ITM Web of Cofeeces, 0509 (07) DOI: 0.05/ mcof/070509 ITA 07 Expoeal Sychozao of he Hopfeld Neual Newos wh New Chaoc Sage Aaco Zha-J GUI, Ka-Hua WANG* Depame of Sofwae Egeeg, Haa College of Sofwae Techology,qogha, 57400, P.R. Cha School of Mahemacs ad Sascs, Haa Nomal Uvesy, Haou, Haa, 5758, P.R. Cha * Coespodg auho: ahuawag@qq.com Absac Ths pape sudes he poblems of expoeal sychozao fo he Hopfeld eual ewos wh mpulsve effecs ad Gu chaoc sage aaco. By employg he Lyapuov fucoal mehod of mpulsve fucoal dffeeal equaos, some cea fo sychozao bewee wo mpulsve eual ewos ae deved. A llusave example s povded o show he effecveess ad feasbly of he poposed mehod ad esuls. Ioduco Recely, he ssue of coolled sychozao complex dyamcal ewos has become a ahe sgfca opc boh heoecal eseach ad paccal applcaos [ 3]. I s ow ha heoy ad pacce, mpulsve cool has bee wdely used o sablze ad sychoze chaoc sysems. Fo example, Yag ad Chua deved some suffce codos fo he sablzao ad sychozao of Chua's oscllaos va mpulsve cool[4]; Xe, We ad L obaed suffce codos fo he sablzao ad sychozao of he Loez sysem va mpulsve cool wh vayg mpulsve evals[5]. The Hopfeld eual ewos wh mpulse effec ae suded, whee he cea o he exsece, uqueess ad global sably of peodc soluo ae obaed. Fuhe, Gu chaos sage aaco was also foud[6-0]. Movaed by he above dscussos, he am of hs pape s o sudy he sychozao of Hopfeld eual ewos wh a Gu chaoc sage aaco. By employg he Lyapuov-le sably heoy of mpulsve fucoal dffeeal equaos, some cea fo sychozao of Hopfeld eual ewos ae deved. The emade of he pape s ogazed as follows: Seco descbes he ssue of sychozao of coupled mpulsve sysems wh a Gu chaoc sage aaco. I Seco 3, some suffce codos fo he sychozao ae deved by cosucg suable Lyapuov-le fuco. I Seco 4, a llusave example s gve o show he effecveess of he poposed mehod. Coclusos ae gve Seco 5. Pelmaes ad Poblem Fomulao I hs pape, we cosde he followg oauoomous Hopfeld eual ewos model wh mpulses x ax() b f ( x()) c(), 0,, () x( ) x( ) x( ) - x( ), 0,, whee x( ) x( ) x( ) ae he mpulses a momes ad s a scly ceasg sequece such ha lm ; x( ) coespods o he sae of he h u a me, a s a posve cosa se; b deoes he segh of he h u o he h u a me, c () deoes he exeal pu o he h euo ad f ( x( )) deoes he oupu of he h u a me. As usual he heoy of mpulsve dffeeal equaos, a he pos of dscouy of he soluo x () we assume ha x( ) x( ). I s clea ha, geeal, he devaves x ( ) do o exs. O he ohe had, accodg o he fs equaly of () hee exs he lms x( ). The Auhos, publshed by EDP Sceces. Ths s a ope access acle dsbued ude he ems of he Ceave Commos Abuo Lcese 4.0 (hp://ceavecommos.og/lceses/by/4.0/).
