Exponential Synchronization for Fractional-order Time-delayed Memristive Neural Networks

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1 Ieraoal Joural of Advaced Nework, Moorg ad Corols Volume 3, No.3, 8 Expoeal Sychrozao for Fracoal-order Tme-delayed Memrsve Neural Neworks Dg Dawe, Zhag Yaq ad Wag Na 3* School of Elecrocs ad Iformao Egeerg, Ahu Uversy Hefe, 36, Cha e-mal: dwdg@ahu.edu.c e-mal: @qq.com 3* e-mal: w_xlb@ahu.edu.c Absrac Cosderg he fac ha he expoeal sychrozao of eural eworks has bee wdely used heorecal research ad praccal applcao of may scefc felds, ad here are a few researches abou he expoeal sychrozao of fracoal-order memrsor-based eural eworks (FMNN). Ths paper coceraes o he FMNN wh me-varyg delays ad vesgaes s expoeal sychrozao. A smple lear error feedback coroller s appled o compel he respose sysem o sychroze wh he drve sysem. Combg he heores of dffereal clusos ad se valued maps, a ew suffce codo cocerg expoeal sychrozao s obaed based o comparso prcple raher ha he radoal Lyapuov heory. The obaed resuls exed expoeal sychrozao of eger-order sysem o fracoal-order memrsor-based eural eworks wh me-varyg delays. Fally, some umercal examples are used o demosrae he effecveess ad correcess of he ma resuls. Keywords-Expoeal Sychrozao; Memrsor-based Neural Neworks; Fracoal-order; Lear Error Feedback Corol; Tme-varyg Delays. I. INTRODUCTION Chua already supposed he exsece of memrsor 97 [], however, he praccal devce of memrsor elecrocs s obaed [] ul 8. I addo o he exsg hree kds of crcu elemes, memrsor s regarded as he fourh basc crcu eleme ad s defed by a olear charge-flux characersc. As everyoe kows, ressors ca be used o work as coeco weghs so ha ca emulae he syapses arfcal eural eworks. However, he eural eworks of bologcal dvdual, log-erm memores s esseal he syapses amog euros, bu for he geeral ressors, s mpossble o have he fuco of memory. Recely, due o he memory characerscs of memrsor, memrsor ca replace he ressor o develop a ew eural eworks ha s memrsor-based eural eworks (MNN) [3-6]. I rece years, more ad more aeos have bee pu o he dyamcal aalyss of memrsor-based eural eworks, such as he vesgao of sably [7-], perodcy [-3], sysem sychrozao [4-], passvy aalyss [3], dsspavy [4-5] ad aracvy [6]. Parcularly, he sably ad sychrozao of MNN has bee wdely suded [7-3]. I fac, sychrozao meas he dyamcs of odes share he commo me-spaal propery. Therefore we ca udersad a ukow dyamcal sysem by achevg he sychrozao wh he well-kow dyamcal sysems [8]. Moreover, he rasmsso of dgal sgals, commucao wll become secury, relable ad secrecy by achevg sychrozao bewee he varous sysems. Therefore, he sychrozao of MNN s sll worh furher research. Moreover, he fracoal-order models ca beer descrbe he memory ad geec properes of varous

