 that is

1205832determines the direction of Hopf bifurcation the Hopf

bifurcation is supercritical (subcritical) if 1205832gt 0 (120583

2lt 0)

and the bifurcating periodic solutions exist for 120591 gt 1205910(120591 lt

Discrete Dynamics in Nature and Society 7

1205910) 1205732determines the stability of the bifurcating periodic

solutions if 1205732lt 0 (120573

2gt 0) the bifurcating periodic

solutions are stable (unstable) and 1198792determines the period

of the bifurcating periodic solutions the period increases(decrease) if 119879

2gt 0 (119879

2lt 0) [8ndash11]

4 Numerical Simulations

In this section in order to illustrate the results above we willgive two examples

Example 1 In system (2) we choose that 1198631= 1198632= 1 119886

1=

11988711= 04 119908

11= 06 119886

2= 03 119887

22= 11990822= 05 119887

12= 03

11988721= 06 119908

12= 24 119908

21= 36 119892

1(119909) = minus01tan(119909) and

1198922(119909) = arctan(119909) then

120597119906

120597119905= Δ119906 minus 04119906 (119905 119909) minus 004 tan (06119906 (119905 119909))

+ 03 arctan (24V (119905 minus 120591 119909))

120597V120597119905= ΔV minus 03V (119905 119909) minus 006 tan (36119906 (119905 minus 120591 119909))

+ 05 arctan (05V (119905 119909))

(64)

in which

0 lt 119909 lt 1

119905 gt 0

(65)

with initial and Neumann boundary conditions

120597119906

120597119897=120597V120597119897= 0 119905 ge 0 119909 = 0 1

119906 (119905 119909) = 05 (1 +119905

120587) sin (120587119909)

V (119905 119909) = (1 +119905

120587) sin (120587119909)

(119905 119909) isin [minus120591 0] times [0 1]

(66)

What should be remarked is that we choose the parametervalues stochastically under the condition 119902

2lt 0 in order to

ensure the existence of Hopf bifurcation at 119864lowast when 120591 = 1205910

So 1205910= 19371 and 119908

0= 02939 Then we can know

on the basis ofTheorem 1 that the trivial uniform steady state119864lowast= (0 0) is asymptotically stable when 0 le 120591 lt 120591

0 When

120591 gt 1205910 the steady state is unstable and a Hopf bifurcation is

arising from the steady state The numerical simulations inFigures 1 and 2 illustrate the facts

When 120591 = 1205910 we get that 119888

1(0) = minus00001 + 00022119894 then

we can acquire that 1205832gt 0 and 120573

2lt 0 Hence when 120591 passes

through 1205910to the right (120591 gt 120591

0) the bifurcation turns up and

the corresponding periodic orbits are orbitally asymptoticallystable

Example 2 In system (2) we choose that 1198631= 1198632= 001

11988721= 09 119886

2= 02 119887

12= 03 119887

11= 11988722= 11990822= 05 119886

1=

11990811= 06 119908

12= 25 119908

21= 36 119892

1(119909) = minus01tan(119909) and

1198922(119909) = arctan(119909) then

120597119906

120597119905= 001Δ119906 minus 06119906 (119905 119909) minus 005 tan (06119906 (119905 119909))

+ 03 arctan (25V (119905 minus 120591 119909))

120597V120597119905= 001ΔV minus 02V (119905 119909) minus 009 tan (36119906 (119905 minus 120591 119909))

+ 05 arctan (05V (119905 119909))

(67)

in which

0 lt 119909 lt 1

119905 gt 0

(68)

with initial and Dirichlet boundary conditions

119906 (119905 0) = 119906 (119905 1) = V (119905 0) = V (119905 1) = 0 119905 ge 0

119906 (119905 119909) = 05 (1 +119905

120587) sin (120587119909)

V (119905 119909) = (1 +119905

120587) sin (120587119909)

(119905 119909) isin [minus120591 0] times [0 1]

(69)

The similar Hopf bifurcation phenomenon is illustratedby the numerical simulations in Figures 3 and 4

5 Discussion and Research Perspective

This section is devoted to a summary of discussion andresearch perspective for the generalized reaction-diffusionneural networkmodelThemodel is based on the assumptionthat the signal transmission is of a digital (McCulloch-Pitts)nature the model then described a combination of analogand digital signal processing in the network [12] Dependingon the modeling approaches neural networks can be mod-eled either as a static neural network model or as a local fieldneural network model In order to let the problem be moregeneral in many aspects we build a generalized reaction-diffusion neural network model which includes reaction-diffusion local field neural networks and reaction-diffusionstatic neural networks For a delayed neural network animportant issue is the dynamical behaviors of the network[13] Thus there has been a large body of work discussing thestability and bifurcation in delayed neural network modelsBy analyzing the characteristic equation we discussed thelocal stability of the trivial uniform of system (2) [14] Itwas shown that when the delay 120591 varies the trivial uniformsteady state exchanges its stability and Hopf bifurcationsoccur Numerical simulations illustrated the occurrence ofthe bifurcate periodic solutions when the delay 120591 passes thecritical value 120591

0

A research perspective includes the problem of deter-mining the bifurcating periodic solutions and the stabilityand directions of the Hopf bifurcation using the normal

8 Discrete Dynamics in Nature and Society

x-axis

05

1

0

minus004

minus002

0

002

004

006

008

01

u-axis

1500 50 100

t-axis(a)

x-axis

05

1

0

100 1500 50

t-axis

minus003

minus002

minus001

0

001

002

003

v-axis

(b)

Figure 1 The temporal solution found by numerical integration of systems (64) and (66) with 120591 = 185 (a) 119906(119905 119909) and (b) V(119905 119909)

minus004

minus002

0

002

004

006

008

01

u-axis

x-axis05

1

50 100 1500

t-axis(a)

minus003

minus002

minus001

0

001

002

003

v-axis

x-axis05

1

00 50

t-axis100 150

(b)

Figure 2 The temporal solution found by numerical integration of systems (64) and (66) with 120591 = 425 (a) 119906(119905 119909) and (b) V(119905 119909)

0

50t-axis

100

150

x-axis

05

1

0

minus02

0

02

04

06

u-axis

(a)0

50t-axis

100

150

x-axis05

1

0

minus05

0

05

1

v-axis

(b)

Figure 3 The temporal solution found by numerical integration of systems (67) and (69) with 120591 = 125 (a) 119906(119905 119909) and (b) V(119905 119909)

Discrete Dynamics in Nature and Society 9

0

50t-axis

100

150

x-axis

05

1

0

minus02

0

02

04

06

u-axis

(a)0

50t-axis

100

150

x-axis

05

1

0

minus05

0

05

1

v-axis

(b)

Figure 4 The temporal solution found by numerical integration of systems (67) and (69) with 120591 = 205 (a) 119906(119905 119909) and (b) V(119905 119909)

form theory and the center manifold reaction A challengingperspective is the comparison of the generalized modelintroduced in the present paper with the experimentallymea-surable quantities Indeed the mathematical models shouldreproduce both qualitatively and quantitatively empiricaldata (see [4])

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (no 61305076) and the ScientificResearch Foundation for the Returned Overseas ChineseScholars State Education Ministry

References

[1] C Bianca M Ferrara and L Guerrini ldquoThe time delaysrsquo effectson the qualitative behavior of an economic growth modelrdquoAbstract and Applied Analysis vol 2013 Article ID 901014 10pages 2013

[2] LV Ballestra LGuerrini andG Pacelli ldquoStability switches andbifurcation analysis of a time delay model for the diffusion of anew technologyrdquo International Journal of Bifurcation amp Chaosvol 24 no 9 Article ID 1450113 2014

[3] Q Gan and R Xu ldquoStability and Hopf bifurcation of a delayedreaction-diffusion neural networkrdquo Mathematical Methods inthe Applied Sciences vol 34 no 12 pp 1450ndash1459 2011

[4] C Bianca and LGuerrini ldquoOn theDalgaard-Strulikmodel withlogistic population growth rate and delayed-carrying capacityrdquoActa Applicandae Mathematicae vol 128 pp 39ndash48 2013

[5] O G Jepps C Bianca and L Rondoni ldquoOnset of diffusivebehavior in confined transport systemsrdquo Chaos vol 18 no 1Article ID 013127 2008

[6] L Gori L Guerrini and M Sodini ldquoEquilibrium and dis-equilibrium dynamics in cobweb models with time delaysrdquo

International Journal of Bifurcation and Chaos vol 25 no 6Article ID 1550088 2015

[7] C Bianca L Guerrini and A Lemarchand ldquoExistence of solu-tions of a partial integrodifferential equation with thermostatand time delayrdquoAbstract and Applied Analysis vol 2014 ArticleID 463409 7 pages 2014

[8] Y Kuang ldquoDelay differential equations with applications inpopulation dynamicsrdquo Discrete amp Continuous Dynamical Sys-tems vol 33 no 4 pp 1633ndash1644 2013

[9] C Huang Y He L Huang and Y Zhaohui ldquoHopf bifurcationanalysis of two neurons with three delaysrdquo Nonlinear AnalysisReal World Applications vol 8 no 3 pp 903ndash921 2007

[10] C Huang L Huang J Feng M Nai and Y He ldquoHopfbifurcation analysis for a two-neuron networkwith four delaysrdquoChaos Solitons amp Fractals vol 34 no 3 pp 795ndash812 2007

[11] Y Song M Han and J Wei ldquoStability and Hopf bifurcationanalysis on a simplified BAM neural network with delaysrdquoPhysica D Nonlinear Phenomena vol 200 no 3-4 pp 185ndash2042005

[12] X-P Yan ldquoHopf bifurcation and stability for a delayed tri-neuron network modelrdquo Journal of Computational and AppliedMathematics vol 196 no 2 pp 579ndash595 2006

[13] H Zhao and L Wang ldquoHopf bifurcation in Cohen-Grossbergneural network with distributed delaysrdquo Nonlinear AnalysisReal World Applications vol 8 no 1 pp 73ndash89 2007

[14] B Zheng Y Zhang and C Zhang ldquoStability and bifurcationof a discrete BAM neural network model with delaysrdquo ChaosSolitons amp Fractals vol 36 no 3 pp 612ndash616 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Discrete Dynamics in Nature and Society

3 Direction and Stability of Hopf Bifurcation

In Section 2 we have demonstrated that systems (2) and (3)undergo a train of periodic solutions bifurcating from thetrivial uniform steady state 119864lowast at the critical value of 120591 Inthis section we derive explicit formulae to determine theproperties of the Hopf bifurcation at critical value 120591

0by using

the normal form theory and center manifold reduction forPFDEs In this section we also let the condition (1198671) holdand 119902

1lt 0 And the work of Bianca and Guerrini in papers

[4ndash7] is the founder of the method in this section

Set 120591 = 120572 + 1205910 We first should normalize the delay 120591 by

the time-scaling 119905 rarr 119905120591 Then (2) can be rewritten in thefixed phase space ℓlowast = 119862([minus1 0] 119883) as

(119905) = 1205910weierpΔ119898 (119905) + 120591

0Z (119864lowast) (119898 (119905))

