Stability of Cohen-Grossberg Neural Networks with Impulsive and Mixed Time Delays

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1 94 IJCSNS Ieraoal Joural of Compuer Scece ad Newor Secury VOL.8 No.2 February 28 Sably of Cohe-Grossberg Neural Newors wh Impulsve ad Mxed Tme Delays Zheag Zhao Qau Sog Deparme of Mahemacs Huzhou Teachers College Huzhou Zheag 33 Cha Deparme of Mahemacs Chogqg Jaoog Uversy Chogqg 474 Cha Absrac I hs paper he problem of sably aalyss for a class of mpulsve Cohe-Grossberg eural ewors wh mxed me delays s cosdered. The mxed me delays comprse boh he me-varyg ad dsrbued delays. By employg a combao of he M -marx heory ad aalyc mehods several suffce codos are obaed o esure he global expoeal sably of equlbrum po for he addressed mpulsve Cohe-Grossberg eural ewor wh mxed delays. The proposed mehod whch does o mae use of he Lyapuov fucoal s show o be smple ye effecve for aalyzg he sably of mpulsve eural ewors wh varable ad/or dsrbued delays. Moreover he expoeal covergece rae s esmaed whch depeds o he sysem parameers. The resuls obaed geeralze a few prevously ow resuls by removg some resrcos or assumpos. A example wh smulao s gve o show he effecveess of he obaed resuls. Key words: Cohe-Grossberg eural ewor global expoeal sably me-varyg delays dsrbued delays mpulsve effec. Iroduco The Cohe-Grossberg eural ewor model frs proposed ad suded by Cohe ad Grossberg 983 [] has araced cosderable aeo due o s poeal applcaos classfcao parallel compug assocave memory sgal ad mage processg especally solvg some dffcul opmzao problems. I such applcaos s of prme mporace o esure ha he desged eural ewors be sable[23]. I pracce due o he fe speeds of he swchg ad rasmsso of sgals me delays do exs a worg ewor ad hus should be corporaed o he model equao [2]. I rece years he dyamcal behavors of Cohe-Grossberg eural ewors wh cosa delays or me-varyg delays or dsrbued delays have bee suded for example see [2-26] ad refereces here. O he oher had mpulsve effec lewse exss a wde varey of evoluoary processes whch saes are chaged abruply a cera momes of me he felds such as medce ad bology ecoomcs elecrocs ad elecommucaos. Neural ewors whch clude Hopfeld eural ewors cellular eural ewors ad Cohe-Grossberg eural ewors are ofe subec o mpulsve perurbaos ha ur affec dyamcal behavors of he sysems. Therefore s ecessary o cosder boh he mpulsve effec ad delay effec whe vesgag he sably of eural ewors [27]. So far several eresg resuls have bee repored ha focusg o he mpulsve effec o delayed eural ewors see [27-39] ad refereces here. To he bes of our owledge few auhors have cosdered he dyamcal behavors of he mpulsve Cohe-Grossberg eural ewor model wh boh me-varyg ad dsrbued delays. Movaed by he above dscussos he obecve of hs paper s o sudy he global expoeal sably of mpulsve Cohe-Grossberg eural ewor wh boh me-varyg ad dsrbued delays ad esmae he expoeal covergece rae dex. By employg a combao of he M-marx heory ad aalyc mehods we oba several suffce codos for esurg he global expoeal sably. Our proposed mehod does o mae use of he Lyapuov fucoal ad s show o be smple ye effecve for aalyzg he sably of mpulsve eural ewors wh varable ad/or dsrbued delays. Our ma resuls geeralze a few prevously ow resuls by removg some resrcos or assumpos. A example wh smulao s gve o show he effecveess of he obaed resuls. Mauscrp receved February 5 28 Mauscrp revsed February 2 28

