Averaging. Chapter Dynamical averaging Time averages CHAPTER 17. AVERAGING 350

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1 CHAPTER 7. AVERAGING 35 an voluion opraor wih a givn obsrvabl, and rlas h xpcaion valu of h obsrvabl o h lading ignvalu of h voluion opraor. Chapr 7 Avraging W Why hink whn you can compu? acij Zworski discuss firs h ncssiy of sudying h avrags of obsrvabls in chaoic dynamics. A im avrag of an obsrvabl is compud by ingraing is valu along a rajcory. Th ingral along rajcory can b spli ino a sum of ovr ingrals valuad on rajcory sgmns; if xponniad, his yilds a muliplicaiv wigh for succssiv rajcory sgmns. This lmnary obsrvaion will nabl us o rcas h formulas for avrags in a muliplicaiv form ha moivas h inroducion of voluion opraors and furhr formal dvlopmns o com. Th main rsul is ha any dynamical avrag masurabl in a chaoic sysm can b xracd from h spcrum of an approprialy consrucd voluion opraor. In ordr o kp our os closr o h ground, in sc. 7.4 w ry ou h formalism on h firs quaniaiv diagnosis whhr a sysm is chaoic, h Lyapunov xponn. 7. Dynamical avraging In chaoic dynamics daild prdicion is impossibl, as any finily spcifid iniial condiion, no mar how prcis, will fill ou h nir accssibl sa spac afr a fini Lyapunov im (.). Hnc for chaoic dynamics on canno follow individual rajcoris for a long im; wha is aainabl, howvr, is a dscripion of h gomry of h s of possibl oucoms, and h valuaion of long-im avrags. Exampls of such avrags ar ranspor cofficins for chaoic dynamical flows, such as scap ras, man drifs and diffusion ras; powr spcra; and a hos of mahmaical consrucs such as gnralizd dimnsions, nropis, and Lyapunov xponns. Hr w oulin how such avrags ar valuad wihin h voluion opraor framwork. Th ky ida is o rplac h xpcaion valus of obsrvabls by h xpcaion valus of gnraing funcionals. This associas 7.. Tim avrags L a=a(x) b any obsrvabl, a funcion ha associas o ach poin in sa spac a numbr, a vcor, or a nsor. Th obsrvabl rpors on a propry of h dynamical sysm. Th obsrvabl is a dvic, such as a hrmomr or lasr Dopplr vlociomr. Th dvic islf dos no chang during h masurmn. Th vlociy fild a i (x)=v i (x) is an xampl of a vcor obsrvabl; h lngh of his vcor, or prhaps a mpraur masurd in an xprimn a insan τ ar xampls of scalar obsrvabls. W dfin h ingrad obsrvabl A as h im ingral of h obsrvabl a valuad along h rajcory of h iniial poin x, A (x )= dτ a[x(τ)], x()= f (x ). (7.) If h dynamics ar givn by an irad mapping and h im is discr, h ingrad obsrvabl afr n iraions is givn by n A n (x )= a(x k ), x k = f k (x )) (7.2) k= (w supprss vcorial indics for h im bing). Exampl 7. Ingrad obsrvabls. (a) If h obsrvabl is h vlociy, a i (x)= v i (x), is im ingral A i (x ) is h rajcory A i (x )= x i (). (b) For Hamilonian flows h acion associad wih a rajcory x() = [q(), p()] passing hrough a phas-spac poin x = [q(), p()] is: A (x )= dτ q(τ) p(τ). (7.3) Th im avrag of h obsrvabl along an orbi is dfind by a(x )= lim A (x ). (7.4) If a dos no bhav oo wildly as a funcion of im for xampl, if a(x) is h Chicago mpraur, boundd bwn 8 o F and+3 o F for all ims A (x ) is xpcd o grow no fasr han, and h limi (7.4) xiss. For an xampl of a im avrag - h Lyapunov xponn - s sc avrag - 27oc22 ChaosBook.org vrsion4, Dc 3 22