+ 119891lowast(119898 (119905) 120572)

(24)

where 119891lowastℓlowast times 119877+ rarr 1198772 is defined by

119891lowast(120601 120572) = 120572weierpΔ120601 (0) + 120591

0Z (119864lowast) (120601) + (120591

0+ 120572)(

1

2119887121199082

1211989210158401015840

2(0) 1206012

2(minus1) +

1

3119887121199083

12119892101584010158401015840

2(0) 1206013

2(minus1) + sdot sdot sdot

1

2119887211199082

2111989210158401015840

1(0) 1206012

1(minus1) +

1

3119887211199083

21119892101584010158401015840

1(0) 1206013

1(minus1) + sdot sdot sdot

) (25)

where 120601 = (1206011 1206012)119879isin ℓlowast

By the discussion in Section 2 we can know that theorigin (0 0) is a steady state of (24) and Λ

0= minus119894120596

01205910 11989412059601205910

are a pair of simple purely imaginary eigenvalues of the linearequation

(119905) = 1205910weierpΔ119898 (119905) + 120591

0Z (119864lowast) (119898 (119905)) (26)

and the functional differential equation

(119905) = 1205910Z (119911119905) (27)

On the basis of the Riesz representation theorem thereexists a function 120578(120579 120591) of bounded variation for 120579 isin [minus1 0]such that

Z (119864lowast) (120593) =

1

1205910

int

0

minus1

119889120578 (120579 1205910) 120593 (120579) where 120593 isin C (28)

Here we choose that

120578 (120579 1205910)

= 1205910(

minus1198861+ 11988711119908111198921015840

1(0) 0

0 minus1198862+ 11988722119908221198921015840

2(0)

)120575 (120579)

minus 1205910(

0 11988712119908121198921015840

2(0)

11988721119908211198921015840

1(0) 0

)120575 (120579 + 1)

(29)

where 120575 is the Dirac delta functionLet 119860(120591

0) denote the infinitesimal generator of the semi-

group induced by the solutions of (27) and let 119860lowast be theformal adjoint of 119860(120591

0) under the bilinear pairing

⟨120595 (119904) 120593 (120579)⟩ = 120595 (0) 120593 (0)

minus int

0

minus1

int

120579

120585=0

120595 (120585 minus 120579) 119889120578 (120579) 120593 (120585) 119889120585

(30)

where 120593 isin 1198621([minus1 0] 119877

2) 120595 isin 119862

1([0 1] (119877

2)lowast) 120578(120579) =

120578(120579 1205910) Then 119860(120591

0) and 119860lowast are a pair of adjoint operators

By the discussions in Section 2 we can realize that 119860(1205910)

has a pair of simple purely imaginary eigenvalues plusmn11989411990801205910and

they are also eigenvalues of119860lowast since119860(1205910) and119860lowast are adjoint

operators Let 119875 and 119875lowast be the center spaces of 119860(1205910) and

119860lowast associated with Λ

0 respectively Hence 119875lowast is the adjoint

space of 119875 and dim119875 = dim119875lowast = 2Let

120574 =11988721119908211198921015840

1(0) 119890minus11989411990801205910

1198862+ 1198941199080minus 119887221199082211989210158402(0)

120581 =11988712119908121198921015840

2(0) 11989011989411990801205910

1198862minus 1198941199080minus 119887221199082211989210158402(0)

(31)

then

1199011(120579) = 119890

11989411990801205910120579 (1 120574)119879

1199012(120579) = 119901

1(120579)

minus 1 le 120579 le 0

(32)

is a basis of 119875 associated with Λ0and

1199021(119904) = (1 120581)

119879119890minus11989411990801205910119904

1199022(119904) = 119902

1(119904)

0 le 119904 le 1

(33)

is a basis of 119876 associated with Λ0

Let Φ = (Φ1 Φ2) where

Φ1(120579) =

1199011(120579) + 119901

2(120579)

2

Φ2(120579) =

1199011(120579) minus 119901

2(120579)

2119894

(34)

Discrete Dynamics in Nature and Society 5

for 120579 isin [minus1 0] and let Ψlowast = (Ψlowast1 Ψlowast

2)119879 where

Ψlowast

1(119904) =

1199021(119904) + 119902

2(119904)

2

Ψlowast

2(119904) =

1199021(119904) minus 119902

2(119904)

2119894

(35)

for 119904 isin [minus1 0]Now we define that (Ψlowast Φ) = (Ψlowast

119895 Φ119896) (119895 119896 = 1 2) and

construct a new basis Ψ for 119876 by

Ψ = (Ψlowast

1 Ψ2)119879

= (Ψlowast Φ)minus1

Ψlowast (36)

Hence (ΨΦ) = 1198682 which is the second-order identity

matrix Moreover we define 1198910for 1198910= (1205731

0 1205732

0) and 119888 sdot 119891

0=

11988801205731

0+ 11988821205732

0for 119888 = (119888

1 1198882)119879isin 119862 Then the center space of

linear equation (26) is given by 119875119862119873ℓlowast where

119875119862119873ℓlowast= Φ (Ψ ⟨120601 119891

0⟩) sdot 1198910 120601 isin ℓ

lowast (37)

and ℓlowast denotes the complementary subspace of119875119862119873ℓlowast where

ℓlowast= 119875119862119873ℓlowastoplus 119876 (38)

Let 1198601205910be defined by

1198601205910120601 (120579)

= 120601 (120579)

+ 1198830(120579) [weierpΔ120601 (0) + 120591

0Z (119864lowast) (120601 (120579)) minus 120601 (0)]

120601 isin ℓlowast

(39)

where1198830 [minus1 0] rarr 119861(119883119883) is given by

1198830(120579) =

0 minus1 le 120579 lt 0

119868 120579 = 0

(40)

Then we have rewritten system (24) and it can berewritten as follows

(119905) = 1198601205910119898(119905) + 119883

0119891lowast(119898 (119905) 120572) (41)

The solution of (24) on the center manifold is given by

119898lowast(119905) = Φ (119909

1 1199092)119879

sdot 1198910+119882(119909

1 1199092 120572) (42)

Letting 119911 = 1199091minus1198941199092119882 = 119882

20(11991122)+119882

11119911119911+119882

02(11991122)+

sdot sdot sdot then

= 11989411990801205910119911 + 119892 (119911 119911) (43)

where

119892 (119911 119911) = (Ψ1(0) minus 119894Ψ

2(0)) ⟨119891

lowast(119898lowast(119905) 0) 119891

0⟩

≜ 11989220

1199112

2+ 11989211119911119911 + 119892

02

1199112

2+ 11989221

1199112119911

2+ sdot sdot sdot

(44)

We can use some easy computations to show that

⟨119891lowast(119898lowast(119905) 0) 119891

0⟩ =

1205910

8(119888111199112+ 119888121199112+ 11988813119911119911

119888211199112+ 119888221199112+ 11988823119911119911

)

+1205910

16(⟨11988801 1⟩

⟨11988802 1⟩) 1199112119911 + sdot sdot sdot

(45)

where

11988811= 11988712119908121198921015840

2(0) (minus119886

1+ 11988711119908111198921015840

1(0)) 120574119890

minus11989411990801205910

11988812= 11988712119908121198921015840

2(0) (minus119886

1+ 11988711119908111198921015840

1(0)) 120574119890

11989411990801205910

11988813= 11988712119908121198921015840

2(0) (minus119886

1+ 11988711119908111198921015840

1(0))

sdot (12057411989011989411990801205910 + 120574119890

minus11989411990801205910)

11988821= 11988721119908211198921015840

1(0) (minus119886

2+ 11988722119908221198921015840

1(0)) 120574119890

minus11989411990801205910

11988822= 11988721119908211198921015840

1(0) (minus119886

2+ 11988722119908221198921015840

2(0)) 120574119890

11989411990801205910

11988823= 11988721119908211198921015840

1(0) (minus119886

2+ 11988722119908221198921015840

2(0))

sdot (12057411989011989411990801205910 + 120574119890

minus11989411990801205910)

11988801= 11988712119908121198921015840

2(0) (minus119886

1+ 11988711119908111198921015840

1(0))

sdot (120574119882(1)

20(0) 11989011989411990801205910 +119882

(2)

20(minus1)) + 2119887

12119908121198921015840

2(0)

sdot (minus1198861+ 11988711119908111198921015840

1(0))

sdot (119882(2)

11(minus1) + 120574119882

(1)

11(0) 119890minus11989411990801205910)

11988802= 11988721119908211198921015840

1(0) (minus119886

2+ 11988722119908221198921015840

2(0))

sdot (120574119882(1)

20(minus1) + 119882

(2)

20(0) 11989011989411990801205910) + 2119887

21119908211198921015840

1(0)

sdot (minus1198862+ 11988722119908221198921015840

2(0))

sdot (120574119882(1)

11(minus1) + 119882

(2)

11(0) 119890minus11989411990801205910)

(46)

Setting (1205951 1205952) = Ψ

1(0) minus 119894Ψ

2(0) by calculating we get

that

11989220=1205910

4(119888111205951+ 119888211205952)

11989202=1205910

4(119888121205951+ 119888221205952)

11989211=1205910

8(119888131205951+ 119888231205952)

11989221=1205910

8(⟨11988801 1⟩ 1205951+ ⟨11988802 1⟩ 1205952)

(47)

Because there are 11988220(120579) and 119882

11(120579) in 119892

21where 120579 isin

[minus1 0] we still need to compute themBy [4] we know that

= 1198601205910119882+119867(119911 119911) (48)

6 Discrete Dynamics in Nature and Society

where

119867(119911 119911) = 11986720

1199112

2+ 11986711119911119911 + 119867

02

1199112

2+ sdot sdot sdot

= 1198830119891lowast(119898lowast(119905) 0)

minus Φ (Ψ ⟨1198830119891lowast(119898lowast(119905) 120572) 119891

0⟩) sdot 1198910

(49)

for119867119894119895isin 119876 with 119894 + 119895 = 2 It follows from (43) (48) and (49)

that

(1198601205910minus 2119894119908

01205910)11988220(120579) = minus119867

20(120579)

119860120591011988211(120579) = minus119867

11(120579)

(50)

By (49) we have that for 120579 isin [minus1 0)

119867 (119911 119911) = minus1

2[119892201199011(120579) + 119892

021199012(120579)] 1199112sdot 1198910

minus [119892111199011(120579) + 119892

111199012(120579)] 119911119911 sdot 119891

0+ sdot sdot sdot

(51)

Comparing the coefficients with (49) we get that for 120579 isin[minus1 0)

11986720(120579) = minus [119892

201199011(120579) + 119892

021199012(120579)] sdot 119891

0 (52)

11986711(120579) = minus [119892

111199011(120579) + 119892

111199012(120579)] sdot 119891

0 (53)

By (50) (52) and the definition of 1198601205910 we get that

20(120579) = 2119894119908

0120591011988220(120579) + [119892

201199011(120579) + 119892

021199012(120579)]

sdot 1198910

(54)