2 Ieraoal Joural of Advaced Nework, Moorg ad Corols Volume 3, No.3, 8 maerals ad process, so he fracoal-order models have receved a lo of research aeos ha eger-order models. I rece years, wh he mproveme of fracoal-order dffereal calculus ad fracoal-order dffereal equaos, s easy o model ad aalyze praccal problems [3, 3]. Therefore, here have bee a lo of researches abou he dyamcal aalyss ad sychrozao of fracoal-order memrsor-based eural eworks (FMNN) [34-39]. Fe-me sychrozao, hybrd proecve sychrozao ad adapve sychrozao of FMNN have all bee researched [34-36]. However, here are oly a very few research resuls o expoeal sychrozao of FMNN. I fac, he expoeal sychrozao of eural eworks has bee wdely used he heorecal research ad praccal applcao of may scefc felds, for example, assocave memory, ecologcal sysem, combaoral opmzao, mlary feld, arfcal ellgece sysem ad so o [4-43]. So he expoeal sychrozao of FMNN s sll worh furher sudyg as s a sgfca academc problem. O he oher had, he sably ad sychrozao of FMNN whou me delay have bee deeply suded such as [33]. However, hardware mplemeao of eural eworks, me delay s uavodable owg o he fe swchg speeds of he amplfers. Ad wll cause sably, oscllao ad chaos pheomea of sysems. So he vesgao for sably ad sychrozao of FMNN cao be depede o he me delay. Movaed by he above dscusso, hs paper sudes he expoeal sychrozao of FMNN wh me-varyg delays. The ma corbuos of hs paper ca be lsed as follow. () Ths s he frs aemp o acheve expoeal sychrozao of FMNN wh me-varyg delays by employg a smple lear error feedback coroller. () The suffce codo for expoeal sychrozao of FMNN wh me delays s obaed based o comparso prcple sead of he radoal Lyapuov heory. (3) Some prevous research resuls of expoeal sychrozao for eger-order memrsor-based sysem are he specal cases of our resuls. Furhermore, some umercal examples are gve o demosrae he effecveess ad correcess of he ma resuls. The res of hs paper s orgazed as follows. Prelmares cludg he roduco of Capuo fracoal-order dervave, model descrpo, assumpos, defos ad lemmas are preseed Seco. Seco 3 roduces he suffce codo for expoeal sychrozao of he FMNN. I Seco4, he umercal smulaos are preseed. Seco5 gves he cocluso of hs paper. II. PRELIMINARIES Compared o he eger-order dervaves, we kow he dsc advaage of Capuo dervave s ha oly requres al codos from he Laplace rasform of fracoal dervave, ad ca represe well-udersood feaures of physcal suaos ad makg more applcable o real world problems [36]. So he res of hs paper, we apply he Capuo fracoal-order dervave for he fracoal-order memrsor-based eural eworks (FMNN) ad vesgae he expoeal sychrozao of FMNN. A. The Capuo fracoal-order dervave Defo [3] The Capuo fracoal-order dervave s defed as follows: q s he order of fracoal dervave, m s he frs eger larger ha q m q m,, s he Gamma fuco,

3 Ieraoal Joural of Advaced Nework, Moorg ad Corols Volume 3, No.3, 8 Parcularly, whe q, B. Model descrpo I hs paper, referrg o some releva works o FMNN [35,36], we cosder a class of FMNN wh me-varyg delays descrbed by he followg equao, x volage of capacor dervave, s he sae varable of he h euro (he c C ), q s he order of fracoal s he self-regulag parameers of he euros, ad ( s a cosa ) represes he rasmsso me-varyg delay. f, g : R R feedback fucos whou ad wh me-varyg delay. a x ad b x are are memrsve coecve weghs, whch deoe he euro ercoeco marx ad he delayed euro ercoeco marx, respecvely. memducaces of memrsors Ad W M ad deoe he R F ad respecvely. R represes he memrsor bewee he feedback f x fuco ad x, F represes he memrsor bewee he feedback fuco g x ad x I. represes he exeral pu. Accordg o he feaure of memrsor, we deoe (5) C. Assumpos, Defos ad Lemmas I he res of paper, we frs make followg assumpo for sysem (4). Assumpo: For N, s, s R, he euro acvao fucos, sasfy f g bouded, f g ad 3

4 Ieraoal Joural of Advaced Nework, Moorg ad Corols Volume 3, No.3, 8 s s, ad are oegave cosas. We cosder sysem (4) as drve sysem ad correspodg respose sysem s gve as follows:3 D y c y a y f y b y g y I u q,, N, (7) Where (8) u ad s a ler error feedback corol fuco whch u y x defed by, e e, e,..., e, e y x T. Accordg o he sysem (4) ad, N are cosas, whch deoes he corol ga. Nex, we defe he sychrozao error e as sysem (7), he sychrozao error sysem ca be descrbed as follows: D e c e a y f y a x f x q b y g y b x g x u ( ),, N a y, b y, a x, b x Accordg o he heores of dffereal clusos ad x se valued maps [4], f ad y are soluos of (4) are he same as hose defed above, () u y x e,, N ad (7) respecvely, sysem (4) ad sysem (7) ca be wre as follow: are cosas, whch deoes he corol ga. 4

5 Ieraoal Joural of Advaced Nework, Moorg ad Corols Volume 3, No.3, 8 Ad Where () Ad () co{ u, v} deoes he closure of covex hull geeraed by real umbers u ad v or real marces u ad v. The he sychrozao error sysem ca be descrbed as follows: 5