2 IJCSNS Ieraoal Joural of Compuer Scece ad Newor Secury VOL.8 No.2 February Model descrpo ad prelmares I hs paper we cosder he followg model. du ( =a( u( b( u( cg ( u( d d f ( u ( τ ( v K ( s h ( u ( s ds I u ( p ( u ( u ( q ( u (( ( for = 2 L ad = 2 L where correspods o he umber of us he eural ewor; u ( correspods o he sae of he h u a me. The frs par s he couous par of model ( whch descrbes he couous evoluo processes of he eural ewor where g f ad h deoe he acvao fucos; τ ( correspods o he rasmsso delay alog he axo of he h u from he h u ad sasfes τ ( τ ( τ s a cosa; a( u( represes a amplfcao fuco a me ; b ( u ( s a appropraely behaved fuco a me = L τ L u(( τ ( J = such ha he soluos of model ( rema bouded; C = ( c D = ( d ad V = ( v are coeco marces; K s he delay erel fuco; I s he cosa pu from ousde of he ewor. The secod par s he dscree par of model ( whch descrbes ha he evoluo processes experece abrup chage of sae a he momes of me (called mpulsve momes where p ( u ( L u ( represes mpulsve perurbaos of he h u a me ad u ( deoes he lef lm of u ( ; q ( u(( τ ( L u (( τ ( represes mpulsve perurbaos of he h u a me whch caused by rasmsso delays; J represes exeral mpulsve pu a me he fxed momes of ( me sasfy < 2 < L lm = ad m { } max { τ } >. 2 Remar. Whe p ( u ( L u ( = u ( τ τ q ( u (( ( L u (( ( = ad J = ( = 2 L ; = 2 L model ( urs o he followg Cohe-Grossberg eural ewor model whou mpulses du ( =a( u( b( u( cg ( u( d for > = 2 L. Noe ha model (2 s a geeral eural ewor ha covers some popular models such as delayed Hopfeld eural ewors delayed cellular eural ewors delayed BAM eural ewors. For coveece we roduce several oaos. u = ( u u L u T R deoes a colum vecor; u 2 d f ( u ( τ ( K ( s h ( u ( s ds I deoes he absolue-value vecor gve by u = ( u u2 L u T R. For marx A= ( a R A deoes he absolue-value marx gve by A = ( a ; ρ( A deoes he specral radus of A ; u deoes a vecor orm defed by u = u max. CXY [ ] deoes he space of couous mappgs from he opologcal space X o he opologcal space Y. PC[ I R ] = { ϕ : I R ϕ( = ϕ( for I ϕ( exss for ( ϕ( = ϕ( for all bu pos ( } where I R s a erval ϕ ( ad ϕ( deoe he lef-had lm ad rgh-had lm of he scalar fuco ϕ ( respecvely. Throughou hs paper we mae he followg assumpos: (H Model ( has a leas oe equlbrum po. (H2 a ( u s a couous fuco ad < a a( u ( a s a cosa for all u R = 2 L. (H3 There exss a posve dagoal marx b B = dag( b b2 L b such ha ( ( u b v b u v for all u v R( u v = 2 L. (2

3 96 IJCSNS Ieraoal Joural of Compuer Scece ad Newor Secury VOL.8 No.2 February 28 (H4 There exs hree posve dagoal marces G = dag( G G2 L G F = dag( F F2 L F ad H = dag( H H2 L H such ha g ( u g ( u2 f ( u f ( u2 G = sup F = sup u u 2 u u2 u u 2 u u2 h( u h( u2 ad H = sup for all u u2 u u u u 2 2 = 2 L. (H5 The delay erel K :[ [ s real valued oegave couous fuco ad sasfes β s e K ( ( s ds = r β where r ( β s couous fuco [ δ δ > ad r ( = = 2 L. ( (H6 There exs oegave marces P ( = p ( Q = ( q such