Noticing that 1199011(120579) = 119901

1(0)11989011989411990801205910120579 hence

11988220(120579)

= [11989411989220

11990801205910

1199011(120579) 11989011989411990801205910120579 +

11989411989202

311990801205910

1199012(120579) 119890minus11989411990801205910120579] sdot 119891

0

+ 1198641119890211989411990801205910120579

(55)

where 1198641= (119864(1)

1 119864(2)

1) isin 1198772 which is a constant vector

In a similar way by (50) and (53) we have that

11988211(120579)

= [minus11989411989211

11990801205910

1199011(0) 11989011989411990801205910120579 +

11989411989211

11990801205910

1199012(0) 119890minus11989411990801205910120579] sdot 119891

0

+ 1198642

(56)

where 1198642= (119864(1)

2 119864(2)

2) isin 1198772 which is also a constant vector

In what follows we seek appropriate 1198641and 119864

2 From the

definition of 1198601205910and (50) we can obtain that

21198941199080120591011988220(0) minus weierpΔ119882

20(0) minus Z (119864

lowast)11988220(120579)

= 11986720(0)

(57)

minus weierpΔ11988211(0) minus Z (119864

lowast)11988211(120579) = 119867

11(0) (58)

where

11986720(0) =

1205910

4(11988811

11988821

) minus [119892201199011(0) + 119892

021199012(0)] sdot 119891

0 (59)

11986711(0) =

1205910

8(11988813

11988823

) minus [119892111199011(0) + 119892

111199012(0)] sdot 119891

0 (60)

Substituting (55) and (59) into (57) we can obtain that

1198641=1

4(

211989411990801205910+ 1198861minus 11988711119908111198921015840

1(0) minus119887

12119908121198921015840

2(0) 119890minus211989411990801205910

minus11988721119908211198921015840

1(0) 119890minus211989411990801205910 2119894119908

01205910+ 1198862minus 11988722119908221198921015840

2(0)

)

minus1

(11988811

11988821

) (61)

In a similar way substituting (56) and (60) into (58) weobtain that

1198642

=1

8(

1198861minus 11988711119908111198921015840

1(0) minus119887

12119908121198921015840

2(0)

1198862minus 11988722119908221198921015840

2(0) minus119887

21119908211198921015840

1(0)

)

minus1

(11988813

11988823

)

(62)

Therefore we can compute the following values

1198881(0) =

119894

21199080

(1198921111989220minus 2100381610038161003816100381611989211

10038161003816100381610038162

minus

10038161003816100381610038161198920210038161003816100381610038162

3) +

11989221

2

1205832= minus

Re 1198881(0)

Re 1205821015840 (1205910)

1205732= 2Re 119888

1(0)

1198792= minus

Im 1198881(0) + 120583

2Im 1205821015840 (120591

0)

1199080

(63)

which determine the quantities of bifurcating periodic solu-tions in the center manifold at the critical value 120591

0 that is

1205832determines the direction of Hopf bifurcation the Hopf

bifurcation is supercritical (subcritical) if 1205832gt 0 (120583

2lt 0)

and the bifurcating periodic solutions exist for 120591 gt 1205910(120591 lt

Discrete Dynamics in Nature and Society 7

1205910) 1205732determines the stability of the bifurcating periodic

solutions if 1205732lt 0 (120573

2gt 0) the bifurcating periodic

solutions are stable (unstable) and 1198792determines the period

of the bifurcating periodic solutions the period increases(decrease) if 119879

2gt 0 (119879

2lt 0) [8ndash11]

4 Numerical Simulations

In this section in order to illustrate the results above we willgive two examples

Example 1 In system (2) we choose that 1198631= 1198632= 1 119886

1=

11988711= 04 119908

11= 06 119886

2= 03 119887

22= 11990822= 05 119887

12= 03

11988721= 06 119908

12= 24 119908

21= 36 119892

1(119909) = minus01tan(119909) and

1198922(119909) = arctan(119909) then

120597119906

120597119905= Δ119906 minus 04119906 (119905 119909) minus 004 tan (06119906 (119905 119909))

+ 03 arctan (24V (119905 minus 120591 119909))

120597V120597119905= ΔV minus 03V (119905 119909) minus 006 tan (36119906 (119905 minus 120591 119909))

+ 05 arctan (05V (119905 119909))

(64)

in which

0 lt 119909 lt 1

119905 gt 0

(65)

with initial and Neumann boundary conditions

120597119906

120597119897=120597V120597119897= 0 119905 ge 0 119909 = 0 1

119906 (119905 119909) = 05 (1 +119905

120587) sin (120587119909)

V (119905 119909) = (1 +119905

120587) sin (120587119909)

(119905 119909) isin [minus120591 0] times [0 1]

(66)

What should be remarked is that we choose the parametervalues stochastically under the condition 119902

2lt 0 in order to

ensure the existence of Hopf bifurcation at 119864lowast when 120591 = 1205910

So 1205910= 19371 and 119908

0= 02939 Then we can know

on the basis ofTheorem 1 that the trivial uniform steady state119864lowast= (0 0) is asymptotically stable when 0 le 120591 lt 120591

0 When

120591 gt 1205910 the steady state is unstable and a Hopf bifurcation is

arising from the steady state The numerical simulations inFigures 1 and 2 illustrate the facts

When 120591 = 1205910 we get that 119888

1(0) = minus00001 + 00022119894 then

we can acquire that 1205832gt 0 and 120573

2lt 0 Hence when 120591 passes

through 1205910to the right (120591 gt 120591

0) the bifurcation turns up and

the corresponding periodic orbits are orbitally asymptoticallystable

Example 2 In system (2) we choose that 1198631= 1198632= 001

11988721= 09 119886

2= 02 119887

12= 03 119887

11= 11988722= 11990822= 05 119886

1=

11990811= 06 119908

12= 25 119908

21= 36 119892

1(119909) = minus01tan(119909) and

1198922(119909) = arctan(119909) then

120597119906

120597119905= 001Δ119906 minus 06119906 (119905 119909) minus 005 tan (06119906 (119905 119909))

+ 03 arctan (25V (119905 minus 120591 119909))

120597V120597119905= 001ΔV minus 02V (119905 119909) minus 009 tan (36119906 (119905 minus 120591 119909))

+ 05 arctan (05V (119905 119909))

(67)

in which

0 lt 119909 lt 1

119905 gt 0

(68)

with initial and Dirichlet boundary conditions

119906 (119905 0) = 119906 (119905 1) = V (119905 0) = V (119905 1) = 0 119905 ge 0

119906 (119905 119909) = 05 (1 +119905

120587) sin (120587119909)

V (119905 119909) = (1 +119905

120587) sin (120587119909)

(119905 119909) isin [minus120591 0] times [0 1]

(69)

The similar Hopf bifurcation phenomenon is illustratedby the numerical simulations in Figures 3 and 4

5 Discussion and Research Perspective

This section is devoted to a summary of discussion andresearch perspective for the generalized reaction-diffusionneural networkmodelThemodel is based on the assumptionthat the signal transmission is of a digital (McCulloch-Pitts)nature the model then described a combination of analogand digital signal processing in the network [12] Dependingon the modeling approaches neural networks can be mod-eled either as a static neural network model or as a local fieldneural network model In order to let the problem be moregeneral in many aspects we build a generalized reaction-diffusion neural network model which includes reaction-diffusion local field neural networks and reaction-diffusionstatic neural networks For a delayed neural network animportant issue is the dynamical behaviors of the network[13] Thus there has been a large body of work discussing thestability and bifurcation in delayed neural network modelsBy analyzing the characteristic equation we discussed thelocal stability of the trivial uniform of system (2) [14] Itwas shown that when the delay 120591 varies the trivial uniformsteady state exchanges its stability and Hopf bifurcationsoccur Numerical simulations illustrated the occurrence ofthe bifurcate periodic solutions when the delay 120591 passes thecritical value 120591

0

A research perspective includes the problem of deter-mining the bifurcating periodic solutions and the stabilityand directions of the Hopf bifurcation using the normal

8 Discrete Dynamics in Nature and Society

x-axis

05

1

0

minus004

minus002

0

002

004

006

008

01

u-axis

1500 50 100

t-axis(a)

x-axis

05

1

0

100 1500 50

t-axis

minus003

minus002

minus001

0

001

002

003

v-axis

(b)

Figure 1 The temporal solution found by numerical integration of systems (64) and (66) with 120591 = 185 (a) 119906(119905 119909) and (b) V(119905 119909)

minus004

minus002

0

002

004

006

008

01

u-axis

x-axis05

1

50 100 1500

t-axis(a)

minus003

minus002

minus001

0

001

002

003

v-axis

x-axis05

1

00 50

t-axis100 150

(b)

Figure 2 The temporal solution found by numerical integration of systems (64) and (66) with 120591 = 425 (a) 119906(119905 119909) and (b) V(119905 119909)

0

50t-axis

100

150

x-axis

05

1

0

minus02

0

02

04

06

u-axis

(a)0

50t-axis

100

150

x-axis05

1

0

minus05

0

05

1

v-axis

(b)

Figure 3 The temporal solution found by numerical integration of systems (67) and (69) with 120591 = 125 (a) 119906(119905 119909) and (b) V(119905 119909)

Discrete Dynamics in Nature and Society 9

0

50t-axis

100

150

x-axis

05

1

0

minus02

0

02

04

06

u-axis

(a)0

50t-axis

100

150

x-axis

05

1

0

minus05

0

05

1

v-axis

(b)

Figure 4 The temporal solution found by numerical integration of systems (67) and (69) with 120591 = 205 (a) 119906(119905 119909) and (b) V(119905 119909)

form theory and the center manifold reaction A challengingperspective is the comparison of the generalized modelintroduced in the present paper with the experimentallymea-surable quantities Indeed the mathematical models shouldreproduce both qualitatively and quantitatively empiricaldata (see [4])

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (no 61305076) and the ScientificResearch Foundation for the Returned Overseas ChineseScholars State Education Ministry

References

[1] C Bianca M Ferrara and L Guerrini ldquoThe time delaysrsquo effectson the qualitative behavior of an economic growth modelrdquoAbstract and Applied Analysis vol 2013 Article ID 901014 10pages 2013

[2] LV Ballestra LGuerrini andG Pacelli ldquoStability switches andbifurcation analysis of a time delay model for the diffusion of anew technologyrdquo International Journal of Bifurcation amp Chaosvol 24 no 9 Article ID 1450113 2014

[3] Q Gan and R Xu ldquoStability and Hopf bifurcation of a delayedreaction-diffusion neural networkrdquo Mathematical Methods inthe Applied Sciences vol 34 no 12 pp 1450ndash1459 2011

[4] C Bianca and LGuerrini ldquoOn theDalgaard-Strulikmodel withlogistic population growth rate and delayed-carrying capacityrdquoActa Applicandae Mathematicae vol 128 pp 39ndash48 2013

[5] O G Jepps C Bianca and L Rondoni ldquoOnset of diffusivebehavior in confined transport systemsrdquo Chaos vol 18 no 1Article ID 013127 2008

