Journal of Computational and Applied Mathematics

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1 Journl of Compuionl n Applie Mhemis 245 (23) Conens liss ville SiVerse SieneDire Journl of Compuionl n Applie Mhemis journl homepge: On exponenil men-squre siliy of wo-sep Mruym mehos for sohsi ely ifferenil equions Wnrong Co,,, Zhongqing Zhng Deprmen of Mhemis, Souhes Universiy, Nnjing 296, PR Chin Division of Applie Mhemis, Brown Universiy, Proviene, RI 292, USA r i l e i n f o s r Arile hisory: Reeive 5 July 2 Reeive in revise form 29 Deemer 22 Keywors: Sohsi moeling Exponenil men-squre siliy Two-sep Mruym mehos We re onerne wih he exponenil men-squre siliy of wo-sep Mruym mehos for sohsi ifferenil equions wih ime ely. We propose fmily of shemes n prove h i n minin he exponenil men-squre siliy of he liner sohsi ely ifferenil equion for every sep size of inegrl frion of he ely in he equion. Numeril resuls for liner n nonliner equions show h his fmily of wo-sep Mruym mehos exhiis eer siliy hn previous wo-sep Mruym mehos. 23 Elsevier B.V. All righs reserve.. Inrouion Sohsi ely ifferenil equions (SDDEs) hve een inresingly use o moel he effes of noise n ime ely on vrious ypes of omplex sysems, suh s elye visul feek sysems [], onrol prolems [2,3] n he ynmis of noisy i-sle sysems wih ely [4]. SDDEs re lso use in moeling iseses, for exmple, epiemi iseses [5], neurologil iseses [6], e., n lso in finne SDDEs pper in moels of sok mrkes [7]. Some numeril mehos n heir onvergene n siliy properies hve een eslishe [8 4] reenly, u mos of hem re on one-sep mehos. Inse of one-sep mehos we here fous on sohsi muli-sep mehos for SDDEs, whih hve een wiely suie for solving sohsi orinry ifferenil equions (SODEs), i.e. wih no ime ely. To exen he muli-sep mehos for SODEs o hose for SDDEs is nonrivil sk n hese exensions hve no een invesige unil reenly. For review of muli-sep mehos for SODEs, we refer o [5,6]. Some more reen suies re s follows. In [7], erin sohsi liner muli-sep mehos re onsrue; n men-squre onvergene res re oine; n onsiseny oniions in he men-squre sense re given for wo-sep Mruym mehos. Ewl n Témm [8] suie he onvergene of sohsi Ams Bshforh sheme wih ppliion o geophysil ppliions. Ams-ype mehos for SODEs re lso nlyze in [9], where firs-orer srong onvergene oniions re given. For some speil SODEs wih iive noise, high orer muli-sep mehos hve een isusse in [2]. In his pper, we follow [2] n suy wo-sep Mruym shemes for he slr equion X() = f, X(), X( τ) + g, X(), X( τ) W(), J, X() = ξ(), [ τ, ], where τ is posiive fixe ely, J = [, T], W() is one-imensionl snr Wiener proess n he funions f : J R R R, g : J R R R. We noe h [2] is perhps he only work on muli-sep mehos for SDDEs, wherein (.) Corresponing uhor : Deprmen of Mhemis, Souhes Universiy, Nnjing 296, PR Chin. Tel.: E-mil resses: wro@seu.eu.n, seu_wr@sin.om (W. Co), Zhongqing_zhng@rown.eu (Z. Zhng) /$ see fron mer 23 Elsevier B.V. All righs reserve. oi:.6/j.m