ITM Web of Cofeeces, 0509 (07) DOI: 0.05/ mcof/070509 ITA 07 Accodg o he above coveo, we assume x ( ) x ( ). Thoughou hs pape, we assume ha: (H) Fucos f ( u ) sasfy he Lpschz codo,.e., hee ae cosas L 0 such ha f ( u ) f ( u ) L u u, fo all u, u R (, ). (H) Thee exss a posve ege p, such ha Fgue. Tme-sees of he x () of sysem () wht,, p ( p),,,,,, (H3) a 0, c( ) ae all couous -peodc fucos. As s ow o all ha () ca exhb chaoc pheomeos [6-0]. I ode o show clealy, we gve he followg example. Example. Cosde a wo-dmesoal eual ewo wh mpulsve effecs, whch ca be descbed by he followg mpulsve dffeeal equaos: Fgue. Tme-sees of he x () of sysem () wh T x x( ) 0.56 f( x( )) 0.48 f( x( )) c (), 0, T, x x( ) 0.6 f( x( )) 0.3 f( x( )) c (), 0, T, x( T ) ( ) x( T ),,,, x( T ) ( ) x( T ),,,, () whee f ( x) 0.5( x x ). Obvously, f ( x ) sasfy (H ). Now we vesgae he fluece of he peod T of mpulsve effec o he sysem (). Se c ( ) 0.5cos(.5 ), c ( ) 0.5s(0.5 ), T 0.4, 0.35, 0.. Accodg o Theoems ad [6], he cellula eual ewo model () has a uque -peodc soluo wh 5-mpulses a peod. Fuhemoe, f T, he (H ) s' sasfed. Peodc oscllao of sysem () wll be desoyed by mpulses effec. Numec esuls show ha sysem () sll has a global aaco whch ca be a Gu chaoc sage aaco. Evey soluos of sysem () wll fally ed o he ew chaoc sage aaco whch s useful expoeal sychozao (see Fg.-3). Fgue 3. Phase poa of Gu chaoc sage aaco of sysem () Fo he pupose of sychozao, we oduce he espose sysem ha s dve by () va a se of sgals y ay() b f ( y()) c(), 0,, (3) y( ) x( ) y( ),,,,,,,, Le e() y() x() be he sychozao eo, x () ad y () ae he sae vaables of dve sysem () ad espose sysem (3). The eo sysem of he mpulsve sychozao s gve by
ITM Web of Cofeeces, 0509 (07) DOI: 0.05/ mcof/070509 ITA 07 e ae ( ) b( e( )), 0,, e( ) e( ),,,,,,,, whee g ( e( )) f ( y( )) f ( x( )). Noe ha he og s he equlbum po of sysem (4). If e () eds expoeally o og evoluo, expoeal sychozao bewee wo sysems would be ealzed. Ou am s o fd some cea o he mpulsve gas such ha dve sysem () ad espod sysem (3) ae expoeally sychozed fo ay al codo. Accodg o he assumpo (H ), g () possesses he followg popees: g ( e( )) L e( ), ad g (0) 0,,,. Thoughou he pape, we deoe (4) y () x () y() x(). Defo. Sysems () ad (3) ae sad o be expoeally sychozed f hee exs cosas M ad 0 such ha y () x () M y(0) x(0) e, fo ay 0. Cosa s sad o be he degee of expoeal sychozao. Lemma. Assume ha x 0, y 0, p, q, p q, he he equaly holds (he equaly s called as Youg equaly). p xy x x p q 3 Ma esuls I hs seco, we vesgae he expoeal sychozao of sysem () ad (3) by usg Lyapuov le fucoal mehod. Theoem. Ude he assumpo (H), (H) ad (H3), he sysem () ad (3) ae expoeally sychozed, f hee exs posve cosas, w 0 ad R,,,,. such ha ( )/( ) ( )/( ) ( ) L b a L b 0,,,,. (5) q ( )/( ) ( )/( ) ( a ) ( ) L b L b 0,,,,. Cosde he followg Lyapuov fuco: Ve (()) e () e. (6) Calculag he uppe gh d-devave D V of V alog he soluo of sysem (4) a he couous pos, 0, we have D e ( ) e e( ) sg( e( )) e D V( e( )) e e ( ) e e ( ) e ( ) ae ( ) b g ( e ( )) b e ( ) g ( e ( )). By Lemma, we have he followg equales: ( )/( ) ( )/( ) DV(()) e e a e() ( ) L b e ( ) L b e( ) ( )/( ) a ( ) L ( )/( ) b L b e ( ) e 0, 0,. Also, we ca calculae gh lms of Lyapuov-le fuco Ve (()) a mpulsve momes as follows Poof. By equaly (5), choose a small 0 such ha 3