[6] L Gori L Guerrini and M Sodini ldquoEquilibrium and dis-equilibrium dynamics in cobweb models with time delaysrdquo

International Journal of Bifurcation and Chaos vol 25 no 6Article ID 1550088 2015

[7] C Bianca L Guerrini and A Lemarchand ldquoExistence of solu-tions of a partial integrodifferential equation with thermostatand time delayrdquoAbstract and Applied Analysis vol 2014 ArticleID 463409 7 pages 2014

[8] Y Kuang ldquoDelay differential equations with applications inpopulation dynamicsrdquo Discrete amp Continuous Dynamical Sys-tems vol 33 no 4 pp 1633ndash1644 2013

[9] C Huang Y He L Huang and Y Zhaohui ldquoHopf bifurcationanalysis of two neurons with three delaysrdquo Nonlinear AnalysisReal World Applications vol 8 no 3 pp 903ndash921 2007

[10] C Huang L Huang J Feng M Nai and Y He ldquoHopfbifurcation analysis for a two-neuron networkwith four delaysrdquoChaos Solitons amp Fractals vol 34 no 3 pp 795ndash812 2007

[11] Y Song M Han and J Wei ldquoStability and Hopf bifurcationanalysis on a simplified BAM neural network with delaysrdquoPhysica D Nonlinear Phenomena vol 200 no 3-4 pp 185ndash2042005

[12] X-P Yan ldquoHopf bifurcation and stability for a delayed tri-neuron network modelrdquo Journal of Computational and AppliedMathematics vol 196 no 2 pp 579ndash595 2006

[13] H Zhao and L Wang ldquoHopf bifurcation in Cohen-Grossbergneural network with distributed delaysrdquo Nonlinear AnalysisReal World Applications vol 8 no 1 pp 73ndash89 2007

[14] B Zheng Y Zhang and C Zhang ldquoStability and bifurcationof a discrete BAM neural network model with delaysrdquo ChaosSolitons amp Fractals vol 36 no 3 pp 612ndash616 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 5

for 120579 isin [minus1 0] and let Ψlowast = (Ψlowast1 Ψlowast

2)119879 where

Ψlowast

1(119904) =

1199021(119904) + 119902

2(119904)

2

Ψlowast

2(119904) =

1199021(119904) minus 119902

2(119904)

2119894

(35)

for 119904 isin [minus1 0]Now we define that (Ψlowast Φ) = (Ψlowast

119895 Φ119896) (119895 119896 = 1 2) and

construct a new basis Ψ for 119876 by

Ψ = (Ψlowast

1 Ψ2)119879

= (Ψlowast Φ)minus1

Ψlowast (36)

Hence (ΨΦ) = 1198682 which is the second-order identity

matrix Moreover we define 1198910for 1198910= (1205731

0 1205732

0) and 119888 sdot 119891

0=

11988801205731

0+ 11988821205732

0for 119888 = (119888

1 1198882)119879isin 119862 Then the center space of

linear equation (26) is given by 119875119862119873ℓlowast where

119875119862119873ℓlowast= Φ (Ψ ⟨120601 119891

0⟩) sdot 1198910 120601 isin ℓ

lowast (37)

and ℓlowast denotes the complementary subspace of119875119862119873ℓlowast where

ℓlowast= 119875119862119873ℓlowastoplus 119876 (38)

Let 1198601205910be defined by

1198601205910120601 (120579)

= 120601 (120579)

+ 1198830(120579) [weierpΔ120601 (0) + 120591

0Z (119864lowast) (120601 (120579)) minus 120601 (0)]

120601 isin ℓlowast

(39)

where1198830 [minus1 0] rarr 119861(119883119883) is given by

1198830(120579) =

0 minus1 le 120579 lt 0

119868 120579 = 0

(40)

Then we have rewritten system (24) and it can berewritten as follows

(119905) = 1198601205910119898(119905) + 119883

0119891lowast(119898 (119905) 120572) (41)

The solution of (24) on the center manifold is given by

119898lowast(119905) = Φ (119909

1 1199092)119879

sdot 1198910+119882(119909

1 1199092 120572) (42)

Letting 119911 = 1199091minus1198941199092119882 = 119882

20(11991122)+119882

11119911119911+119882

02(11991122)+

sdot sdot sdot then

= 11989411990801205910119911 + 119892 (119911 119911) (43)

where

119892 (119911 119911) = (Ψ1(0) minus 119894Ψ

2(0)) ⟨119891

lowast(119898lowast(119905) 0) 119891

0⟩

≜ 11989220

1199112

2+ 11989211119911119911 + 119892

02

1199112

2+ 11989221

1199112119911

2+ sdot sdot sdot

(44)

We can use some easy computations to show that

⟨119891lowast(119898lowast(119905) 0) 119891

0⟩ =

1205910

8(119888111199112+ 119888121199112+ 11988813119911119911

119888211199112+ 119888221199112+ 11988823119911119911

)

+1205910

16(⟨11988801 1⟩

⟨11988802 1⟩) 1199112119911 + sdot sdot sdot

(45)

where

11988811= 11988712119908121198921015840

2(0) (minus119886

1+ 11988711119908111198921015840

1(0)) 120574119890

minus11989411990801205910

11988812= 11988712119908121198921015840

2(0) (minus119886

1+ 11988711119908111198921015840

1(0)) 120574119890

11989411990801205910

11988813= 11988712119908121198921015840

2(0) (minus119886

1+ 11988711119908111198921015840

1(0))

sdot (12057411989011989411990801205910 + 120574119890

minus11989411990801205910)

11988821= 11988721119908211198921015840

1(0) (minus119886

2+ 11988722119908221198921015840

1(0)) 120574119890

minus11989411990801205910

11988822= 11988721119908211198921015840

1(0) (minus119886

2+ 11988722119908221198921015840

2(0)) 120574119890

11989411990801205910

11988823= 11988721119908211198921015840

1(0) (minus119886

2+ 11988722119908221198921015840

2(0))

sdot (12057411989011989411990801205910 + 120574119890

minus11989411990801205910)

11988801= 11988712119908121198921015840

2(0) (minus119886

1+ 11988711119908111198921015840

1(0))

sdot (120574119882(1)

20(0) 11989011989411990801205910 +119882

(2)

20(minus1)) + 2119887

12119908121198921015840

2(0)

sdot (minus1198861+ 11988711119908111198921015840

1(0))

sdot (119882(2)

11(minus1) + 120574119882

(1)

11(0) 119890minus11989411990801205910)

11988802= 11988721119908211198921015840

1(0) (minus119886

2+ 11988722119908221198921015840

2(0))

sdot (120574119882(1)

20(minus1) + 119882

(2)

20(0) 11989011989411990801205910) + 2119887

21119908211198921015840

1(0)

sdot (minus1198862+ 11988722119908221198921015840

2(0))

sdot (120574119882(1)

11(minus1) + 119882

(2)

11(0) 119890minus11989411990801205910)

(46)

Setting (1205951 1205952) = Ψ

1(0) minus 119894Ψ

2(0) by calculating we get

that

11989220=1205910

4(119888111205951+ 119888211205952)

11989202=1205910

4(119888121205951+ 119888221205952)

11989211=1205910

8(119888131205951+ 119888231205952)

11989221=1205910

8(⟨11988801 1⟩ 1205951+ ⟨11988802 1⟩ 1205952)

(47)

Because there are 11988220(120579) and 119882

11(120579) in 119892

21where 120579 isin

[minus1 0] we still need to compute themBy [4] we know that

= 1198601205910119882+119867(119911 119911) (48)

6 Discrete Dynamics in Nature and Society

where

119867(119911 119911) = 11986720

1199112

2+ 11986711119911119911 + 119867

02

1199112

2+ sdot sdot sdot

= 1198830119891lowast(119898lowast(119905) 0)

minus Φ (Ψ ⟨1198830119891lowast(119898lowast(119905) 120572) 119891

0⟩) sdot 1198910

(49)

for119867119894119895isin 119876 with 119894 + 119895 = 2 It follows from (43) (48) and (49)

that

(1198601205910minus 2119894119908

01205910)11988220(120579) = minus119867

20(120579)

119860120591011988211(120579) = minus119867

11(120579)

(50)

By (49) we have that for 120579 isin [minus1 0)

119867 (119911 119911) = minus1

2[119892201199011(120579) + 119892

021199012(120579)] 1199112sdot 1198910

minus [119892111199011(120579) + 119892

111199012(120579)] 119911119911 sdot 119891

0+ sdot sdot sdot

(51)

Comparing the coefficients with (49) we get that for 120579 isin[minus1 0)

11986720(120579) = minus [119892

201199011(120579) + 119892

021199012(120579)] sdot 119891

0 (52)

11986711(120579) = minus [119892

111199011(120579) + 119892

111199012(120579)] sdot 119891

0 (53)

By (50) (52) and the definition of 1198601205910 we get that

20(120579) = 2119894119908

0120591011988220(120579) + [119892

201199011(120579) + 119892

021199012(120579)]

sdot 1198910

(54)

Noticing that 1199011(120579) = 119901

1(0)11989011989411990801205910120579 hence

11988220(120579)

= [11989411989220

11990801205910

1199011(120579) 11989011989411990801205910120579 +

11989411989202

311990801205910

1199012(120579) 119890minus11989411990801205910120579] sdot 119891

0

+ 1198641119890211989411990801205910120579

(55)

where 1198641= (119864(1)

1 119864(2)

1) isin 1198772 which is a constant vector

In a similar way by (50) and (53) we have that

11988211(120579)

= [minus11989411989211

11990801205910

1199011(0) 11989011989411990801205910120579 +

11989411989211

11990801205910

1199012(0) 119890minus11989411990801205910120579] sdot 119891

0

+ 1198642

(56)

where 1198642= (119864(1)

2 119864(2)

2) isin 1198772 which is also a constant vector

In what follows we seek appropriate 1198641and 119864

2 From the

definition of 1198601205910and (50) we can obtain that

21198941199080120591011988220(0) minus weierpΔ119882

20(0) minus Z (119864

lowast)11988220(120579)

= 11986720(0)

(57)

minus weierpΔ11988211(0) minus Z (119864

lowast)11988211(120579) = 119867

11(0) (58)

where

11986720(0) =

1205910

4(11988811

11988821

) minus [119892201199011(0) + 119892

021199012(0)] sdot 119891

0 (59)

11986711(0) =

1205910

8(11988813

11988823

) minus [119892111199011(0) + 119892

111199012(0)] sdot 119891

0 (60)

Substituting (55) and (59) into (57) we can obtain that

1198641=1

4(

211989411990801205910+ 1198861minus 11988711119908111198921015840

1(0) minus119887

12119908121198921015840

2(0) 119890minus211989411990801205910

minus11988721119908211198921015840

1(0) 119890minus211989411990801205910 2119894119908

01205910+ 1198862minus 11988722119908221198921015840

2(0)

)

minus1

(11988811

11988821

) (61)

In a similar way substituting (56) and (60) into (58) weobtain that

1198642

=1

8(

1198861minus 11988711119908111198921015840

1(0) minus119887

12119908121198921015840

2(0)

