An impact of noise on invariant manifolds in nonlinear dynamical systems

Transcription

1 JOURNAL OF MATHEMATICAL PHYSICS 51, An impac of noise on invarian manifolds in nonlinear dynamical sysems X Sn, a Jinqiao Dan, and Xiaofan Li Deparmen of Applied Mahemaics, Illinois Insie of Technology, Chicago, Illinois 6616, USA Received 3 Jly 29; acceped 25 Febrary 21; pblished online 15 April 21 Invarian manifolds provide geomeric srcres for ndersanding dynamical behavior of nonlinear sysems. However, hese nonlinear sysems are ofen sbjec o random flcaions or noises. I is hs desirable o qanify he impac of noises on he invarian manifolds. When he noise inensiy is small, his impac is esimaed via asympoic analysis in he conex of Liapnov Perron formlaion. Namely, he random invarian manifold is represened as a perrbaion of he deerminisic invarian manifold, wih a well-defined bond for he deviaion. 21 American Insie of Physics. doi:1.163/ I. INTRODUCTION AND MOTIVATION Invarian manifolds for deerminisic dynamical sysems have been exensively sdied. They are special invarian ses in he sae space and carry essenial dynamical informaion. Their exisence and reglariy properies have been reasonably ndersood see Ref. 6 and references herein. Dynamical sysems, as mahemaical models for nonlinear phenomena in engineering and science, are ofen sbjec o random flcaions or noises. I is desirable o ndersand he impac of noises on he invarian manifolds. In his paper, we sdy he impac of small noises on he invarian sable and nsable manifolds, i.e., we compare he invarian manifolds for he original deerminisic sysem and for he randomly perrbed sysem and qanify heir difference when he noise is small. Sable and nsable manifolds for sochasic ordinary differenial eqaions SDEs and sochasic parial differenial eqaions SPDEs have been sdied recenly see Refs. 12, 1, and 8 and Refs. 4, 5, 9, and 3, respecively. We consider he following nonlinear sochasic evolionary eqaion wih a mliplicaive noise, in a separable Hilber space H wih scalar prodc, and indced norm : du = AU + FU + U Ẇ, d U = X H, 1 where A is a linear operaor, is in he sense of Sraonovich sochasic calcls, W=W, is a scalar Brownian moion defined on a probabiliy space,f,p, and is a posiive parameer represening he inensiy of he noise. Noe ha Io s form of 1 is du d = AU + U 2 + FU + UẆ, U = X H. The nonlineariy FU saisfies F= and DF= and is Lipschiz coninos on H, a Elecronic mail: xsn15@ii.ed /21/514/4272/12/$3. 51, American Insie of Physics

2 Sn, Dan, and Li J. Mah. Phys. 51, FU 1 FU 2 L F U 1 U 2, where L F is he Lipschiz consan and is he normal of he space H. When he nonlineariy FU is locally Lipschiz coninos, he approximaion resl in his paper can be applied o he modified sochasic eqaion where he nonlineariy is appropriaely coff and hs obain approximaion informaion for he local random invarian manifolds. The sae space H is he Eclidean space R n when he above eqaion is a SDE or a fncion space if he above eqaion is a SPDE. When =, Eq. 1 redces o a deerminisic evolionary eqaion, du = AU + FU. d 2 We compare he invarian manifolds for he original deerminisic sysem 2 and for he randomly perrbed sysem 1 and qanify heir difference when he noise inensiy is small. Noe ha he exisence of random sable and nsable manifolds for 1 was proved sing he Liapnov Perron mehod. 5 Here we condc an asympoic analysis of hese random sable and nsable manifolds in he Liapnov Perron seing when is sfficienly small. Noe ha random cenerlike manifolds were esimaed or approximaed for some SPDEs by Wang and Dan 11 and Blomker and Wang. 2 This paper is organized as follows. In Sec. II, we review some basic conceps of random dynamical sysems and recall he exisence resl for random invarian manifolds. The main resl on asympoic analysis for random invarian manifolds is described in Sec. III, and wo illsraive examples are presened in Sec. IV. II. RANDOM INVARIANT MANIFOLDS Following Ref. 5, we assme hrogho he paper ha he linear operaor A:DA H generaes a srongly coninos semigrop e A on H, which saisfies he psedo exponenial dichoomy wih exponens and a bond K, i.e., here exiss a coninos projecion P on H sch ha i P e A =e A P. ii The resricion e A RP,, is an isomorphism of he range RP of P ono iself, and we define e A for as he inverse map. iii The following esimaes hold: e A P x Ke x,, e A P s x Ke x,, 3 where P s =I P. Denoe H s = P s H, H = P H, and hence H=H s H. A. Random dynamical sysems A measrable random dynamical sysem on Hilber space H,B over a driving sysem T wih ime T is a mapping :T H H,,,x,,x, wih he following properies. 1 i Measrabiliy: is BT F B-measrable. ii The mappings,=,, :H H form a cocycle over, i.e., hey saisfy,=id X for all and +s,=,ss, for all s, T, and. For SDEs and SPDEs, 1 we idenify =W, and define he driving sysem is he Wiener shif, i.e., =+.