ha ( p ( u u p ( v v p u v = ( q ( u L u q ( v L v q u v L L for all ( u L u T R ( v L v T R = 2 L = 2 L. Defo : The equlbrum po u = ( u u2 L u T of model ( s sad o be globally expoeally sable f here exs cosas ε > ad M > such ha (3 for all > where u ( = ( u( u2( L u( T s ay soluo of model ( wh al value u ( s = φ ( s PC(( ] R = 2 L ad ( ( u u M φ u e ε φ u = max sup φ ( s u. s ( ] Defo 2[8]: A real marx A= ( a s sad o be a M -marx f a ( = 2 L ad successve prcple mors of A are posve. To prove our resuls he followg lemmas ha ca be foud [8 33] are ecessary. Lemma [8]: Le Q be marx wh o-posve off-dagoal elemes he Q s a M -marx f ad oly f oe of he followg codos holds. ( The real pars of all egevalues of Q are posve. ( There exss a vecor ξ > such ha ξ T Q >. Whe A s a M -marx deog Ω ( A = { ξ R Aξ > ξ > } we ow from Lemma ha Ω ( A s oempy. Lemma 2[33]: Le A be a oegave marx he A has a oegave egevalue ha s ρ ( A ad s egevecors are oegave. Le Γ ( A = { ξ R Aξ = ρ( A ξ}. Whe A s a oegave marx follows from Lemma 2 ha Γ ( A s oempy. 3. Ma Resuls Theorem : Uder assumpos (H-(H6 he equlbrum po of model ( s globally expoeally sable ad he expoeal covergece rae equals ε α f he followg codos are sasfed ( W = B C G D F V H s a M -marx. ( [ ] Δ= I Γ( P IΓ( Q I Ω( W s oempy. = ( There exss a cosa α such ha lα α < ε = 2 L (4 where he sequece α sasfes α max{ ρ( P e ρ( Q} (5 ad he scalar ε > s deermed by he equaly ξ b ξ ( c G e d F v r( ε H < a = (6 for a gve ξ = ( ξ ξ2 L ξ T Δ τ = max { τ }. Proof. From assumpo (H we le u = ( u u2 L u T be a equlbrum po of model (. By deog y( = u( u a% ( y( = a( y( u b % ( y ( = b( y ( u b( u ( ( = ( ( ( g% y g y u g u f% y f y u f u ( ( = ( ( ( h % ( y( = h( y( u h( u p% ( y( L y ( = p ( y( u L y ( u p ( u L u q% ( y( L y ( = q ( y( u L y ( u q ( u L u we ca rewre model ( as follows:

4 IJCSNS Ieraoal Joural of Compuer Scece ad Newor Secury VOL.8 No.2 February dy ( =a ( y ( b ( y ( c g ( y ( d v K ( s h% ( u ( s ds (7 y( = p% ( y( L y( q% ( y(( τ ( L y(( τ ( = for = 2 L = 2 L. Sce W s a M -marx ad he se Δ s oempy from lemma here exss T ξ = ( ξ ξ2 L ξ Δ Ω( W such ha ξb ξ ( cg df vh < = 2 L. (8 We ca choose a suffcely small posve cosa ε > such ha τε ξ b ξ ( c G e d F v r( ε H < a = 2 L. (9 Le ( x( e ε = y( = 2 L. Calculag he upper rgh dervave Dx ( of x ( alog he soluos of (7 from he assumpo (H2 (H3 (H4 ad (H5 we ca ge ε( ε( D x( = εe y( εe sg( y( a% ( y ( b% ( y ( c g% ( y ( = d f% ( y ( τ ( = v K ( s h% ( y ( s ds = a% ( y ( εe y ( εe ε( ε( a a% ( y( b y( c G y( % % % d f% ( y ( τ ( = d F y ( τ ( v K ( s y ( H ds = a% ( y( = ε x( a% ( y( bx ( a ( τ c G x ( d F e x ( ( v K ( s x ( H ds a% ( y ( b x ( c G x ( a = ( τ ( = e d F x v H K ( s x ( ds = for = 2 L ; < < = 2 L. Leg φ u l = m{ ξ } he we have ε ( ( s x s = e y( s y( s = φ( s u φ u ξl < s = 2 L. Le us prove x( ξl < = 2 L. ( If ( s o rue he here exs some ad [ ] such ha x( = ξl Dx ( ad x ( ξ l for < = 2 L. However from (9 ( ad (H5 we ge D x( a% ( y( b ξ a c G ξ e d F ξ vhr ( εξ l < ad hs s a coradco. So x( ξl < = 2 L whch s ( y( ξle ε < = 2 L. (2 I he followg we wll use he mahemacal duco o prove ha ( ( αα α ξle ε L < 2 y = L = 2 L (3 holds for α =.