2 W. Co, Z. Zhng / Journl of Compuionl n Applie Mhemis 245 (23) muli-sep mehos re propose for m-imensionl sysems of Iô SDDEs wih riving Wiener proesses n muli-ely, n heir properies re suie onerning onsiseny, numeril siliy n onvergene. Inse of working wih he generl SDDE (.), we suy he liner es moel (.2), whih n she some insigh on he generl SDDE (.), X() = [X() + X( τ)] + [X() + X( τ)]w(),, X() = ξ(), [ τ, ], (.2) where,,, R, τ is posiive fixe ely, W() is one-imensionl snr Wiener proess n ξ() is C([ τ, ]; R)-vlue iniil segmen. In his work, we im o erive men-squre sle wo-sep Mruym mehos for he SDDE (.2). The pper is orgnize in he following wy. In Seion 2 we provie some neessry noions n preliminries on SDDEs, inluing some properies of nlyil soluions o Eq. (.2). Also, in his seion he wo-sep Mruym mehos n heir onvergene properies re inroue. In Seion 3 we erive series of wo-sep Mruym mehos n prove h he numeril soluion is exponenilly sle for he exponenilly eying liner SDDE in men-squre sense. Seion 4 illusres he men-squre siliy of hese wo-sep Mruym mehos wih numeril exmples for he es moel (.2) n nonliner equion. 2. Noions n preliminries Le (Ω, F, P) e proiliy spe wih filrion (F ), whih sisfies he usul oniions (inresing n righoninuous; eh {F }, onins ll P-null ses in F ). Le W(), in Eq. (.2) e F -pe n inepenen of F. Assume ξ(), [ τ, ] o e F -mesurle n righ oninuous, n E ξ 2 <. Here ξ is efine y ξ = sup τ ξ() n is he Eulien norm in R. Throughou he pper, Eqs. (.) n (.2) re inerpree in he Iô sense. Uner hese usul oniions, Eq. (.2) hs unique srong soluion X() : [ τ, + ) R, whih sisfies Eq. (.2) n X() is mesurle, smple-oninuous n F -pe proess; see [22,23]. Definiion ([24]). The rivil soluion of Eq. (.) is si o e exponenilly men-squre sle, if here exiss pir of onsns λ > n C >, suh h, whenever E ξ 2 <, E X(, ξ) 2 CE ξ 2 e λ,. (2.) The inequliy (2.) implies h E X() 2 goes o exponenilly in s we ssume E ξ 2 <. Lemm 2 ([25]). If he oniion < ( + ) 2 (2.2) hols, hen he rivil soluion of Eq. (.2) is exponenilly men-squre sle. Applying he wo-sep Mruym mehos o Eq. (.) les o he following α j X i j = h j= β j f ( i j, X i j, X i m j ) + j= γ j g( i j, X i j, X i m j )W i j, i = 2, 3,..., N, (2.3) j= where α j, β j, γ j, (j {,, }) re prmeers; h > is he sepsize in ime whih sisfies τ = mh for posiive ineger m, n n = nh, N = T/h. The inremens W i := W( i+ ) W( i ), re inepenen N (, h)-isriue Gussin rnom vriles. Suppose h X i is F i -mesurle he mesh-poin i. Then X i is n pproximion o X( i ), where for i, X i re given y he iniil funion. Definiion 3. The funion u : R + R R R is si o e uniform Lipshiz oninuous if here exiss posiive onsn L u, suh h he funion u sisfies u(, x, x 2 ) u(, y, y 2 ) L u ( x y + x 2 y 2 ) (2.4) for every x, x 2, y, y 2 R n, When here exiss posiive onsn K, suh h u(, x, y) K( + x 2 + y 2 ) 2 (2.5) for x, y R n, we sy h u sisfies liner growh oniion.