3 Impac of noise on invarian manifolds J. Mah. Phys. 51, To faciliae random dynamical sysem sdy of 1, we conver i ino a parial differenial eqaion wih random coefficiens, called a random parial differenial eqaion RPDE. To his end, we inrodce zw as he saionary solion of he following Langevin eqaion: dz + zd = dw. Then zw=z, where Z is he saionary solion of dz+zd=dw and can be expressed as Z = e dw. Moreover, Define a ransform wih is inverse ransform Z = e Z + e e dw. x ª T,X = X e z X ª T 1,x = x e z. Denoe U,,X as he solion of 1 wih iniial vale X. Inrodcing 4 = T,U,,X = e z U,,X, hen he new sysem sae saisfies he following RPDE: 5 where d d = A + z + G,, = x H, 5 x = T,X = e z X and G, ª e z Fe z. The solion mapping of 5, i.e.,,x,,x, generaes a random dynamical sysem. Ths see Ref. 5,,X T 1,,,T,X ª U,,X is also a random dynamical sysem. In fac, he relaionship beween solions of 1 and 5 is described by U,,X = T 1,,,T,X, 6,,x = T,U,,T 1,x. B. Definiion of random invarian manifolds A random se M is called an invarian se for a random dynamical sysem,,x if 1,5

4 Sn, Dan, and Li J. Mah. Phys. 51, ,,M M for. If we can represen M as a graph of a C k or Lipschiz mapping sch ha h s,:h s H M = M s = + h s, H s, hen M s is called a C k or Lipschiz sable manifold, where H s is he sable sbspace and H is he nsable sbspace. Similarly, if we can represen M as a graph of a C k or Lipschiz mapping sch ha h,:h H s M = M = + h, H, hen M is called a C k or Lipschiz nsable manifold. C. Exisence of random invarian manifolds Denoe,,x he solion of 5 in H wih he iniial daa,, =. Define he Banach space for each,, Ĉ + =:, H is coninos and sp e z d, wih he norm Ĉ + = sp e z d., We recall from Refs. 4 and 5 he following resl for random invarian manifolds. If KL F , where K,, and are from he psedoexponenial dichoomy condiion in he beginning of his secion and L F is he Lipschiz consan of F, as discssed in Sec. I, hen here exiss a Lipschiz invarian sable manifold for he random evolionary eqaion 5 which is given by M s = + h s, H s, where h s :H s H is a Lipschiz coninos mapping and saisfies h s,=. If we frher assme ha F is C 1 in he sense of Fréche differeniabiliy, hen he mapping h s :H s H is C 1, i.e., he invarian manifold M s is C 1. I is shown 5 ha M s if and only if here exiss a fncion Ĉ + wih = and saisfies,, = e A+ z s ds e + A s+ s z r dr P s G s,sds 7 + e A s+ s z r dr P G s,sds, 8 where = P s. I follows from he above eqaion ha