5 98 IJCSNS Ieraoal Joural of Compuer Scece ad Newor Secury VOL.8 No.2 February 28 Whe = from (2 we ow ha (3 holds. Suppose ha he equales ( y( αα α ξle ε L < = 2 L holds for = 2 L m. From assumpo (H6 ad (4 he dscree par of model (7 sasfes ha y ( p% ( y ( L y ( m m m m q% ( y (( τ ( L y (( τ ( m m m m m ( m ( m p y ( m q y (( m τ ( m ( m ε ( m p αα Lαmξ le ( m ε( mτ( m q αα Lαmξ le ( m ( m ( m p e q m le ε ξ ξ α α Lα (5 for = 2 L. From ξ = ( ξ ξ2 L ξ T Δ ad Lemma 2 we ow ha ξ Γ ( P m ad ξ Γ ( Q m. Thus Pmξ = ρ( Pm ξ Qmξ = ρ( Qm ξ.e. From (5 (6 ad (4 we ge ε ( m y( m ( ρ( Pm e ρ( Qm ααlαm ξle (7 ε ( m αα Lαmαmξle for = 2 L. Ths ogeher wh boh (4 ad (2 lead o ( y( m αα αm αmξle ε L = 2 L ; ( m ] (8.e. x( ααl αm αmξl = 2 L ; ( m ] (9 I he followg we wll prove ha x( ααl αm αmξl = 2 L ; [ m m (2 holds. If (2 s o rue he here exs some ad [ m m such ha x( = ααl αm αmξl D x ( ad x ( ααl αm αmξl For < = 2 L. However from (9 ( ad (H5 we ge D x a% y b ξ ( ( ( a c Gξ e d Fξ vhr ( εξ αα Lαmαml < whch s a coradco. Ths dcaes ha (2 holds. To hs ed by he mahemacal duco we ca coclude ha (3 holds. From (5 we have ( α e α = 2 L. I follows from (3 ha α( α ( 2 α( 2 ε( y( e e Le ξle ξ α( ε( = φ u e e m{ ξ} ξ α( ε( φ u e e m{ ξ} ξ ( εα( = φ u e m{ ξ} for ay [ = 2 L ha s ξ ( ( u ( u u e ε α φ m{ ξ} for So ( ( u ( u M φ u e ε α ξ for where M =. Ths meas ha he m{ ξ} equlbrum po u of model ( s globally expoeally sable ad he expoeal covergece rae equals ε α. The proof s compleed. Remar 2. We may choose approprae marces P ad Q assumpo (H6 o guaraee ha he se Δ Theorem s oempy. I parcular whe P = p E ad Q = qe ( p q are oegave cosas ad E s a u marx Δ s ceraly oempy. So by usg Theorem we ca oba he followg corollary easly. Corollary : Uder assumpos (H-(H5 he equlbrum po of model ( s globally expoeally sable f he followg codos are sasfed ( There exs oegave cosas such ha p ( u Lu p ( v L v p u v p ad q

6 IJCSNS Ieraoal Joural of Compuer Scece ad Newor Secury VOL.8 No.2 February q ( u Lu q ( v L v q u v for all ( u L u T R ( v L v T R = 2 L = 2 L. ( W = B C G D F V H s M -marx. ( Le α max{ p qe }. Assume ha here exss a cosa α such ha lα α < ε = 2 L where he scalar ε s deermed by he equaly ξ ξ ( ε < ( b a = c G e d F v r H T for a gve ξ = ( ξ ξ2 L ξ Ω( W. Proof. Nocg codo ( s a specal case of (H6 wh P = pe ad Q = q E we ow ha (H6 s sasfed. I ca be easly compued ha Δ= I Γ( P IΓ( Q I Ω ( W =Ω( W. From codo = [ ] ( we ow ha Ω ( W s oempy ad herefore he codo ( of Theorem s sasfed. By usg Theorem we ca deduce he cocluso ad he proof s complee. From Theorem of [8] ad Corollary of hs paper we ca prove he followg resul. For smplcy of he preseao he proof s spped. Corollary 2: Uder assumpos (H2-(H5 model (2 has a uque equlbrum po whch s globally expoeally sable f W = B C G D F V H s a M -marx. Remar 3. I [2-6] [ ] he amplfcao fucos were requred o sasfy < a a( u a < for all u R = 2 L. I s worh pog ou our paper he upper boud cosra o he amplfcao fucos