3 84 W. Co, Z. Zhng / Journl of Compuionl n Applie Mhemis 245 (23) The hrerisi polynomil of (2.3) is given y ρ(λ) = α λ 2 + α λ + α. Definiion 4. The meho (2.3) is si o fulfill Dhlquis s roo oniion if (i) he roos of ρ(λ) lie on or wihin he uni irle; (ii) he roos on he uni irle re simple. Here re he onsiseny n onvergene properies of numeril mehos (2.3). Lemm 5 ([2]). Assume h he oeffiiens f n g of he SDDE (.) re Lipshiz oninuous in he sense of (2.4) n hve firs-orer oninuous pril erivives wih respe o he firs vrile n seon-orer oninuous pril erivives wih respe o he seon n hir vriles; hese pril erivives sisfy he liner growh oniion (2.5); he oeffiiens of he sohsi liner wo-sep Mruym sheme (2.3) sisfy Dhlquis s roo oniion, n he onsiseny oniions α j =, 2α + α = j= β j, α = γ, α + α = γ. (2.6) j= Then he glol error of he sheme (2.3) pplie o (.) sisfies mx E X( i ) X i 2 = O(h /2 ). i=2,...,n 3. Men-squre-siliy of he wo-sep Mruym mehos In his seion, we will erive he shemes in he lss of wo-sep Mruym mehos for Eq. (.2) n eermine series of wo-sep Mruym shemes, whih re exponenilly men-squre sle, see Theorem 7. Bse on he resuls in [8,26,27], we give he following efiniion. Definiion 6. A numeril meho is si o e exponenilly men-squre sle, if here exis onsn C > n λ h >, suh h, whenever E ξ 2 <, he numeril soluion X n of Eq. (.2), whih hs n exponenilly men-squre sle rivil soluion, he mesh n = nh, n, sisfies E(X n ) 2 CE ξ 2 e λ h n, s n, for fixe sep size h uner he onsrin h = τ/m, where m is posiive ineger. Applying he wo-sep Mruym mehos (2.3) o Eq. (.2) gives α j X i j = h j= β j [X i j + X i m j ] + j= γ j [X i j + X i m j ]W i j, i = 2, 3,... ; (3.) j= for i, we hve X i = ξ( i ). For eer ury, we ompue X using he Milsein meho wih smll sep size, whih hs onvergene re O(h) in men-squre sense (see []). We lso suppose h X is F -mesurle he mesh-poin. By hoosing he prmeers of he wo-sep Mruym meho o sisfy he onsiseny oniion (2.6) n hen we ge α =, α <, β = β =, (3.2) α = α, β = 2 + α, γ =, γ = + α. (3.3) Thus, we oin fmily of wo-sep Mruym shemes from (3.): X i+ + α X i + ( α )X i = h(2 + α )(X i+ + X i m+ ) + (X i + X i m )W i + ( + α )(X i + X i m )W i, (3.4) where prmeer α <. Nex we eermine he oniions on he prmeers for men squre siliy. I n e heke h he shemes (3.4) sisfy Dhlquis s roo oniion n ll ssumpions in Lemm 5. Thus, we hve he following onlusion on men-squre exponenil siliy. Theorem 7. Assume h he oniion (2.2) hols. If he prmeers of he wo-sep Mruym meho (3.) sisfy he resrie oniions (3.2) n (3.3), hen he meho is exponenilly men-squre sle.