5 Impac of noise on invarian manifolds J. Mah. Phys. 51, h s, = and h s,=. This is he so-called Liapnov Perron eqaion. Given M s, he sable manifold of 1 can be expressed as 5 e As+ s zr dr P G s,s,,ds 9 where T 1 is defined as in 4. M s = T 1,M s, 1 III. ASYMPTOTIC ANALYSIS FOR RANDOM INVARIANT MANIFOLDS In his secion, we propose an approach o approximae he random sable and nsable manifolds by asympoic analysis for sfficienly small. Only he sable manifolds are considered, as he nsable manifolds can be reaed similarly. Denoe he sable manifold for 5 1 as M s = + h s, H s. Le he deerminisic sable manifold i.e., = be represened as M = + h d H s, where h s,:h s H and h d :H s H are Lipschiz mappings. We expend h s, = h d + h 1, + 2 h 2, + + k h k, +. Wrie he solion of 5 in he form wih he iniial condiion = k k + 14 By Taylor expansion, we obain = + h s, = + h d + h 1, and e z = e Z =1+Z + + k Z k + 16 k! e s zr dr = e s Zr dr Z r dr + + k s Z r dr k =1+s k! Sppose F is sfficienly smooh wih respec o. Wih 16, i follows from 6 ha G, = e z Fe z = e Z Fe Z = 1 Z + F1 + Z = F + Z F + F 1 + Z +, where F represens he firs order Fréche derivaive of he fncion F wih respec o and evalaed a. 7 In Eclidean space, he Fréche derivaive redces o he classical derivaive. Sbsiing 13, 14, and 18 ino 5 and eqaing he erms wih he same power of, we ge 18

6 Sn, Dan, and Li J. Mah. Phys. 51, and d = A + F d 19 = + h d, Solve for and 1, d1 = A + F d 1 Z + F F 2 1 = h 1,. = e A e + A s F sds, 21 1 = e A+ F s dsh 1, Wih he righ hand side of 9 can be wrien as e As+ s F r dr Z s + F s F s sds. 22 e As+ s zr dr P G s,sds = I + I 1 + R 2, 23 where R 2 represens he remainder erm, I = e As P F sds, 24 I 1 e = Ass Z r dr Z s P F + P F s 1 s + Z s ds. 25 Sbsiing 13 and 23 ino 9 and maching he powers in, wege and h d e = As P F sds h 1, e = Ass Z r dr Z s P F + P F s 1 s + Z s s ds. Iniively, if he sable manifold 9 for 5 exiss and is sfficien smooh fncion of, hen h d and h 1,, as obained by power expansion wih respec o, shold be well defined. In he following, we make his iniive idea rigoros. A. Main resl We firs sae a resl abo approximaing sable manifolds for RPDEs inclding random ordinary differenial eqaions. Theorem 1: Approximae sable manifold for random evolionary eqaions. Le M s

7 Impac of noise on invarian manifolds J. Mah. Phys. 51, =+h s,h s represen he sable manifold for d/d=a+z +G,. Assme ha (i) F is wice coninosly Fréche differeniable wih respec o. (ii) For some, he Lipschiz consan of F saisfies KL F Then as is sfficienly small, he random sable manifold M s can be approximaed as M s = + h d + h 1, + R 2 H s, where R 2 C 2 wih C, almos srely, 26 h d e = As P F sds, 27 and h 1, = e Ass Z r dr Z s P F + P F s 1 s + Z s sds. 28 Proof: Expressions 27 and 28 are derived a he beginning of his secion. Acally, he asympoic expansions sed o obain 27 and 28 are validaed by he smoohness of h s,. I is shown 1 ha nder assmpions i and ii, h s, is C 2 wih respec o. We now need o show ha h d and h 1, as in 27 and 28 are well defined. Noe ha h d is he mapping whose graph is he deerminisic sable manifold of 2, whose exisence and niqeness are well esablished nder condiion 26. I remains o show ha here exiss h 1,H saisfying 28 and h 1,. To his end, sbsiing 14 and ino 8 and eqaing he erms wih he same power of, wege 1,, = e A Z s ds + e A ss + P s F s Z s s ds + Z r dr Z s P s F s e A ss Z r dr Z s P F s + P F s Z s s ds + e A s P s F s 1 sds e + A s P F s 1 sds. 29 Denoe he righ hand side of 29 as J 1 ;,. Then i can be checked ha J ;, is a well-defined mapping from C + 1 H s o C + 1, where C + 1 is a weighed Banach space defined as C + 1 = :, H is coninos and sp e 1, wih he norm