s o loger eeded. I addo assumpo (H3 o he behaved fucos our resuls s he same as ha [5 8 8] ad he codo for dffereably mposed o behaved fucos [2-4] [6] [7] [9-] s removed our resuls. Remar 4. Corollary 2 of hs paper shows ha here s a uque equlbrum po u of he couous par of he sysem ( uder assumpos (H2-(H5. I may cases u may o be a soluo of he dscree par of he sysem ( whou he exeral mpulsve pu. I oher words he ere sysem ( may have o equlbrum po. I order o guaraee ha he ere sysem ( has a equlbrum po as [33] we ca roduce he exeral mpulsve pu J so ha u s also a equlbrum po of he dscree par of he sysem (. 4. Example Example. Cosder he followg model du ( = (2 cos u( [ u( f( u( f2( u2( d g( u( τ ( 2 K( sh ( sds K2 ( sh 2 ( sds > du2 ( =(3 s u2( [ 2 u2( 2 f2( u2( d 2 g( u( τ( 3 g2( u2( τ( (3 K( s h ( s ds 2 K2 ( s h2 ( s ds 23 >.5.5 u( =.2 e u(. e u2( e u (( τ (.4 e =.5.5 u2( =.4 e u(.2 e u2( e u2 (( τ ( 2.8 e = g ( x = g ( x = f ( x = f ( x = h ( x = h ( x = x where K ( x = K ( x = K ( x = K ( x = e τ ( = cos. = =.5 = 2 L. Oe ca verfy ha he po ( 2 T s a equlbrum po of model (3 ad model (3 sasfes assumpos (H2-(H6 wh a = a 2 = 2 B = 2 C = D 2 = V = F G H.5 2 = = = P =.e Q =.4e τ =.. I ca be easly checed ha 7 2 W=B- C F D G V H = s a M -marx ad ρ( P = ρ( Q =.4e. Γ ( P = T ( z z z = 2z 2 Γ ( Q = R Therefore { 2 2 } ( W {( z z2 T.6z z2 3.5 z z z2 } {( z z2 T z2 2 z z z2 } Ω = < < > > so Δ = = > > s oempy. Le ξ = ( 2 T Δ ad ε =.264 so ha he followg equales

7 IJCSNS Ieraoal Joural of Compuer Scece ad Newor Secury VOL.8 No.2 February 28 ( ξ b ξ c G e d F v r( ε H < a =.5 hold for = 2. Tag α = e α =. we ow ha he equales α max{.4e.4e e } = 2 L ad.5 lα l e = α < ε = 2 L.5 are sasfed. Clearly all codos of Theorem are sasfed. From Theorem we ow ha he uque equlbrum po ( 2 T of model (3 s globally expoeally sable ad he expoeal covergece rae equals.64. The global expoeal sably of equlbrum po ( 2 T of model (3 s furher verfed by he smulao gve Fgure where he al sae s ae as u ( 2 s(3 u ( cos(4 2 s ( ]. u Fg. Quazao procedure for measureme of ose level flucuao. 5. Cocluso I hs paper he problem o expoeal sably has bee vesgaed for a class of mpulsve Cohe-Grossberg eural ewors wh boh he me-varyg ad dsrbued delays. Several suffce codos for checg he global expoeal sably of equlbrum po have bee esablshed by usg he M-marx heory ad aalyc mehods. Moreover he expoeal covergece rae dex has bee esmaed whch depeds o he sysem parameers. The proposed resuls have geeralzed some recely ow oes he leraure ad removed some resrcos o he eural ewors. A example wh smulao has bee gve o show he effecveess of he obaed resuls. u u2 Acowledgme Ths wor s suppored by he Naoal Naural Scece Foudao of Cha uder Gra he Naural Scece Foudao of CQ CSTC uder gra 27BB43 he Scefc Research Fud of Chogqg Mucpal Educao Commsso uder Gra KJ74. Refereces [] M. A. Cohe Grossberg S. Absolue sably of global paer formao ad parallel memory sorage by compeve eural ewors. IEEE Trasacos