ITM Web of Cofeeces, 0509 (07) DOI: 0.05/ mcof/070509 ITA 07 whch mples ha Ve (( )) e ( ) e e ( ) e Ve (()) Ve ((0)) e(0) y (0) x (0) max{ } y(0) x(0), e ( ) e V( e( )), fo 0. (7) e e( ) 0.56 g ( e( )) 0.48 g( e( )), e e( ) 0.6 g ( e( )) 0.3 g( e( )), T, (0) e( T ) e( T ),,,, e( T ) e( T ),,,. Numec esuls show ha sysem (9) sll has a Gu chaoc sage aaco [6-0]. The phase plo of Gu chaoc sage aaco of espose sysem s show Fg. 4. I s easy o chec he codos Theoem ae sasfed. Theefoe, sysems () ad (9) exhb expoeal sychozao. Defe e() y() x() ad he eos (0) bewee sysems () ad (9) ae depced Fg. 5. Accodg o (6), we have Ve (()) m{ } e () e m{ } e y( ) x( ). (8) Fom (7)(8), we ca oba y ( ) x ( ) M y(0) x(0) e. whee M max { } m { }. Accodg o Defo, we coclude ha he dve sysem () ad he espose sysem (3) ae expoeally sychozed. Ths complees he poof. Fgue 4. Phase poa of Gu chaoc sage aaco of sysem (9) 4 A Smulao Example I hs seco, we gve a example o llusae he effecveess of he esuls obaed he pevous secos. Cosde a wo-dmesoal eual ewos wh mpulsve effecs. Tag () as he dve sysem Example. The espose sysem s cosuced as follows: y y( ) 0.56 f( y( )) 0.48 f( y( )) c( ), T, y y( ) 0.6 f( y( )) 0.3 f( y( )) c( ), T, y( T ) x( T ) y( T),,,, y( T ) x( T ) y( T),,,. The he eo sysem of dve sysem () ad espod sysem (9) s cosuced as follows: (9) Fgue 5. Sychozao eos bewee dve sysem () ad espose sysem (9) Coclusos I hs pape, he codos fo he expoeal sychozao of a class of eual ewos wh mpulsve effecs ae deved by ulzg Lyapuov fucoal mehod. A umecal smulao s gve o show he effecveess ad feasbly of he poposed mehod. As fa as we ow, hee s o pape o deal wh such a poblem. 4
ITM Web of Cofeeces, 0509 (07) DOI: 0.05/ mcof/070509 ITA 07 Acowledgmes Ths eseach was oly suppoed by he Naue Scece Foudao of Haa Povce ude Ga No. 764, 7097 ad Foudao of Haa Educaoal Commee ude Ga No. Hy07ZD-0. Refeeces [] T. Hayaawa, W.M. Haddad, K.Y. Volyasyy. Neual ewo hybd adapve cool fo olea ucea mpulsve dyamcal sysems, Nolea Aal. Hybd Sys. (008):86 874. [] G. Agaovch, E. Lsy, A. Slavova. Impulsve cool of a hyseess cellula eual ewo model, Nolea Aal. Hybd Sys. 3(009):65 73. [3] J. Zhou, L. Xag, Z.R. Lu. Sychozao complex delayed dyamcal ewos wh mpulsve effecs, Physca A. 384(007):684 69. [4] T. Yag, L.O. Chua. Impulsve sablzao fo cool ad sychozao of chaoc sysem, IEEE Tas. Ccus Sys. I. 44(997):976-988. [5] W. Xe, C. We, Z. L.Impulsve cool fo he sablzao ad sychozao of Loez sysem, Phys. Le. A. 75(000): 67-7. [6] Z.J. Gu, W.G. Ge. Exsece ad uqueess ad peodc soluos of oauoomous cellula eual ewos wh mpulses, Phys. Le. A. 354(006): 84-94. [7] J. Zhag, Z.J. Gu. Exsece ad sably of peodc soluos of hgh-ode Hopfeld eual ewos wh mpulses ad delays. Joual of Compuaoal ad Appled Mahemacs. 4()(009): 60-63. [8] J. Zhag, Z.J. Gu. Peodc soluos of oauoomous cellula eual ewos wh mpulses ad delays. Nolea Aalyss: Real Wold Applcaos, 0(3)(009):89-903 [9] Z.J. Gu, W.G. Ge. Peodc soluo ad chaoc sage aaco fo shug hboy cellula eual ewos wh mpulses. I Chaos. 6 (006): 336-- 0. [0] Z.J. Gu, W.G. Ge. Impulsve effec of couous me eual ewos ude pue sucual vaaos.i Ieaoal Joual of Bfucao ad Chaos. 7(007): 7-39. 5