1198862minus 11988722119908221198921015840

2(0) minus119887

21119908211198921015840

1(0)

)

minus1

(11988813

11988823

)

(62)

Therefore we can compute the following values

1198881(0) =

119894

21199080

(1198921111989220minus 2100381610038161003816100381611989211

10038161003816100381610038162

minus

10038161003816100381610038161198920210038161003816100381610038162

3) +

11989221

2

1205832= minus

Re 1198881(0)

Re 1205821015840 (1205910)

1205732= 2Re 119888

1(0)

1198792= minus

Im 1198881(0) + 120583

2Im 1205821015840 (120591

0)

1199080

(63)

which determine the quantities of bifurcating periodic solu-tions in the center manifold at the critical value 120591

0 that is

1205832determines the direction of Hopf bifurcation the Hopf

bifurcation is supercritical (subcritical) if 1205832gt 0 (120583

2lt 0)

and the bifurcating periodic solutions exist for 120591 gt 1205910(120591 lt

Discrete Dynamics in Nature and Society 7

1205910) 1205732determines the stability of the bifurcating periodic

solutions if 1205732lt 0 (120573

2gt 0) the bifurcating periodic

solutions are stable (unstable) and 1198792determines the period

of the bifurcating periodic solutions the period increases(decrease) if 119879

2gt 0 (119879

2lt 0) [8ndash11]

4 Numerical Simulations

In this section in order to illustrate the results above we willgive two examples

Example 1 In system (2) we choose that 1198631= 1198632= 1 119886

1=

11988711= 04 119908

11= 06 119886

2= 03 119887

22= 11990822= 05 119887

12= 03

11988721= 06 119908

12= 24 119908

21= 36 119892

1(119909) = minus01tan(119909) and

1198922(119909) = arctan(119909) then

120597119906

120597119905= Δ119906 minus 04119906 (119905 119909) minus 004 tan (06119906 (119905 119909))

+ 03 arctan (24V (119905 minus 120591 119909))

120597V120597119905= ΔV minus 03V (119905 119909) minus 006 tan (36119906 (119905 minus 120591 119909))

+ 05 arctan (05V (119905 119909))

(64)

in which

0 lt 119909 lt 1

119905 gt 0

(65)

with initial and Neumann boundary conditions

120597119906

120597119897=120597V120597119897= 0 119905 ge 0 119909 = 0 1

119906 (119905 119909) = 05 (1 +119905

120587) sin (120587119909)

V (119905 119909) = (1 +119905

120587) sin (120587119909)

(119905 119909) isin [minus120591 0] times [0 1]

(66)

What should be remarked is that we choose the parametervalues stochastically under the condition 119902

2lt 0 in order to

ensure the existence of Hopf bifurcation at 119864lowast when 120591 = 1205910

So 1205910= 19371 and 119908

0= 02939 Then we can know

on the basis ofTheorem 1 that the trivial uniform steady state119864lowast= (0 0) is asymptotically stable when 0 le 120591 lt 120591

0 When

120591 gt 1205910 the steady state is unstable and a Hopf bifurcation is

arising from the steady state The numerical simulations inFigures 1 and 2 illustrate the facts

When 120591 = 1205910 we get that 119888

1(0) = minus00001 + 00022119894 then

we can acquire that 1205832gt 0 and 120573

2lt 0 Hence when 120591 passes

through 1205910to the right (120591 gt 120591

0) the bifurcation turns up and

the corresponding periodic orbits are orbitally asymptoticallystable

Example 2 In system (2) we choose that 1198631= 1198632= 001

11988721= 09 119886

2= 02 119887

12= 03 119887

11= 11988722= 11990822= 05 119886

1=

11990811= 06 119908

12= 25 119908

21= 36 119892

1(119909) = minus01tan(119909) and

1198922(119909) = arctan(119909) then

120597119906

120597119905= 001Δ119906 minus 06119906 (119905 119909) minus 005 tan (06119906 (119905 119909))

+ 03 arctan (25V (119905 minus 120591 119909))

120597V120597119905= 001ΔV minus 02V (119905 119909) minus 009 tan (36119906 (119905 minus 120591 119909))

+ 05 arctan (05V (119905 119909))

(67)

in which

0 lt 119909 lt 1

119905 gt 0

(68)

with initial and Dirichlet boundary conditions

119906 (119905 0) = 119906 (119905 1) = V (119905 0) = V (119905 1) = 0 119905 ge 0

119906 (119905 119909) = 05 (1 +119905

120587) sin (120587119909)

V (119905 119909) = (1 +119905

120587) sin (120587119909)

(119905 119909) isin [minus120591 0] times [0 1]

(69)

The similar Hopf bifurcation phenomenon is illustratedby the numerical simulations in Figures 3 and 4

5 Discussion and Research Perspective

This section is devoted to a summary of discussion andresearch perspective for the generalized reaction-diffusionneural networkmodelThemodel is based on the assumptionthat the signal transmission is of a digital (McCulloch-Pitts)nature the model then described a combination of analogand digital signal processing in the network [12] Dependingon the modeling approaches neural networks can be mod-eled either as a static neural network model or as a local fieldneural network model In order to let the problem be moregeneral in many aspects we build a generalized reaction-diffusion neural network model which includes reaction-diffusion local field neural networks and reaction-diffusionstatic neural networks For a delayed neural network animportant issue is the dynamical behaviors of the network[13] Thus there has been a large body of work discussing thestability and bifurcation in delayed neural network modelsBy analyzing the characteristic equation we discussed thelocal stability of the trivial uniform of system (2) [14] Itwas shown that when the delay 120591 varies the trivial uniformsteady state exchanges its stability and Hopf bifurcationsoccur Numerical simulations illustrated the occurrence ofthe bifurcate periodic solutions when the delay 120591 passes thecritical value 120591

0

A research perspective includes the problem of deter-mining the bifurcating periodic solutions and the stabilityand directions of the Hopf bifurcation using the normal

8 Discrete Dynamics in Nature and Society

x-axis

05

1

0

minus004

minus002

0

002

004

006

008

01

u-axis

1500 50 100

t-axis(a)

x-axis

05

1

0

100 1500 50

t-axis

minus003

minus002

minus001

0

001

002

003

v-axis

(b)

Figure 1 The temporal solution found by numerical integration of systems (64) and (66) with 120591 = 185 (a) 119906(119905 119909) and (b) V(119905 119909)

minus004

minus002

0

002

004

006

008

01

u-axis

x-axis05

1

50 100 1500

t-axis(a)

minus003

minus002

minus001

0

001

002

003

v-axis

x-axis05

1

00 50

t-axis100 150

(b)

Figure 2 The temporal solution found by numerical integration of systems (64) and (66) with 120591 = 425 (a) 119906(119905 119909) and (b) V(119905 119909)

0

50t-axis

100

150

x-axis

05

1

0

minus02

0

02

04

06

u-axis

(a)0

50t-axis

100

150

x-axis05

1

0

minus05

0

05

1

v-axis

(b)

Figure 3 The temporal solution found by numerical integration of systems (67) and (69) with 120591 = 125 (a) 119906(119905 119909) and (b) V(119905 119909)

Discrete Dynamics in Nature and Society 9

0

50t-axis

100

150

x-axis

05

1

0

minus02

0

02

04

06

u-axis

(a)0

50t-axis

100

150

x-axis

05

1

0

minus05

0

05

1

v-axis

(b)

Figure 4 The temporal solution found by numerical integration of systems (67) and (69) with 120591 = 205 (a) 119906(119905 119909) and (b) V(119905 119909)

form theory and the center manifold reaction A challengingperspective is the comparison of the generalized modelintroduced in the present paper with the experimentallymea-surable quantities Indeed the mathematical models shouldreproduce both qualitatively and quantitatively empiricaldata (see [4])

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (no 61305076) and the ScientificResearch Foundation for the Returned Overseas ChineseScholars State Education Ministry

References

[1] C Bianca M Ferrara and L Guerrini ldquoThe time delaysrsquo effectson the qualitative behavior of an economic growth modelrdquoAbstract and Applied Analysis vol 2013 Article ID 901014 10pages 2013

[2] LV Ballestra LGuerrini andG Pacelli ldquoStability switches andbifurcation analysis of a time delay model for the diffusion of anew technologyrdquo International Journal of Bifurcation amp Chaosvol 24 no 9 Article ID 1450113 2014

[3] Q Gan and R Xu ldquoStability and Hopf bifurcation of a delayedreaction-diffusion neural networkrdquo Mathematical Methods inthe Applied Sciences vol 34 no 12 pp 1450ndash1459 2011

[4] C Bianca and LGuerrini ldquoOn theDalgaard-Strulikmodel withlogistic population growth rate and delayed-carrying capacityrdquoActa Applicandae Mathematicae vol 128 pp 39ndash48 2013

[5] O G Jepps C Bianca and L Rondoni ldquoOnset of diffusivebehavior in confined transport systemsrdquo Chaos vol 18 no 1Article ID 013127 2008

[6] L Gori L Guerrini and M Sodini ldquoEquilibrium and dis-equilibrium dynamics in cobweb models with time delaysrdquo

International Journal of Bifurcation and Chaos vol 25 no 6Article ID 1550088 2015

[7] C Bianca L Guerrini and A Lemarchand ldquoExistence of solu-tions of a partial integrodifferential equation with thermostatand time delayrdquoAbstract and Applied Analysis vol 2014 ArticleID 463409 7 pages 2014

[8] Y Kuang ldquoDelay differential equations with applications inpopulation dynamicsrdquo Discrete amp Continuous Dynamical Sys-tems vol 33 no 4 pp 1633ndash1644 2013

[9] C Huang Y He L Huang and Y Zhaohui ldquoHopf bifurcationanalysis of two neurons with three delaysrdquo Nonlinear AnalysisReal World Applications vol 8 no 3 pp 903ndash921 2007

[10] C Huang L Huang J Feng M Nai and Y He ldquoHopfbifurcation analysis for a two-neuron networkwith four delaysrdquoChaos Solitons amp Fractals vol 34 no 3 pp 795ndash812 2007

[11] Y Song M Han and J Wei ldquoStability and Hopf bifurcationanalysis on a simplified BAM neural network with delaysrdquoPhysica D Nonlinear Phenomena vol 200 no 3-4 pp 185ndash2042005

[12] X-P Yan ldquoHopf bifurcation and stability for a delayed tri-neuron network modelrdquo Journal of Computational and AppliedMathematics vol 196 no 2 pp 579ndash595 2006

[13] H Zhao and L Wang ldquoHopf bifurcation in Cohen-Grossbergneural network with distributed delaysrdquo Nonlinear AnalysisReal World Applications vol 8 no 1 pp 73ndash89 2007