o Sysems Ma ad Cyberecs 983; 3(5: [2] H Ye A.N. Mchel K.N. Wag. Qualave aalyss of Cohe-Grossberg eural ewors wh mulple delays. Physcal Revew E 995; 5(3: [3] L. Wag X.F. Zou. Harmless delays Cohe-Grossberg eural ewors. Physca D 22; 7 (2: [4] C.C. Hwag C.J. Cheg T.L. Lao. Globally expoeal sably of geeralzed Cohe-Grossberg eural ewors wh delays. Physcs Leers A 23; 39 (-2: [5] X.F. Lao C.G. L K.W. Wog. Crera for expoeal sably of Cohe-Grossberg eural ewors. Neural Newors 24; 7 (: [6] J.D. Cao J.L. Lag. Boudedess ad sably for Cohe-Grossberg eural ewor wh me-varyg delays. Joural of Mahemacal Aalyss ad Applcaos 24; 296 (2: [7] C.H. Feg R. Plamodo. Sably aalyss of bdrecoal assocave memory ewors wh me delays. IEEE Trasacos o Neural Newors 23; 4 (6: [8] S. Ar Z. Orma. Global sably aalyss of Cohe-Grossberg eural ewors wh me varyg delays. Physcs Leers A 25; 34 (5-6: [9] K. Yua J.D. Cao. A aalyss of global asympoc sably of delayed Cohe-Grossberg eural ewors va osmooh aalyss. IEEE Trasacos o Crcus ad Sysems I 25; 52(9: [] J.D. Cao X.L. L. Sably delayed Cohe-Grossberg eural ewors: LMI opmzao approach. Physca D 25; 22 (-2: [] J.H. Su L. Wa. Global expoeal sably ad perodc soluos of Cohe-Grossberg eural ewors wh couously dsrbued delays. Physca D 25; 28 (-2: -2. [2] L. Wag. Sably of Cohe-Grossberg eural ewors wh dsrbued delays. Appled Mahemacs ad Compuao 25; 8(:93-. [3] J. Lu. Global expoeal sably of Cohe-Grossberg eural ewors wh me-varyg delays. Chaos Solos ad Fracals 25; 26 (3: [4] F.H. Tu X.F. Lao. Harmless delays for global asympoc sably of Cohe-Grossberg eural ewors. Chaos Solos ad Fracals 25; 26(3: [5] Z.D. Wag Y.R. Lu M.Z. L X.H. Lu. Sably aalyss for sochasc Cohe-Grossberg eural ewors wh mxed me delays. IEEE Trasacos o Neural Newors 26; 7:

8 IJCSNS Ieraoal Joural of Compuer Scece ad Newor Secury VOL.8 No.2 February 28 [6] C. Wu J. Rua W. L. O he exsece ad sably of he perodc soluo he Cohe-Grossberg eural ewor wh me delay ad hgh-order erms. Appled Mahemacs ad Compuao 26; 77(: [7] K.N. Lu D.Y. Xu Z.C. Yag. Global araco ad sably for Cohe-Grossberg eural ewors wh delays. Neural Newors 26; 9(: [8] Q.K. Sog J.D. Cao. Sably aalyss of Cohe-Grossberg eural ewor wh boh me-varyg ad couously dsrbued delays. Joural of Compuaoal ad Appled Mahemacs 26 97(: [9] Q.K. Sog J.D. Cao. Global expoeal robus sably of Cohe-Grossberg eural ewor wh me-varyg delays ad reaco-dffuso erms. Joural of he Fral Isue 26; 343(7: [2] T.W. Huag C.D. L G. Che. Sably of Cohe-Grossberg eural ewors wh ubouded dsrbued delays. Chaos Solos ad Fracals 27; 34(3: [2] C.H. L S.Y. Yag. A furher aalyss o harmless delays Cohe-Grossberg eural ewors. Chaos Solos ad Fracals 27; 34(2: [22] F. Log Y.X. Wag S.Z. Zhou. Exsece ad expoeal sably of perodc soluos for a class of Cohe-Grossberg eural ewors wh bouded ad ubouded delays. Nolear Aalyss: Real World Applcaos 27; 8(3: [23] Z. Che D.H. Zhao J. Rua. Dyamc aalyss of hgh-order Cohe-Grossberg eural ewors wh me delay. Chaos