8 Sn, Dan, and Li J. Mah. Phys. 51, C1 + = sp e 1, and 1 is any vale sch ha 1, and 1 saisfies 26, i.e., KL F In he following, we show ha J ;, is a conracion mapping from C + 1 o C + 1. Noe ha J 1 ;, Jũ 1 ;, C1 + = sp, e 1 sp, e 1 sp, e 1 sp F 1 ũ 1 C1,KL + e A s P s F s 1 ũ 1 ds e + A s P F s 1 ũ 1 ds e A s P s F s 1 ũ 1 ds e + A s P F s 1 ũ 1 ds Ke s F s 1 ũ 1 ds Ke + s F s 1 ũ 1 ds e 1 s ds e + 1 ds s KL F ũ 1 C We have sed he ineqaliy F s L F o ge 3. Therefore, i follows ha J ;, is a conracion mapping from C + 1 o C + 1, and hs here exiss a niqe 1,,C + 1 saisfying 29. Se h 1,= P 1,,= 1,, in 29. We see ha h 1, saisfies 28 and h 1,H. The propery h 1, follows from he fac ha 1,,C + 1. This complees he proof of his heorem. Now we firs sae a resl abo approximaing sable manifolds for SPDEs inclding SDEs. Theorem 2: Approximae sable manifold for sochasic evolionary eqaions. Le M s =+h s,h s represen he sable manifold of du = AU + FU + U Ẇ, d hen nder he assmpions of Theorem 1 above, M s can be esimaed as M s = + h d, + h 1, + R 2 H s, where R 2 C 2 wih C, a.s., and h 1, is a zero-mean Gassian random variable defined as h 1, = h 1, Zh d + Zh d wih h d v being he Fréche derivaive of h d wih respecive o and evalaed a v. Proof: Assmpions i and ii in Theorem 1 imply ha h 1,:H s H is a C 1 mapping, as saed in Sec. II C. By 1, we conclde ha 31

9 Impac of noise on invarian manifolds J. Mah. Phys. 51, M s = T 1,M s = = T 1, + h s, H s = = e z + h s, H s = = e z + h s, H s = = e z + e z h s, H s = = + e z h s e z H s. 32 Using h s,=h d +h 1,+ and e z =1+Z+, we can expand h s e z as h s e z = h s, Zh s + R 2 = h d + h 1, Zh d + R 2, where R 2 is he remainder erm on he order of 2. I follows from 33 and 32 ha 33 M s = = + h d, + h 1, + Zh d Zh d + R 2 H s. Seing h 1, = h 1, + Zh d Zh d, we hen have M s = + h d + h 1, + R 2 H s. I remains o show ha h 1, is a zero-mean Gassian random variable. I follows from 2 and 28 ha h 1, can be explicily expressed as where h 1, = I V 1 B + C, V e = As P F s e As+ s F r dr ds, 34 B e = Ass Z r dr Z s P F + P F s Z s s ds, C e = As P F s e As+ s F d s e Ar+ r.f d Z r r + F r F r rdrds. By he definiion of Z see Sec. II, he erms sch as Z, s Z r dr, and Z s can be expressed as Io inegral wih respec o he Brownian moion. As a conseqence, he h 1, can be regarded as an inegral wih respec o he Brownian moion wih deerminisic inegrand and hs is zero-mean Gassian. This complees he proof of his heorem. Remark 1: Eqaion 34 provides an explici expression for h 1,, which depends only on he nperrbed deerminisic sysem. IV. ILLUSTRATIVE EXAMPLES Le s look a wo examples, in Eclidean space and Hilber space, respecively. Example 1: Consider a SDE sysem

10 Sn, Dan, and Li J. Mah. Phys. 51, Ẋ = X + X Ẇ Ẏ = Y + X 2 + Y Ẇ, 35 where and W is a scalar Brownian moion. We consider he sable manifold in a neighborhood arond,. In his example, A= 1 1, H=R 2 a finie dimensional Hilber space, H s = x x R, and H = ȳ ȳr. The ransformed differenial eqaion wih random coefficiens is, ẋ = x + Z x ẏ = y + Z y + e Z x 2 where Z is he saionary solion of dz+zd=dw, i.e., Z= e dw and Z =e Z+e e dw. Denoe = x H s, h d = ȳ H, and h 1,= ȳ 1 H. Solving he corresponding eqaion 19, wege x e = ȳ e + 1 x 2e 3 e Noe ha he projecion operaor from H o H is P = 1, F = x 2e 2, and he firs order Fréche derivaive i.e., Jacobian marix in R 2 is F = 2x e 2. I follows from 27 ha and h d = ȳ = x h d = 2 x 2. 3 Sbsiing ȳ = 1 3 x 2 ino 36, wege = x e 1 3x 2e 2. Sbsiing P, A, F, F, and ino 31 and 34, and by direc compaion, we ge = h 1, x 2 3 e 3 dw. By he propery of Gassian Brownian moion W and he Io isomery, we conclde ha h 1, is a zero-mean Gassian random variable wih covariance marix