6 Ieraoal Joural of Advaced Nework, Moorg ad Corols Volume 3, No.3, 8 Defo [8] For, he expoeal al codo e s s,, R sychrozao of sysem (4) ad sysem (7) ca be rasformed o he expoeal sably of he error sysem (9) (error approaches o zero). The error sysem (9) s sad o sasfes e Q max sup s exp P,, s be expoeally sable, f here exs cosa Q, P,,,..., e e e e such ha he soluo T of error sysem (9) wh,,..., expoeal covergece. P, s called he esmaed rae of Lemma [4] Uder he assumpo, he followg esmao ca be obaed: co a y f y co a x f x A F e (), co b y g y co b x g x () B G e, A max a, a, B max b, b,, N, F e f y f x, G e g y g x, N. Proof: If y () For y, x, N we ca easly have par() hold. From (9) ad(), we ca ge, x, he co a y f y co a x f x a f y a f x a F e A F e. y () For, x, he co a y f y co a x f x a f y a f x 6

7 Ieraoal Joural of Advaced Nework, Moorg ad Corols Volume 3, No.3, 8 a F e A F e. x y y x (3) For or, he co a y f y co a x f x a ( f y f ()) a ( f () f x ) A f y f () A ( f () f x ) A ( f y f x ) A F e. The complee he proof of par (). I he smlar way, par() ca be easly hold. III. MAIN RESULTS We prese he expoeal sably resuls for he sychrozao error sysem of FMNN,whe he error sysem (9) s expoeally sable, he sysem (4) ad sysem (7) wll acheve he expoeal sychrozao. Theorem If here exs posve cosa,,,..., such ha for ay,,,..., () he he error sysem (9) s globally expoeally sable. Proof: Cosder followg equaly,,,..., W e, accordg o he error sysem (9) or (4) ad lemma, we ca ge he (6) Evaluag he fracoal order dervave of W alog he raecory of error sysem, he Defe W W W exp,,,,...,, 7

8 Ieraoal Joural of Advaced Nework, Moorg ad Corols Volume 3, No.3, 8 W e s max sup. s W, for ay,,,..., We wll prove ha, for, here mus exs W. Oherwse, sce, q ad some D W such ha,,..., W ad. The exp D q W c W A W B W W c W exp A exp W B W exp W exp c W exp A exp W B exp W c W exp A exp exp. W B W Moreover, from equaly(5), we have ( ) c A B exp,,,,...,, Therefore q so s easy o fd ha D W W W W exp,,,,...,. Thus, whch coradcs D q W. Tha shows e max sup e s exp,,,,...,. s I shows 8

9 Ieraoal Joural of Advaced Nework, Moorg ad Corols Volume 3, No.3, 8 e max sup e s exp,,,,...,. s Ths complees he proof. IV. NUMERICAL RESULTS Example Cosder wo-dmeso fracoal-order memrsor-based eural eworks I hs seco, we wll gve wo umercal examples o demosrae our aalyss o expoeal sychrozao of FMNN. q D x cx a x f x a x f x b x g x b x g x I q D x cx a x f x a x f x b x g x b x g x I c c, a x, a x.8,, x,., x, a x a x 4, x,.5, x,., x,.8, x, b x b x.5, x,., x,.5, x,.6, x, b x b x., x,.4, x, T T e e, I ( I, I ) (,), q.9 s( ), f x x ad ake he acvao fuco as g x.5 x x,,,. The model () has chaoc aracors wh al values x.45,.65 T whch ca be see Fgure. We cosder sysem () as he drve sysem ad correspodg respose sysem s defed as Eq.(7). Ad for u y x, he coroller he parameer s chose as 9.5,.5. From Theorem, whe we ake., we ca easly kow.7,, 9

10 x () x (),y () x (),y () Ieraoal Joural of Advaced Nework, Moorg ad Corols Volume 3, No.3, 8 ( c ) A B exp 9.5,.5, we ca ge.73,.3 s rue whe. So whe ( c ) A A B exp B exp.798, ( c ) A A B exp B exp.7. I sasfes he codo of Theorem, he he expoeal sychrozao of drve-respose sysem s 6 a x () acheved. 4 y () Whe he respose sysem wh hs coroller, we ge sae raecores of varable, x y ad x, y are depced Fgurea ad b. Moreover, - Fgure3a ad 3b depc he sychrozao error e, e curves bewee he drve sysem ad respose sysem. These umercal smulaos show he sae x, y raecores of varable ad, sychroous ad sychrozao error x y are e, e coverge o zero. These prove he correcess of he Theorem..8.6 are b x () y () Fgure. Expoeal sychrozao of sae varable wh a croller : x, y, b : x, y x () Fgure. The chaoc aracors of fracoal-order memrsor-based eural eworks(8)