[14] B Zheng Y Zhang and C Zhang ldquoStability and bifurcationof a discrete BAM neural network model with delaysrdquo ChaosSolitons amp Fractals vol 36 no 3 pp 612ndash616 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Discrete Dynamics in Nature and Society

where

119867(119911 119911) = 11986720

1199112

2+ 11986711119911119911 + 119867

02

1199112

2+ sdot sdot sdot

= 1198830119891lowast(119898lowast(119905) 0)

minus Φ (Ψ ⟨1198830119891lowast(119898lowast(119905) 120572) 119891

0⟩) sdot 1198910

(49)

for119867119894119895isin 119876 with 119894 + 119895 = 2 It follows from (43) (48) and (49)

that

(1198601205910minus 2119894119908

01205910)11988220(120579) = minus119867

20(120579)

119860120591011988211(120579) = minus119867

11(120579)

(50)

By (49) we have that for 120579 isin [minus1 0)

119867 (119911 119911) = minus1

2[119892201199011(120579) + 119892

021199012(120579)] 1199112sdot 1198910

minus [119892111199011(120579) + 119892

111199012(120579)] 119911119911 sdot 119891

0+ sdot sdot sdot

(51)

Comparing the coefficients with (49) we get that for 120579 isin[minus1 0)

11986720(120579) = minus [119892

201199011(120579) + 119892

021199012(120579)] sdot 119891

0 (52)

11986711(120579) = minus [119892

111199011(120579) + 119892

111199012(120579)] sdot 119891

0 (53)

By (50) (52) and the definition of 1198601205910 we get that

20(120579) = 2119894119908

0120591011988220(120579) + [119892

201199011(120579) + 119892

021199012(120579)]

sdot 1198910

(54)

Noticing that 1199011(120579) = 119901

1(0)11989011989411990801205910120579 hence

11988220(120579)

= [11989411989220

11990801205910

1199011(120579) 11989011989411990801205910120579 +

11989411989202

311990801205910

1199012(120579) 119890minus11989411990801205910120579] sdot 119891

0

+ 1198641119890211989411990801205910120579

(55)

where 1198641= (119864(1)

1 119864(2)

1) isin 1198772 which is a constant vector

In a similar way by (50) and (53) we have that

11988211(120579)

= [minus11989411989211

11990801205910

1199011(0) 11989011989411990801205910120579 +

11989411989211

11990801205910

1199012(0) 119890minus11989411990801205910120579] sdot 119891

0

+ 1198642

(56)

where 1198642= (119864(1)

2 119864(2)

2) isin 1198772 which is also a constant vector

In what follows we seek appropriate 1198641and 119864

2 From the

definition of 1198601205910and (50) we can obtain that

21198941199080120591011988220(0) minus weierpΔ119882

20(0) minus Z (119864

lowast)11988220(120579)

= 11986720(0)

(57)

minus weierpΔ11988211(0) minus Z (119864

lowast)11988211(120579) = 119867

11(0) (58)

where

11986720(0) =

1205910

4(11988811

11988821

) minus [119892201199011(0) + 119892

021199012(0)] sdot 119891

0 (59)

11986711(0) =

1205910

8(11988813

11988823

) minus [119892111199011(0) + 119892

111199012(0)] sdot 119891

0 (60)

Substituting (55) and (59) into (57) we can obtain that

1198641=1

4(

211989411990801205910+ 1198861minus 11988711119908111198921015840

1(0) minus119887

12119908121198921015840

2(0) 119890minus211989411990801205910

minus11988721119908211198921015840

1(0) 119890minus211989411990801205910 2119894119908

01205910+ 1198862minus 11988722119908221198921015840

2(0)

)

minus1

(11988811

11988821

) (61)

In a similar way substituting (56) and (60) into (58) weobtain that

1198642

=1

8(

1198861minus 11988711119908111198921015840

1(0) minus119887

12119908121198921015840

2(0)

1198862minus 11988722119908221198921015840

2(0) minus119887

21119908211198921015840

1(0)

)

minus1

(11988813

11988823

)

(62)

Therefore we can compute the following values

1198881(0) =

119894

21199080

(1198921111989220minus 2100381610038161003816100381611989211

10038161003816100381610038162

minus

10038161003816100381610038161198920210038161003816100381610038162

3) +

11989221

2

1205832= minus

Re 1198881(0)

Re 1205821015840 (1205910)

1205732= 2Re 119888

1(0)

1198792= minus

Im 1198881(0) + 120583

2Im 1205821015840 (120591

0)

1199080

(63)

which determine the quantities of bifurcating periodic solu-tions in the center manifold at the critical value 120591

0 that is

1205832determines the direction of Hopf bifurcation the Hopf

bifurcation is supercritical (subcritical) if 1205832gt 0 (120583

2lt 0)

and the bifurcating periodic solutions exist for 120591 gt 1205910(120591 lt

Discrete Dynamics in Nature and Society 7

1205910) 1205732determines the stability of the bifurcating periodic

solutions if 1205732lt 0 (120573

2gt 0) the bifurcating periodic

solutions are stable (unstable) and 1198792determines the period

of the bifurcating periodic solutions the period increases(decrease) if 119879

2gt 0 (119879

2lt 0) [8ndash11]

4 Numerical Simulations

In this section in order to illustrate the results above we willgive two examples

Example 1 In system (2) we choose that 1198631= 1198632= 1 119886

1=

11988711= 04 119908

11= 06 119886

2= 03 119887

22= 11990822= 05 119887

12= 03

11988721= 06 119908

12= 24 119908

21= 36 119892

1(119909) = minus01tan(119909) and

1198922(119909) = arctan(119909) then

120597119906

120597119905= Δ119906 minus 04119906 (119905 119909) minus 004 tan (06119906 (119905 119909))

+ 03 arctan (24V (119905 minus 120591 119909))

120597V120597119905= ΔV minus 03V (119905 119909) minus 006 tan (36119906 (119905 minus 120591 119909))

+ 05 arctan (05V (119905 119909))

(64)

in which

0 lt 119909 lt 1

119905 gt 0

(65)

with initial and Neumann boundary conditions

120597119906

120597119897=120597V120597119897= 0 119905 ge 0 119909 = 0 1

119906 (119905 119909) = 05 (1 +119905

120587) sin (120587119909)

V (119905 119909) = (1 +119905

120587) sin (120587119909)

(119905 119909) isin [minus120591 0] times [0 1]

(66)

What should be remarked is that we choose the parametervalues stochastically under the condition 119902

2lt 0 in order to

ensure the existence of Hopf bifurcation at 119864lowast when 120591 = 1205910

So 1205910= 19371 and 119908

0= 02939 Then we can know

on the basis ofTheorem 1 that the trivial uniform steady state119864lowast= (0 0) is asymptotically stable when 0 le 120591 lt 120591

0 When

120591 gt 1205910 the steady state is unstable and a Hopf bifurcation is

arising from the steady state The numerical simulations inFigures 1 and 2 illustrate the facts

When 120591 = 1205910 we get that 119888

1(0) = minus00001 + 00022119894 then

we can acquire that 1205832gt 0 and 120573

2lt 0 Hence when 120591 passes

through 1205910to the right (120591 gt 120591

0) the bifurcation turns up and

the corresponding periodic orbits are orbitally asymptoticallystable

Example 2 In system (2) we choose that 1198631= 1198632= 001

11988721= 09 119886

2= 02 119887

12= 03 119887

11= 11988722= 11990822= 05 119886

1=

11990811= 06 119908

12= 25 119908

21= 36 119892

1(119909) = minus01tan(119909) and

1198922(119909) = arctan(119909) then

120597119906

120597119905= 001Δ119906 minus 06119906 (119905 119909) minus 005 tan (06119906 (119905 119909))

+ 03 arctan (25V (119905 minus 120591 119909))

120597V120597119905= 001ΔV minus 02V (119905 119909) minus 009 tan (36119906 (119905 minus 120591 119909))

+ 05 arctan (05V (119905 119909))

(67)

in which

0 lt 119909 lt 1

119905 gt 0

(68)

with initial and Dirichlet boundary conditions

119906 (119905 0) = 119906 (119905 1) = V (119905 0) = V (119905 1) = 0 119905 ge 0

119906 (119905 119909) = 05 (1 +119905

120587) sin (120587119909)

V (119905 119909) = (1 +119905

120587) sin (120587119909)

(119905 119909) isin [minus120591 0] times [0 1]

(69)

The similar Hopf bifurcation phenomenon is illustratedby the numerical simulations in Figures 3 and 4

5 Discussion and Research Perspective

This section is devoted to a summary of discussion andresearch perspective for the generalized reaction-diffusionneural networkmodelThemodel is based on the assumptionthat the signal transmission is of a digital (McCulloch-Pitts)nature the model then described a combination of analogand digital signal processing in the network [12] Dependingon the modeling approaches neural networks can be mod-eled either as a static neural network model or as a local fieldneural network model In order to let the problem be moregeneral in many aspects we build a generalized reaction-diffusion neural network model which includes reaction-diffusion local field neural networks and reaction-diffusionstatic neural networks For a delayed neural network animportant issue is the dynamical behaviors of the network[13] Thus there has been a large body of work discussing thestability and bifurcation in delayed neural network modelsBy analyzing the characteristic equation we discussed thelocal stability of the trivial uniform of system (2) [14] Itwas shown that when the delay 120591 varies the trivial uniformsteady state exchanges its stability and Hopf bifurcationsoccur Numerical simulations illustrated the occurrence ofthe bifurcate periodic solutions when the delay 120591 passes thecritical value 120591

0

A research perspective includes the problem of deter-mining the bifurcating periodic solutions and the stabilityand directions of the Hopf bifurcation using the normal

8 Discrete Dynamics in Nature and Society

x-axis

05

1

0

minus004

minus002

0

002

004

006

008

01

u-axis

1500 50 100

t-axis(a)

x-axis

05

1

0

100 1500 50

t-axis

minus003

minus002

minus001

0

001

002

003

v-axis

(b)

Figure 1 The temporal solution found by numerical integration of systems (64) and (66) with 120591 = 185 (a) 119906(119905 119909) and (b) V(119905 119909)

minus004

minus002

0

002

004

006

008

01

u-axis

x-axis05

1

50 100 1500

t-axis(a)

minus003

minus002

minus001

0

001

002

003

v-axis

x-axis05

1

00 50

t-axis100 150

(b)

Figure 2 The temporal solution found by numerical integration of systems (64) and (66) with 120591 = 425 (a) 119906(119905 119909) and (b) V(119905 119909)

0

50t-axis

100

150

x-axis

05

1

0

minus02

0

02

04

06

u-axis

(a)0

50t-axis

100

150

x-axis05

1

0

minus05

0

05

1

v-axis

(b)

Figure 3 The temporal solution found by numerical integration of systems (67) and (69) with 120591 = 125 (a) 119906(119905 119909) and (b) V(119905 119909)

Discrete Dynamics in Nature and Society 9

0

50t-axis

100

150

x-axis

05

1

0

minus02

0

02

04

06

u-axis

(a)0

50t-axis

100

150

x-axis

05

1

0

minus05

0

05

1

v-axis

(b)

Figure 4 The temporal solution found by numerical integration of systems (67) and (69) with 120591 = 205 (a) 119906(119905 119909) and (b) V(119905 119909)

form theory and the center manifold reaction A challengingperspective is the comparison of the generalized modelintroduced in the present paper with the experimentallymea-surable quantities Indeed the mathematical models shouldreproduce both qualitatively and quantitatively empiricaldata (see [4])