4 W. Co, Z. Zhng / Journl of Compuionl n Applie Mhemis 245 (23) The proof of his heorem nees he following lemm. Lemm 8. Uner oniion (2.2), he numeril soluions {X i, i 2} proue y he wo-sep Mruym meho (3.4) sisfy E(X i X i W i ) < he(x 2 i ) + he( X i X i m ), E(X i X i m W i ) < he( X i X i m ) + he(x 2 i m ). (3.5) (3.5) Proof. Noe h E(W k ) =, E (W k ) 2 = h n X j is F i -mesurle n inepenen of W i if j i. Hene from properies of oniionl expeion we n ge E(X i X j W k ) =, i, j k, E(X i X j W k W k ) =, i, j k, E(X i X j W 2 k ) = he(x ix j ), i, j k. (3.6) (3.7) (3.8) Now we prove he inequliy (3.5). From he sheme (3.4) we hve X i+ = (2 + α )hx i m+ + ( + α )X i α X i ( (2 + α )h) + (X i + X i m )W i + ( + α )(X i + X i m )W i n hen X i = ( (2 + α )h) (2 + α )hx i m + ( + α )X i 2 α X i + (X i + X i m )W i + ( + α )(X i 2 + X i m 2 )W i 2. Due o he oniion (2.2) n α <, we ge (2 + α )h >. Using (3.6) (3.8), i hols h E(X i X i W i ) = (2 + α )he(x i m X i W i ) + ( + α )E(X i 2 X i W i ) ( (2 + α )h) α E(X 2 i W i ) + E(X 2 i W 2 ) + i E(X i m X i W 2 ) i + ( + α )E(X i 2 W i 2 X i W i ) + ( + α )E(X i m 2 W i 2 X i W i ) = he(x 2 i ( (2 + α )h) ) + he(x i m X i ) < he(x 2 i ) + he( X i m X i ). Inequliy (3.5) n e prove in he sme wy. This proves he lemm. Proof of Theorem 7. The explii form of he sheme (3.4) is (2 + α )h X i+ = (2 + α )hx i m+ + ( + α )X i α X i + (X i + X i m )W i + ( + α )(X i + X i m )W i. We squre oh sies of he ls ifferene equion o oin 2 ( (2 + α )h) 2 X 2 = i+ (2 + α ) 2 2 h 2 X 2 + i m+ ( + α )X i α X i + (Xi + X i m ) 2 W 2 i + ( + α ) 2 (X i + X i m ) 2 W 2 + i 2(2 + α )hx i m+ ( + α )X i α X i + 2(2 + α )hx i m+ (X i + X i m )W i + 2(2 + α )hx i m+ ( + α )(X i + X i m )W i + 2 ( + α )X i α X i (X i + X i m )W i + 2 ( + α )X i α X i ( + α )(X i + X i m )W i + 2( + α )(X i + X i m )(X i + X i m )W i W i.

5 86 W. Co, Z. Zhng / Journl of Compuionl n Applie Mhemis 245 (23) Then ( (2 + α )h) 2 X 2 = i+ α2 X 2 i + 2 W 2 i X 2 i + ( + α ) 2 X 2 + i ( + α ) 2 2 W 2 i X 2 + i (2 + α ) 2 h 2 2 X 2 i m+ + 2 W 2 i X 2 + i m ( + α ) 2 2 W 2 i X 2 2α i m ( + α )X i X i 2α ( + α )W i X i X i 2α (2 + α )hx i X i m+ 2α ( + α )W i X i X i m + 2( + α )(2 + α )hx i X i m+ where we efine + 2W 2 i X ix i m + 2( + α ) 2 W 2 i X i X i m + O i (W i, W i ), (3.9) O i (W i, W i ) = 2( + α ) 2 (X i + X i m )X i W i + 2(2 + α )hx i m+ (X i + X i m )W i + 2(2 + α )hx i m+ ( + α )(X i + X i m )W i + 2 ( + α )X i α X i (X i + X i m )W i + 2( + α )(X i + X i m )(X i + X i m )W i W i. By he lineriy of expeion n (3.6) (3.7) in Lemm 8, we ge E (O i (W i, W i )) =. Le Y i = E(X 2 i ), i =,, 2,.... Tking he expeion over oh sies of (3.9), n using he inequliy n (3.5), (3.5) n (3.8) in Lemm 8, i follows h where P Y i+ P Y i + P 2 Y i + P 3 Y i m+ + P 4 Y i m + P 5 Y i m, i = 2, 3,..., (3.) P = ( (2 + α )h) 2, P = α h α ( + α ) α (2 + α ) h + h, P 2 = ( + α ) 2 2 h + ( + α ) 2 α ( + α ) 2α ( + α ) 2 h 2α ( + α ) h + ( + α )(2 + α ) h + ( + α ) 2 h, P 3 = (2 + α ) 2 2 h 2 α (2 + α ) h + ( + α )(2 + α ) h, P 4 = 2 h + h, P 5 = ( + α ) 2 ( 2 + )h 2α ( + α ) 2 h 2α ( + α ) h. Le P = P(,,,, α, h) = P + P 2 + P 3 + P 4 + P 5 /P. I is ovious h Y i+ P mx Y i, Y i, Y i m+, Y i m, Y i m, i = 2, 3,.... (3.) We now lim h, for ny sepsize h = τ/m, i m Y i+ mx P i+, P i +,..., P E ξ 2. (3.2) Thus, if P <, hen we n ge lim i Y i =, s E ξ 2 <. In f, we use reurrene meho o inequliy (3.) n ge Y i+ P mx{y i, Y i, Y i m+, Y i m, Y i m } P 2 mx{y i, Y i 2,..., Y i 2m 3 } P P i m + mx i m +2 mx P i+ mx mx Y i i m,..., Y, E ξ 2 Y,..., i i m E ξ 2, P E ξ 2 E ξ 2, P E ξ 2,..., P i+, P i,..., P P i i m E ξ 2 i m + E ξ 2. (3.3)

6 W. Co, Z. Zhng / Journl of Compuionl n Applie Mhemis 245 (23) Tle Prmeers for ifferen wo-sep Mruym shemes. Sheme α β β Two-sep meho (TS) /2 /4 5/4 Two-sep meho 2 (TS2) 3/2 /2 Two-sep meho 3 (TS3) /2 3/2 Two-sep meho 4 (TS4) /3 /3 4/3 Two-sep meho 5 (TS5) 2/3 4/3 Two-sep meho 6 (TS6) 4/3 2/3 The hir inequliy is oine euse he inex of Y i m in (3.) keeps reuing y m 2 fer eh reurrene proess. The fifh inequliy hols sine he inex of Y i in (3.) elines y fer eh reurrene. For he reson h E ξ 2 is elimine from he reurrene proess, we ivie i y P for eh ime n evenully we ge i [ i m E ξ 2. P ] In he proess of ierion, Y i m (i m + ) will e he firs erm own o Y = E ξ 2 n i lls for is previous ] + seps, where [z] mens he lrges ineger no more hn rel numer z. This proves he lim (3.2). [ i m I is essenil o verify h P <, i.e. P + P 2 + P 3 + P 4 + P 5 < P. Rell h α < n hus 2 α 2 2(2 + α ), hen from (3.), we hve P + P 2 + P 3 + P 4 + P 5 P = + 2(2 + α ) + (2 α 2 )( + )2 h + (2 + α ) 2 2 h 2 ( (2 + α )h) 2 (2 + α ) + (2 + α ) + 2 (2 α2 )( + )2 = (2 + α ) 2 ( 2 2 )h (2 + α ) 2 ( 2 2 )h 2 + 2(2 + α ) + + ( + ) 2 h. (3.4) Bse on oniion (2.2), we oin h P + P 2 + P 3 + P 4 + P 5 P < hols for eh sepsize h = τ/m. Due o P <, we oin Y i+ P [ i m ]+ E ξ 2 from (3.3) n hus i m + Y i+ P E ξ 2 P i+ E ξ 2 = e λ h i+ E ξ 2, h where λ h = ln P >. The proof ens. ()h 4. Numeril exmples In ll our numeril exmples, E(X 2 ) = 2 n X n (ω i ) 2, 2 i= re he smple verge over 2 rjeories in Ml. Tle liss numer of wo-sep Mruym shemes wih ifferen prmeers uner es here. Noe h only he shemes in ol (TS3 n TS5) sisfy he require oniions in Theorem 7 n hene re exponenilly men-squre sle. However, he oher Mruym shemes TS, TS2, TS4, TS6 o no sisfy he oniions in Theorem 7 n hus my e only oniionlly sle n even no sle s we show ler on. Exmple. We onsier he liner es moel X() = [X() + X( τ)] + [X() + X( τ)]w(),, X() = + τ, [ τ, ] (4.) o illusre he men-squre siliy of he wo-sep Mruym shemes in Tle. We hoose he prmeers s = 4, = 2, =.5, =.5 n τ =, whih ensures h he ex soluion of he Eq. (4.) is men-squre sle y Lemm 2. From Fig., we oserve h TS is no men-squre sle for oh lrge sepsize h = /4 n smll sepsize h = /64. From Figs. 2 n 3, we see h TS3 n TS5 re men-squre sle even for lrge sepsize h = /4; Fig. 2 illusres h TS4 is oniionlly men-squre sle; TS4 is men squre sle if he sepsize h is smll enough (like h = /64 here). In Fig. 3, we fix he sepsize h = /8 n show h o gre exen he men squre siliy of he implii sheme TS3 is eer hn he explii wo-sep sheme TS6. If we es TS6 for rher long ime inervl, hen he numeril soluion

7 88 W. Co, Z. Zhng / Journl of Compuionl n Applie Mhemis 245 (23) Fig.. Simulions wih TS n TS3. (): TS, h = /4; (): TS3, h = /4; (): TS, h = /64; (): TS3, h = / Fig. 2. Simulions wih TS4 n TS5. (): TS4, h = /4; (): TS5, h = /4; (): TS4, h = /64; (): TS5, h = /64. will osille n will finlly iverge. From Figs. 3, we oserve h he numeril soluion from TS6 lows up erlier hn ny oher unsle implii mehos shown in Figs. n 2. On he oher hn, numeril soluions of oh TS3 n TS5 onverge. o zero very fs.

8 W. Co, Z. Zhng / Journl of Compuionl n Applie Mhemis 245 (23) Fig. 3. Simulions wih TS3 n he explii meho TS6. (): TS6; (): TS3. h = / TS3 TS2 TS Fig. 4. Simulions wih TS, TS2 n TS3 for fixe sepsize h = /2. In Fig. 4 we hoose ifferen prmeers for he Eq. (4.): = 3, =, =.5, =.5 n τ = when (4.) is men squre sle. Here we use very lrge sepsize h = /2 for TS, TS2 n TS3. The resuls show h he sheme TS3 minins he men-squre siliy even wih lrge sepsize h. Exmple 2. We es he propose wo-sep Mruym mehos for he following nonliner SDDE (Exmple 5.2., [28]): X() = + X() + X() sin(x( τ))w(),, + X() = + τ, [ τ, ]. (4.2) The soluion of Eq. (4.2) is men squre sle if 2 2 n [28].

9 9 W. Co, Z. Zhng / Journl of Compuionl n Applie Mhemis 245 (23) e f Fig. 5. Simulions wih TS n TS3. (): TS, h = /2; (): TS3, h = /2; (): TS, h = /6; (): TS3, h = /6; (e): TS, h = /64; (f): TS3, h = / e f Fig. 6. Simulions wih TS4 n TS5. (): TS4, h = /2; (): TS5, h = /2; (): TS4, h = /6; (): TS5, h = /6; (e): TS4, h = /64; (f): TS5, h = /64.

10 W. Co, Z. Zhng / Journl of Compuionl n Applie Mhemis 245 (23) X n X n X n X n Fig. 7. Simulions wih TS2 n TS5. (): TS2, h = /4; (): TS5, h = /4; (): TS2, h = /6; (): TS5, h = /6. 4 X n X n X n X n Fig. 8. Simulions wih TS3 n TS4. (): TS4, h = /6; (): TS3, h = /4; (): TS4, h = /64; (): TS3, h = /64. We ke =, = n τ =. I is shown from Figs. 5 n 6 h he shemes TS3 n TS5 minin heir men-squre siliy for nonliner SDDE (4.2); TS n TS4 re oniionlly men-squre sle. In some rnge of sepsize h, smller h les o greer insiliy, ompring o (), () in Fig. 5 n (), () in Fig. 6.