11 e () e () Ieraoal Joural of Advaced Nework, Moorg ad Corols Volume 3, No.3, 8. a. b Fgure 3. Sychrozao error bewee he drve ad respose sysem a : e, b : e Example Cosder hree-dmeso fracoal-order memrsor-based eural eworks q D x cx a x f x a x f x a3 x3 f3 x3 b x g x b x g x b3 x3 3 g3 x3 3 I b x g x b x g x b3 x3 3 g3 x3 3 I q D x3 c3x3 a3 x f x a3 x f x a33 x3 f3 x3 b3 x g x b3 x g x b33 x3 3 g3 x3 3 I 3 q D x cx a x f x a x f x a3 x3 f3 x3 c c c, , x,, x,, x, a x a x a3 x, x,, x,, x,, x,, x,, x, a x a x a3 x, x,, x,, x, a x, x,, x3,, x3, a3 x3 a33 x3, x3,, x3,, x3,

12 Ieraoal Joural of Advaced Nework, Moorg ad Corols Volume 3, No.3, 8, x, b3, x x,, x3 3,, x3 3,, x3 3,, x3 3,, x3 3,, x3 3. 3, x,, x, b x b x, x,, x,, x,, x, b x b x, x,, x, b x b x b x b x , x,, x, e e, I ( I, I, I ) (,,) T T Ad 3 q.9 ad ake he acvao fuco as f x g x ah( x),,,3.we cosder sysem() as he drve sysem ad he correspodg respose sysem s defed Eq.(7). Ad for he coroller u y x, s chose as 9.5,.5, 3. From Theorem, we ake.. Accordg A max a, a, B max b, b,,,3 A B, we ca easly kow ( c ) A B exp s rue whe ,.5, 3 we ca ge So o whe.7, ad choose. ( c ) A A A B B B exp.89, ( c ) A A A B B B exp.99, ( c ) A A A B B B exp.4. I suggess he codo of Theorem s sasfed, he drve-respose sysem acheves he sychrozao. Whe he respose sysem wh hs coroller, we ge x, y x sae raecores of varable ad, y

13 x 3 (),y 3 () x (),y () x (),y () Ieraoal Joural of Advaced Nework, Moorg ad Corols Volume 3, No.3, 8 3 ad, 3 x y are depced Fgure 4a,4b,4c. Moreover, Fgure 5a,5b,5c depc he sychrozao error curves. a x () y (),, e e e 3 bewee he drve sysem ad respose.8.6 sysem. I s easy o see ha he sae raecores of varable x, y, x, y x3, y3 ad e 3 ad sychrozao error, e, e o zero. So he Theorem s proved o be correc. are sychroous are coverge I addo, we choose 9.5,.5, 3, accordg o he Theorem, eeds he followg equales o hold: b x () y () 7 l 3 3 l 3 3 l So, we us eed l 3 3 holds. We have he expoeal covergece rae, fgure 6 depcs he relao of me-varyg delay ad expoeal c x 3 () y 3 () covergece rae Fgure 4. Sychrozao of sae varable wh a 3 3 coroller : x, y, b : x, y, c : x, y 3

14 e 3 () e () e () expoeal covergece rae Ieraoal Joural of Advaced Nework, Moorg ad Corols Volume 3, No.3, 8. a b me-varyg delay Fgure 6. The relao of me-varyg delay ad expoeal covergece rae.. V. CONCLUSION c Ths paper acheves he expoeal sychrozao of a class of FMNN wh me-varyg delays by usg lear error feedback coroller. Based o comparso prcple, he ew heorem s derved o guaraee he expoeal sychrozao bewee he drve sysem ad respose sysem. The mehods proposed for sychrozao s effecve ad s easy o acheve ha oher complex corol mehods. Moreover, ca be exeded o vesgae oher dyamcal behavors of fracoal-order memrsve eural eworks, such as realzg he lag sychrozao or a-sychrozao of hs sysem based o he suable coroller. These ssues wll be he opc of fuure research. Fally, umercal examples are gve o llusrae he effecveess of he proposed heory Fgure 5. Sychrozao error bewee he drve ad respose a : e, b : e, c : e3 sysem REFERENCES [] L.O. Chua, Memrsor-he mssg crcu eleme, IEEE Tras Crcu Theory, vol.8, pp.57 59, 97. [] D.B.Srukov, G.S.Sder ad D.R.Sewar, The mssg memrsor foud, Naure, vol.453, pp.8-83, 8. [3] L.O.Chua, Ressace swchg memores are memrsors, Appled Physcs A Maerals Scece ad Processg, vol.,pp ,. [4] M.J.Sharfy ad Y.M.Baadak, Geeral spce models for memrsor ad applcao o crcu smulao of memrsor-based syapses ad memory cells, J Crcu Sys Comp, vol.9, pp.47-44,. 4

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