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (no 61305076) and the ScientificResearch Foundation for the Returned Overseas ChineseScholars State Education Ministry

References

[1] C Bianca M Ferrara and L Guerrini ldquoThe time delaysrsquo effectson the qualitative behavior of an economic growth modelrdquoAbstract and Applied Analysis vol 2013 Article ID 901014 10pages 2013

[2] LV Ballestra LGuerrini andG Pacelli ldquoStability switches andbifurcation analysis of a time delay model for the diffusion of anew technologyrdquo International Journal of Bifurcation amp Chaosvol 24 no 9 Article ID 1450113 2014

[3] Q Gan and R Xu ldquoStability and Hopf bifurcation of a delayedreaction-diffusion neural networkrdquo Mathematical Methods inthe Applied Sciences vol 34 no 12 pp 1450ndash1459 2011

[4] C Bianca and LGuerrini ldquoOn theDalgaard-Strulikmodel withlogistic population growth rate and delayed-carrying capacityrdquoActa Applicandae Mathematicae vol 128 pp 39ndash48 2013

[5] O G Jepps C Bianca and L Rondoni ldquoOnset of diffusivebehavior in confined transport systemsrdquo Chaos vol 18 no 1Article ID 013127 2008

[6] L Gori L Guerrini and M Sodini ldquoEquilibrium and dis-equilibrium dynamics in cobweb models with time delaysrdquo

International Journal of Bifurcation and Chaos vol 25 no 6Article ID 1550088 2015

[7] C Bianca L Guerrini and A Lemarchand ldquoExistence of solu-tions of a partial integrodifferential equation with thermostatand time delayrdquoAbstract and Applied Analysis vol 2014 ArticleID 463409 7 pages 2014

[8] Y Kuang ldquoDelay differential equations with applications inpopulation dynamicsrdquo Discrete amp Continuous Dynamical Sys-tems vol 33 no 4 pp 1633ndash1644 2013

[9] C Huang Y He L Huang and Y Zhaohui ldquoHopf bifurcationanalysis of two neurons with three delaysrdquo Nonlinear AnalysisReal World Applications vol 8 no 3 pp 903ndash921 2007

[10] C Huang L Huang J Feng M Nai and Y He ldquoHopfbifurcation analysis for a two-neuron networkwith four delaysrdquoChaos Solitons amp Fractals vol 34 no 3 pp 795ndash812 2007

[11] Y Song M Han and J Wei ldquoStability and Hopf bifurcationanalysis on a simplified BAM neural network with delaysrdquoPhysica D Nonlinear Phenomena vol 200 no 3-4 pp 185ndash2042005

[12] X-P Yan ldquoHopf bifurcation and stability for a delayed tri-neuron network modelrdquo Journal of Computational and AppliedMathematics vol 196 no 2 pp 579ndash595 2006

[13] H Zhao and L Wang ldquoHopf bifurcation in Cohen-Grossbergneural network with distributed delaysrdquo Nonlinear AnalysisReal World Applications vol 8 no 1 pp 73ndash89 2007

[14] B Zheng Y Zhang and C Zhang ldquoStability and bifurcationof a discrete BAM neural network model with delaysrdquo ChaosSolitons amp Fractals vol 36 no 3 pp 612ndash616 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 7

1205910) 1205732determines the stability of the bifurcating periodic

solutions if 1205732lt 0 (120573

2gt 0) the bifurcating periodic

solutions are stable (unstable) and 1198792determines the period

of the bifurcating periodic solutions the period increases(decrease) if 119879

2gt 0 (119879

2lt 0) [8ndash11]

4 Numerical Simulations

In this section in order to illustrate the results above we willgive two examples

Example 1 In system (2) we choose that 1198631= 1198632= 1 119886

1=

11988711= 04 119908

11= 06 119886

2= 03 119887

22= 11990822= 05 119887

12= 03

11988721= 06 119908

12= 24 119908

21= 36 119892

1(119909) = minus01tan(119909) and

1198922(119909) = arctan(119909) then

120597119906

120597119905= Δ119906 minus 04119906 (119905 119909) minus 004 tan (06119906 (119905 119909))

+ 03 arctan (24V (119905 minus 120591 119909))

120597V120597119905= ΔV minus 03V (119905 119909) minus 006 tan (36119906 (119905 minus 120591 119909))

+ 05 arctan (05V (119905 119909))

(64)

in which

0 lt 119909 lt 1

119905 gt 0

(65)

with initial and Neumann boundary conditions

120597119906

120597119897=120597V120597119897= 0 119905 ge 0 119909 = 0 1

119906 (119905 119909) = 05 (1 +119905

120587) sin (120587119909)

V (119905 119909) = (1 +119905

120587) sin (120587119909)

(119905 119909) isin [minus120591 0] times [0 1]

(66)

What should be remarked is that we choose the parametervalues stochastically under the condition 119902

2lt 0 in order to

ensure the existence of Hopf bifurcation at 119864lowast when 120591 = 1205910

So 1205910= 19371 and 119908

0= 02939 Then we can know

on the basis ofTheorem 1 that the trivial uniform steady state119864lowast= (0 0) is asymptotically stable when 0 le 120591 lt 120591

0 When

120591 gt 1205910 the steady state is unstable and a Hopf bifurcation is

arising from the steady state The numerical simulations inFigures 1 and 2 illustrate the facts

When 120591 = 1205910 we get that 119888

1(0) = minus00001 + 00022119894 then

we can acquire that 1205832gt 0 and 120573

2lt 0 Hence when 120591 passes

through 1205910to the right (120591 gt 120591

0) the bifurcation turns up and

the corresponding periodic orbits are orbitally asymptoticallystable

Example 2 In system (2) we choose that 1198631= 1198632= 001

11988721= 09 119886

2= 02 119887

12= 03 119887

11= 11988722= 11990822= 05 119886

1=

11990811= 06 119908

12= 25 119908

21= 36 119892

1(119909) = minus01tan(119909) and

1198922(119909) = arctan(119909) then

120597119906

120597119905= 001Δ119906 minus 06119906 (119905 119909) minus 005 tan (06119906 (119905 119909))

+ 03 arctan (25V (119905 minus 120591 119909))

120597V120597119905= 001ΔV minus 02V (119905 119909) minus 009 tan (36119906 (119905 minus 120591 119909))

+ 05 arctan (05V (119905 119909))

(67)

in which

0 lt 119909 lt 1

119905 gt 0

(68)

with initial and Dirichlet boundary conditions

119906 (119905 0) = 119906 (119905 1) = V (119905 0) = V (119905 1) = 0 119905 ge 0

119906 (119905 119909) = 05 (1 +119905

120587) sin (120587119909)

V (119905 119909) = (1 +119905

120587) sin (120587119909)

(119905 119909) isin [minus120591 0] times [0 1]

(69)

The similar Hopf bifurcation phenomenon is illustratedby the numerical simulations in Figures 3 and 4

5 Discussion and Research Perspective

This section is devoted to a summary of discussion andresearch perspective for the generalized reaction-diffusionneural networkmodelThemodel is based on the assumptionthat the signal transmission is of a digital (McCulloch-Pitts)nature the model then described a combination of analogand digital signal processing in the network [12] Dependingon the modeling approaches neural networks can be mod-eled either as a static neural network model or as a local fieldneural network model In order to let the problem be moregeneral in many aspects we build a generalized reaction-diffusion neural network model which includes reaction-diffusion local field neural networks and reaction-diffusionstatic neural networks For a delayed neural network animportant issue is the dynamical behaviors of the network[13] Thus there has been a large body of work discussing thestability and bifurcation in delayed neural network modelsBy analyzing the characteristic equation we discussed thelocal stability of the trivial uniform of system (2) [14] Itwas shown that when the delay 120591 varies the trivial uniformsteady state exchanges its stability and Hopf bifurcationsoccur Numerical simulations illustrated the occurrence ofthe bifurcate periodic solutions when the delay 120591 passes thecritical value 120591

0

A research perspective includes the problem of deter-mining the bifurcating periodic solutions and the stabilityand directions of the Hopf bifurcation using the normal

8 Discrete Dynamics in Nature and Society

x-axis

05

1

0

minus004

minus002

0

002

004

006

008

01

u-axis

1500 50 100

t-axis(a)

x-axis

05

1

0

100 1500 50

t-axis

minus003

minus002

minus001

0

001

002

003

v-axis

(b)

Figure 1 The temporal solution found by numerical integration of systems (64) and (66) with 120591 = 185 (a) 119906(119905 119909) and (b) V(119905 119909)

minus004

minus002

0

002

004

006

008

01

u-axis

x-axis05

1

50 100 1500

t-axis(a)

minus003

minus002

minus001

0

001

002

003

v-axis

x-axis05

1

00 50

t-axis100 150

(b)

Figure 2 The temporal solution found by numerical integration of systems (64) and (66) with 120591 = 425 (a) 119906(119905 119909) and (b) V(119905 119909)

0

50t-axis

100

150

x-axis

05

1

0

minus02

0

02

04

06

u-axis

(a)0

50t-axis

100

150

x-axis05

1

0

minus05

0

05

1

v-axis

(b)

Figure 3 The temporal solution found by numerical integration of systems (67) and (69) with 120591 = 125 (a) 119906(119905 119909) and (b) V(119905 119909)

Discrete Dynamics in Nature and Society 9

0

50t-axis

100

150

x-axis

05

1

0

minus02

0

02

04

06

u-axis

(a)0

50t-axis

100

150

x-axis

05

1

0

minus05

0

05

1

v-axis

(b)

Figure 4 The temporal solution found by numerical integration of systems (67) and (69) with 120591 = 205 (a) 119906(119905 119909) and (b) V(119905 119909)

form theory and the center manifold reaction A challengingperspective is the comparison of the generalized modelintroduced in the present paper with the experimentallymea-surable quantities Indeed the mathematical models shouldreproduce both qualitatively and quantitatively empiricaldata (see [4])

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (no 61305076) and the ScientificResearch Foundation for the Returned Overseas ChineseScholars State Education Ministry

References

[1] C Bianca M Ferrara and L Guerrini ldquoThe time delaysrsquo effectson the qualitative behavior of an economic growth modelrdquoAbstract and Applied Analysis vol 2013 Article ID 901014 10pages 2013

[2] LV Ballestra LGuerrini andG Pacelli ldquoStability switches andbifurcation analysis of a time delay model for the diffusion of anew technologyrdquo International Journal of Bifurcation amp Chaosvol 24 no 9 Article ID 1450113 2014

[3] Q Gan and R Xu ldquoStability and Hopf bifurcation of a delayedreaction-diffusion neural networkrdquo Mathematical Methods inthe Applied Sciences vol 34 no 12 pp 1450ndash1459 2011

[4] C Bianca and LGuerrini ldquoOn theDalgaard-Strulikmodel withlogistic population growth rate and delayed-carrying capacityrdquoActa Applicandae Mathematicae vol 128 pp 39ndash48 2013