11 92 W. Co, Z. Zhng / Journl of Compuionl n Applie Mhemis 245 (23) TS,Xn TS,X2n TS4,Xn TS4,X2n TS6,Xn TS6,X2n TS6 TS TS4 5 5 TS,Xn TS,X2n TS4,Xn TS4,X2n TS6,Xn TS6,X2n TS TS6 TS Fig. 9. Simulions wih TS, TS4 n TS6. (): h = /4; (): h = /8. X n X n X n X n Fig.. Simulions wih TS3 n TS5. (): TS3, h = /2; (): TS5, h = /2; (): TS3, h = ; (): TS5, h =. Exmple 3. We es he propose wo-sep Mruym mehos for he following nonliner sohsi ely ifferenil 2-imensionl sysem: X () X () sin(x ( τ)) X () W () = A + B + C, (4.3) X 2 () X 2 () os(x 2 ( τ)) X 2 () W 2 () where A = 28, B = 3 2 /2 3/2, C =. /4 5/2 /2 When [ τ, ], X () = + τ n X 2 () = e. Bse on Corollry 2.2 [24], we know h soluions of sysem (4.3) re exponenilly men-squre sle. From Fig. 7, we oserve h he soluions oine y sheme TS2 low up for h = /4 n re exponenilly mensqure sle when h = /6 (() n ()) n he soluions oine from exponenilly men-squre sle sheme TS5 re men-squre sle for oh h = /4 n h = /6, see Fig. 7() n (). I is shown from Fig. 8 h TS3 performs goo exponenilly men-squre siliy. Menwhile, TS4 is oniionlly men-squre sle, n he smller sep size h = /64 is neee (ompre o TS2).

12 W. Co, Z. Zhng / Journl of Compuionl n Applie Mhemis 245 (23) We ompre soluions oine y he shemes TS, TS4 n TS6 in Fig. 9. Fig. 9() shows h neiher of hem re mensqure sle for h = /4, n he soluion from he explii sheme TS6 lows up fser hn h from implii sheme TS n TS4. On he oher hn, he explii sheme TS6 performs eer hn he sheme TS n TS4 when h = /8 s we n see from Fig. 9() h TS6 is men-squre sle for h = /8 u TS n TS4 re no. Fig. 9 inies h he explii sheme TS6 requires less resrie ime sep size h for men-squre siliy hn he implii sheme TS n TS4 o. In Fig., we es he propose shemes TS3 n TS5 wih very lrge sep size h = /2 n h =, he shemes TS3 n TS5 re minining heir exponenil siliy in men-squre sense for nonliner sohsi ely ifferenil sysem (4.3). 5. Conlusion We hve propose fmily of exponenilly men-squre sle wo-sep Mruym shemes. I hs een prove h he propose shemes n minin he exponenil men-squre siliy of he liner SDDE for every sep size of inegrl frion of he ely in he equion. Numeril exmples show h his fmily of numeril mehos exhiis exponenil men-squre siliy for oh liner n some priulr nonliner SDDE n 2-imensionl SDDEs. The numeril resuls sugges h our propose sheme n e ope for generl nonliner SDDEs, u furher numeril suies re require. Aknowlegmens The uhors hnk he referees n Dr. Tkeomo Misui for heir helpful suggesions for improving he pper. The firs uhor lso woul like o hnk Professor George Em Krnikis for his hospiliy when she ws visiing Division of Applie Mhemis Brown Universiy. Firs uhor s work ws suppore y he NSF of Chin (No. 936). Seon uhor s work ws suppore y AIRFORCE MURI. Referenes [] A. Beuer, K. Vsilkos, Effes of noise on elye visul feek sysem, J. Theore. Biol. 65 (993) [2] M. Grigoriu, Conrol of ime ely liner sysems wih Gussin whie noise, Pro. Eng. Meh. 2 (997) [3] M.D. Pol, A. Pirro, Time ely inue effes on onrol of liner sysems uner rnom exiion, Pro. Eng. Meh. 6 (2) [4] L.S. Tsimring, A. Pikovsky, Noise-inue ynmis in isle sysems wih ely, Phys. Rev. Le. 87 (2) 2562 [4 pges]. [5] E. Bere, V.B. Kolmnovskii, L. Shikhe, Siliy of epiemi moel wih ime elys influene y sohsi perurions, Mh. Compu. Simul. 