11 Impac of noise on invarian manifolds J. Mah. Phys. 51, x Ths, he random sable manifold for he SDE sysem 35 above is represened as he graph of h s, = h d + h 1, + R 2, where R 2 C 2 and = x H s, i.e., he random sable manifold can be expressed as M = s xȳ ȳ x 2 x 2 = 3 e 3 dw R + O 2, x. 3 When =, he above random sable manifold redces o he sable manifold a, for he nperrbed deerminisic sysem, as seen from 37, M = xȳ ȳ x 2 = 3, x R. I is clear ha wih accracy of O 2, he mean of he random sable manifold is he deerminisic sable manifold, wih variance of 1 x Example 2: Consider he following SPDE: U = U xx +1U U 3 + U Ẇ, x,1 U, = U1, =, where and W is a scalar Brownian moion. In his example, A=+1, H=L 2,1, DA =H 2,1, and F= 3. Noe ha he eigenvales of A are n =1 n 2, and he corresponding normalized eigenfncions are e n = 2sinnx, n=1,2,... Here H s =Spane 1 and H =Spane 2,e 3,...,e n,... The ransformed RPDE is = xx +1 + Z e 2Z 3, x,1, = 1, =. We calclae ha P =,e 1 = 1 2 xsinxdx, e A = n=1 e n,e n e n, and Fréche derivaive F =3 2 wih being he solion of he following eqaion: = xx +1 3, x,1 x, = + h d 38, = 1, =. According o he exisence heorem of he random sable invarian manifold see Sec. II above, we conclde ha in a sfficienly small neighborhood arond = in H, here exiss a local sable manifold, which, according o Theorem 2, can be approximaed as where M s = + h d + h 1, + R 2 H s, h 1, = h 1, + Zh d Zh d. Unlike Example 1, here we canno solve h d and h 1, analyically. Noe ha h d is he sable manifold of he deerminisic sysem =. Once is fond e.g., by nmerically

12 Sn, Dan, and Li J. Mah. Phys. 51, solving he deerminisic parial differenial eqaion 38, we can esimae h 1, via 28 or 34, and hs esimae h 1,. ACKNOWLEDGMENTS We wold like o hank Kening L, Brigham Yong Universiy, for helpfl sggesions and discssions. This work was parly sppored by he NSF nder Gran Nos , 62539, and Arnold, L., Random Dynamical Sysems Springer-Verlag, New York, Blomker, D. and Wang, W., Qaliaive properies of local random invarian manifolds for SPDEs wih qdraic nonlineariy, e-prin arxiv:812.39v2, Caraballo, T., Dan, J., L, K., and Schmalfss, B., Invarian manifolds for random and sochasic parial differenial eqaions, Adv. Nonlinear Sd. 1, Dan, J., L, K., and Schmalfss, B., Invarian manifolds for sochasic parial differenial eqaions, Ann. Probab. 31, Dan, J., L, K., and Schmalfss, B., Smooh sable and nsable manifolds for sochasic evolionary eqaions, J. Dyn. Differ. Eq. 16, Gckenheimer, J. and Holmes, P., Nonlinear Oscillaions, Dynamical Sysems, and Bifrcaion of Vecor Fields Springer-Verlag, New York, Hner, J. and Nachergaele, B., Applied Analysis World Scienific, Singapore, Mohammed, S.-E. A. and Schezow, M., The sable manifold heorem for sochasic differenial eqaions, Ann. Probab. 27, Mohammed, S.-E. A., Zhang, T., and Zhao, H., The sable manifold heorem for semilinear sochasic evolion eqaions and sochasic parial differenial eqaions, Mem. Am. Mah. Soc. 196, Sn, X., Topics in inerfacial dynamics and sochasic dynamics, Ph.D. hesis, Illinois Insie of Technology, Wang, W. and Dan, J., A dynamical approximaion for sochasic parial differenial eqaions, J. Mah. Phys. 48, Wanner, T., in Dynamics Repored, edied by C. Jones, U. Kirchgraber, and H. O. Walher Springer-Verlag, New York, 1995, Vol. 4, pp

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