[5] O G Jepps C Bianca and L Rondoni ldquoOnset of diffusivebehavior in confined transport systemsrdquo Chaos vol 18 no 1Article ID 013127 2008

[6] L Gori L Guerrini and M Sodini ldquoEquilibrium and dis-equilibrium dynamics in cobweb models with time delaysrdquo

International Journal of Bifurcation and Chaos vol 25 no 6Article ID 1550088 2015

[7] C Bianca L Guerrini and A Lemarchand ldquoExistence of solu-tions of a partial integrodifferential equation with thermostatand time delayrdquoAbstract and Applied Analysis vol 2014 ArticleID 463409 7 pages 2014

[8] Y Kuang ldquoDelay differential equations with applications inpopulation dynamicsrdquo Discrete amp Continuous Dynamical Sys-tems vol 33 no 4 pp 1633ndash1644 2013

[9] C Huang Y He L Huang and Y Zhaohui ldquoHopf bifurcationanalysis of two neurons with three delaysrdquo Nonlinear AnalysisReal World Applications vol 8 no 3 pp 903ndash921 2007

[10] C Huang L Huang J Feng M Nai and Y He ldquoHopfbifurcation analysis for a two-neuron networkwith four delaysrdquoChaos Solitons amp Fractals vol 34 no 3 pp 795ndash812 2007

[11] Y Song M Han and J Wei ldquoStability and Hopf bifurcationanalysis on a simplified BAM neural network with delaysrdquoPhysica D Nonlinear Phenomena vol 200 no 3-4 pp 185ndash2042005

[12] X-P Yan ldquoHopf bifurcation and stability for a delayed tri-neuron network modelrdquo Journal of Computational and AppliedMathematics vol 196 no 2 pp 579ndash595 2006

[13] H Zhao and L Wang ldquoHopf bifurcation in Cohen-Grossbergneural network with distributed delaysrdquo Nonlinear AnalysisReal World Applications vol 8 no 1 pp 73ndash89 2007

[14] B Zheng Y Zhang and C Zhang ldquoStability and bifurcationof a discrete BAM neural network model with delaysrdquo ChaosSolitons amp Fractals vol 36 no 3 pp 612ndash616 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Discrete Dynamics in Nature and Society

x-axis

05

1

0

minus004

minus002

0

002

004

006

008

01

u-axis

1500 50 100

t-axis(a)

x-axis

05

1

0

100 1500 50

t-axis

minus003

minus002

minus001

0

001

002

003

v-axis

(b)

Figure 1 The temporal solution found by numerical integration of systems (64) and (66) with 120591 = 185 (a) 119906(119905 119909) and (b) V(119905 119909)

minus004

minus002

0

002

004

006

008

01

u-axis

x-axis05

1

50 100 1500

t-axis(a)

minus003

minus002

minus001

0

001

002

003

v-axis

x-axis05

1

00 50

t-axis100 150

(b)

Figure 2 The temporal solution found by numerical integration of systems (64) and (66) with 120591 = 425 (a) 119906(119905 119909) and (b) V(119905 119909)

0

50t-axis

100

150

x-axis

05

1

0

minus02

0

02

04

06

u-axis

(a)0

50t-axis

100

150

x-axis05

1

0

minus05

0

05

1

v-axis

(b)

Figure 3 The temporal solution found by numerical integration of systems (67) and (69) with 120591 = 125 (a) 119906(119905 119909) and (b) V(119905 119909)

Discrete Dynamics in Nature and Society 9

0

50t-axis

100

150

x-axis

05

1

0

minus02

0

02

04

06

u-axis

(a)0

50t-axis

100

150

x-axis

05

1

0

minus05

0

05

1

v-axis

(b)

Figure 4 The temporal solution found by numerical integration of systems (67) and (69) with 120591 = 205 (a) 119906(119905 119909) and (b) V(119905 119909)

form theory and the center manifold reaction A challengingperspective is the comparison of the generalized modelintroduced in the present paper with the experimentallymea-surable quantities Indeed the mathematical models shouldreproduce both qualitatively and quantitatively empiricaldata (see [4])

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (no 61305076) and the ScientificResearch Foundation for the Returned Overseas ChineseScholars State Education Ministry

References

[1] C Bianca M Ferrara and L Guerrini ldquoThe time delaysrsquo effectson the qualitative behavior of an economic growth modelrdquoAbstract and Applied Analysis vol 2013 Article ID 901014 10pages 2013

[2] LV Ballestra LGuerrini andG Pacelli ldquoStability switches andbifurcation analysis of a time delay model for the diffusion of anew technologyrdquo International Journal of Bifurcation amp Chaosvol 24 no 9 Article ID 1450113 2014

[3] Q Gan and R Xu ldquoStability and Hopf bifurcation of a delayedreaction-diffusion neural networkrdquo Mathematical Methods inthe Applied Sciences vol 34 no 12 pp 1450ndash1459 2011

[4] C Bianca and LGuerrini ldquoOn theDalgaard-Strulikmodel withlogistic population growth rate and delayed-carrying capacityrdquoActa Applicandae Mathematicae vol 128 pp 39ndash48 2013

[5] O G Jepps C Bianca and L Rondoni ldquoOnset of diffusivebehavior in confined transport systemsrdquo Chaos vol 18 no 1Article ID 013127 2008

[6] L Gori L Guerrini and M Sodini ldquoEquilibrium and dis-equilibrium dynamics in cobweb models with time delaysrdquo

International Journal of Bifurcation and Chaos vol 25 no 6Article ID 1550088 2015

[7] C Bianca L Guerrini and A Lemarchand ldquoExistence of solu-tions of a partial integrodifferential equation with thermostatand time delayrdquoAbstract and Applied Analysis vol 2014 ArticleID 463409 7 pages 2014

[8] Y Kuang ldquoDelay differential equations with applications inpopulation dynamicsrdquo Discrete amp Continuous Dynamical Sys-tems vol 33 no 4 pp 1633ndash1644 2013

[9] C Huang Y He L Huang and Y Zhaohui ldquoHopf bifurcationanalysis of two neurons with three delaysrdquo Nonlinear AnalysisReal World Applications vol 8 no 3 pp 903ndash921 2007

[10] C Huang L Huang J Feng M Nai and Y He ldquoHopfbifurcation analysis for a two-neuron networkwith four delaysrdquoChaos Solitons amp Fractals vol 34 no 3 pp 795ndash812 2007

[11] Y Song M Han and J Wei ldquoStability and Hopf bifurcationanalysis on a simplified BAM neural network with delaysrdquoPhysica D Nonlinear Phenomena vol 200 no 3-4 pp 185ndash2042005

[12] X-P Yan ldquoHopf bifurcation and stability for a delayed tri-neuron network modelrdquo Journal of Computational and AppliedMathematics vol 196 no 2 pp 579ndash595 2006

[13] H Zhao and L Wang ldquoHopf bifurcation in Cohen-Grossbergneural network with distributed delaysrdquo Nonlinear AnalysisReal World Applications vol 8 no 1 pp 73ndash89 2007

[14] B Zheng Y Zhang and C Zhang ldquoStability and bifurcationof a discrete BAM neural network model with delaysrdquo ChaosSolitons amp Fractals vol 36 no 3 pp 612ndash616 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 9

0

50t-axis

100

150

x-axis

05

1

0

minus02

0

02

04

06

u-axis

(a)0

50t-axis

100

150

x-axis

05

1

0

minus05

0

05

1

v-axis

(b)

Figure 4 The temporal solution found by numerical integration of systems (67) and (69) with 120591 = 205 (a) 119906(119905 119909) and (b) V(119905 119909)

form theory and the center manifold reaction A challengingperspective is the comparison of the generalized modelintroduced in the present paper with the experimentallymea-surable quantities Indeed the mathematical models shouldreproduce both qualitatively and quantitatively empiricaldata (see [4])

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (no 61305076) and the ScientificResearch Foundation for the Returned Overseas ChineseScholars State Education Ministry

References

[1] C Bianca M Ferrara and L Guerrini ldquoThe time delaysrsquo effectson the qualitative behavior of an economic growth modelrdquoAbstract and Applied Analysis vol 2013 Article ID 901014 10pages 2013

[2] LV Ballestra LGuerrini andG Pacelli ldquoStability switches andbifurcation analysis of a time delay model for the diffusion of anew technologyrdquo International Journal of Bifurcation amp Chaosvol 24 no 9 Article ID 1450113 2014

[3] Q Gan and R Xu ldquoStability and Hopf bifurcation of a delayedreaction-diffusion neural networkrdquo Mathematical Methods inthe Applied Sciences vol 34 no 12 pp 1450ndash1459 2011

[4] C Bianca and LGuerrini ldquoOn theDalgaard-Strulikmodel withlogistic population growth rate and delayed-carrying capacityrdquoActa Applicandae Mathematicae vol 128 pp 39ndash48 2013

[5] O G Jepps C Bianca and L Rondoni ldquoOnset of diffusivebehavior in confined transport systemsrdquo Chaos vol 18 no 1Article ID 013127 2008

[6] L Gori L Guerrini and M Sodini ldquoEquilibrium and dis-equilibrium dynamics in cobweb models with time delaysrdquo

International Journal of Bifurcation and Chaos vol 25 no 6Article ID 1550088 2015

[7] C Bianca L Guerrini and A Lemarchand ldquoExistence of solu-tions of a partial integrodifferential equation with thermostatand time delayrdquoAbstract and Applied Analysis vol 2014 ArticleID 463409 7 pages 2014

[8] Y Kuang ldquoDelay differential equations with applications inpopulation dynamicsrdquo Discrete amp Continuous Dynamical Sys-tems vol 33 no 4 pp 1633ndash1644 2013

[9] C Huang Y He L Huang and Y Zhaohui ldquoHopf bifurcationanalysis of two neurons with three delaysrdquo Nonlinear AnalysisReal World Applications vol 8 no 3 pp 903ndash921 2007

[10] C Huang L Huang J Feng M Nai and Y He ldquoHopfbifurcation analysis for a two-neuron networkwith four delaysrdquoChaos Solitons amp Fractals vol 34 no 3 pp 795ndash812 2007

[11] Y Song M Han and J Wei ldquoStability and Hopf bifurcationanalysis on a simplified BAM neural network with delaysrdquoPhysica D Nonlinear Phenomena vol 200 no 3-4 pp 185ndash2042005

[12] X-P Yan ldquoHopf bifurcation and stability for a delayed tri-neuron network modelrdquo Journal of Computational and AppliedMathematics vol 196 no 2 pp 579ndash595 2006

[13] H Zhao and L Wang ldquoHopf bifurcation in Cohen-Grossbergneural network with distributed delaysrdquo Nonlinear AnalysisReal World Applications vol 8 no 1 pp 73ndash89 2007

[14] B Zheng Y Zhang and C Zhang ldquoStability and bifurcationof a discrete BAM neural network model with delaysrdquo ChaosSolitons amp Fractals vol 36 no 3 pp 612ndash616 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of