Solos ad Fracals 27; 32(4: [24] Z.S. Mao H.Y. Zhao. Dyamcal aalyss of Cohe-Grossberg eural ewors wh dsrbued delays. Physcs Leers A 27; 364(: [25] Q.H. Zhou L. Wa L. Su. Expoeal sably of reaco-dffuso geeralzed Cohe-Grossberg eural ewors wh me-varyg delays. Chaos Solos ad Fracals 27; 32(5: [26] W. Wu B.T. Cu M. Huag. Global asympoc sably of delayed Cohe-Grossberg eural ewors. Chaos Solos ad Fracals 27; 34(3: [27] Z.H. Gua G.R. Che. O delayed mpulsve Hopfeld eural ewors. Neural Newors 999; 2(2: [28] K. Gopalsamy. Sably of arfcal eural ewors wh mpulses. Appled Mahemacs ad Compuao 24; 54(3: [29] Y.K. L. Exsece ad sably of perodc soluos for Cohe-Grossberg eural ewors wh mulple delays. Chaos Solos ad Fracals 24; 2(3: [3] Y.K. L. Global expoeal sably of BAM eural ewors wh delays ad mpulse. Chaos Solos ad Fracals 25; 24(: [3] X.F. Yag X.F. Lao D.J. Evas Y.Y. Tag. Exsece ad sably of perodc soluo mpulsve Hopfeld eural ewors wh fe dsrbued delays. Physcs Leers A 25; 343(-3: 8-6. [32] Y. Zhag J.T. Su. Sably of mpulsve eural ewors wh me delays. Physcs Leers A 25; 348(-2: [33] D.Y. Xu Z.C. Yag. Impulsve delay dffereal equaly ad sably of eural ewors. Joural of Mahemacal Aalyss ad Applcaos 25; 35 (: 7-2. [34] Z. Che J. Rua. Global sably aalyss of mpulsve Cohe-Grossberg eural ewors wh delay. Physcs Leers A 25; 345 (-3: -. [35] Z.C. Yag D.Y. Xu. Exsece ad expoeal sably of perodc soluo for mpulsve delay dffereal equaos ad applcaos. Nolear Aalyss 26; 64(: [36] Z.C. Yag D.Y. Xu. Impulsve effecs o sably of Cohe-Grossberg eural ewors wh varable delays. Appled Mahemacs ad Compuao 26; 77(: [37] D.W.C. Ho J.L. Lag J. Lam. Global expoeal sably of mpulsve hgh-order BAM eural ewors wh me-varyg delays. Neural Newors 26; 9(: [38] Z. Che J. Rua. Global dyamc aalyss of geeral Cohe-Grossberg eural ewors wh mpulse. Chaos Solos ad Fracals 27; 32(5: [39] F.J. Yag C.L. Zhag D.Q. Wu. Global sably aalyss of mpulsve BAM ype Cohe-Grossberg eural ewors wh delays. Appled Mahemacs ad Compuao 27; 86(: Zheag Zhao receved he M.S. degrees Deparme of Mahemacs OKAYAMA Uversy of Japa 996 ad 998. Durg he sayed TSS Sofware Co. Ld. Japa o develop sofware of compuer. Sce 22 He has wored Deparme of Mahemacs Huzhou Teachers College Zheag Cha. Qau Sog was bor 964. He receved he B.S. degree Mahemacs 986 from Schua Normal Uversy Chegdu Cha ad he M.S. degree Appled Mahemacs 996 from Norhweser Polyechcal Uversy X a Cha. He was a sude a refresher class he Deparme of Mahemacs Schua Uversy Chegdu Cha from Sepember 989 o July 99. From July 986 o December 2 he was wh Deparme of MahemacsSchua Uversy of Scece ad Egeerg Schua Cha. From Jauary 2 o Jue 26 he was wh he Deparme of Mahemacs Huzhou Uversy Zheag Cha. I July 26 he moved o he Deparme of Mahemacs Chogqg Jaoog Uversy Chogqg Cha. He s currely a Professor a Chogqg Jaoog Uversy. He s he auhor or coauhor of more ha 4 oural papers ad oe eded boo. Hs curre research eress clude eural ewors chaos sychrozao ad sably heory.