45 (998) [6] A. Beuer, J. Belir, Feek n elys in neurologil iseses: moeling suy using ynmil sysems, Bull. Mh. Biol. 55 (993) [7] D.G. Hoson, L.C.G. Rogers, Complee moels wih sohsi liliy, Mh. Finne 8 (998) [8] C.T.H. Bker, E. Bukwr, Exponenil siliy in p-h men of soluions, n of onvergen Euler-ype soluions, of sohsi ely ifferenil equions, J. Compu. Appl. Mh. 84 (25) [9] Z. Fn, Wveform relxion meho for sohsi ifferenil equions wih onsn ely, Appl. Numer. Mh. 6 (2) [] Y. Hu, S.A. Mohmme, F. Yn, Disree-ime pproximions of sohsi ely equions: he Milsein sheme, Ann. Pro. 32 (24) [] R. Li, Convergene n siliy of numeril soluions o SDDEs wih Mrkovin swihing, Appl. Mh. Compu. 75 (26) 8 9. [2] M. Liu, W. Co, Z. Fn, Convergene n siliy of he semi-implii Euler meho for liner sohsi ifferenil ely equion, J. Compu. Appl. Mh. 7 (24) [3] X. Mo, Numeril soluions of sohsi ifferenil ely equions uner he generlize Khsminskii-ype oniions, Appl. Mh. Compu. 27 (2) [4] S. Zhou, F. Wu, Convergene of numeril soluions o neurl sohsi ely ifferenil equions wih Mrkovin swihing, J. Compu. Appl. Mh. 229 (29) [5] P.E. Kloeen, E. Plen, Numeril Soluion of Sohsi Differenil Equions, Springer, Berlin, 992. [6] G.N. Milsein, Numeril Inegrion of Sohsi Differenil Equions, Kluwer Aemi Pulishers Group, Dorreh, 995. [7] E. Bukwr, R. Winkler, Muli-sep mehos for SDEs n heir ppliion o prolems wih smll noise, SIAM J. Numer. Anl. 44 (26) [8] B.D. Ewl, R. Témm, Numeril nlysis of sohsi shemes in geophysis, SIAM J. Numer. Anl. 42 (25) [9] L. Brugnno, K. Burrge, P.M. Burrge, Ams-ype mehos for he numeril soluion of sohsi orinry ifferenil equions, BIT 4 (2) [2] G. Denk, S. Shäffler, Ams mehos for he effiien soluion of sohsi ifferenil equions wih iive noise, Compuing 59 (996) [2] E. Bukwr, R. Winkler, Muli-sep Mruym mehos for sohsi ely ifferenil equions, Soh. Anl. Appl. 25 (27) [22] X. Mo, Sohsi Differenil Equions n Appliions, Horwoo, 997. [23] S.E.A. Mohmme, Sohsi Funionl Differenil Equions, in: Reserh Noes in Mhemis, vol. 99, Pimn, Lonon, 984. [24] X. Mo, Exponenil Siliy of Sohsi Differenil Equions, Mrel Dekker, In., New York, 994. [25] X. Mo, Rzumikhin-ype heorems on exponenil siliy of sohsi funionl ifferenil equions, Soh. Pro. Appl. 65 (996) [26] V.K. Brwell, Speil siliy prolems for funionl equions, BIT 5 (975) [27] L. Shikhe, Lypunov Funionls n Siliy of Sohsi Differene Equions, Springer, Lonon, Dorreh, Heielerg, New York, 2. [28] X. Mo, Siliy of Sohsi Differenil Equions wih Respe o Semimringles, Longmn Sienifi & Tehnil, 99.