2 CHAPTER 7. AVERAGING 35 CHAPTER 7. AVERAGING 352 Th im avrag is a propry of h orbi, indpndn of h iniial poin on ha orbi: if w sar a a lar sa spac poin f T (x ) w g a coupl of xra fini conribuions ha vanish in h limi: a[ f T (x )] = lim +T T = a(x ) lim = a(x ). dτ a[ f τ (x )] ( T dτ a[ f τ (x )] +T ) dτ a[ f τ (x )] Exampl 7.2 Lyapunov xponn. Givn a -dimnsional map, considr obsrvablλ(x)=ln f (x) and ingrad obsrvabl n n A n (x )= ln f (x k ) =ln f (x k ) = ln f n x (x ). k= k= Th Lyapunov xponn is h avrag ra of h xpansion n λ(x )= lim ln f (x k ). n n k= S sc for furhr dails. Th ingrad obsrvabl A (x ) and h im avrag a(x ) ak a paricularly simpl form whn valuad on a priodic orbi. Dfin xrcis 4.6 a A p = p T p a p n p = T p dτ a[ f τ (x )] for a flow = np i= a[ f i (x )] for a map, x p, (7.5) whr p is a prim cycl, T p is is priod, and n p is is discr im priod in h cas of irad map dynamics. Th quaniy A p is a loop ingral of h obsrvabl along a singl ravrsal of a prim cycl p, so i is an inrinsic propry of h cycl, indpndn of h saring poin x p. (If h obsrvabl a is no a scalar bu a vcor or marix w migh hav o b mor carful in dfining an avrag which is indpndn of h saring poin on h cycl). If h rajcory rracs islf r ims, w jus obain A p rpad r ims. Evaluaion of h asympoic im avrag (7.4) hrfor rquirs only a singl ravrsal of h cycl: a p = A p /T p. (7.6) Howvr, a(x ) is in gnral a wild funcion of x ; for a hyprbolic sysm i aks h sam valua for almos all iniial x, bu a diffrn valu (7.6) on any priodic orbi, i.., on a dns s of poins (figur 7. (b)). Figur 7.: (a) A ypical chaoic rajcory xplors h sa spac wih h long im visiaion frquncy building up h naural masurρ (x). (b) im avrag valuad along an aypical rajcory such as a priodic orbi fails o xplor h nir accssibl sa spac. (A. Johansn) (a) x Exampl 7.3 Drminisic diffusion. Th phas spac of an opn sysm such as h Sinai gas (an infini 2-dimnsional priodic array of scaring disks, s sc. 25.) is dns wih iniial poins ha corrspond o priodic runaway rajcoris. Th man disanc squard ravrsd by any such rajcory grows as x() 2 2, and is conribuion o h diffusion ra D x() 2 /, (7.4) valuad wih a(x)= x() 2, divrgs. Smingly hr is a paradox; vn hough inuiion says h ypical moion should b diffusiv, w hav an infiniy of ballisic rajcoris. For chaoic dynamical sysms, his paradox is rsolvd by also avraging ovr h iniial x and worrying abou h masur of h pahological rajcoris. (coninud in xampl 7.4) scion Spaial avrags Th spac avrag of a quaniy a valuad ovr all sa spac rajcoris x() a im is givn by h d-dimnsional ingral ovr all iniial poins x a im =: a () = = dx a[x()], x()= f (x ) dx = volum of. (7.7) Th spac is assumd o hav fini volum - opn sysms lik h 3-disk gam of pinball ar discussd in sc Wha is i w rally do in xprimns? W canno masur h im avrag (7.4), as hr is no way o prpar a singl iniial condiion wih infini prcision. Th bs w can do is prpar an iniial dnsiy ρ(x), prhaps concnrad on som small (bu always fini) nighborhood. Thn w can abandon h uniform spac avrag (7.7) and considr insad h wighd spaial avrag a ρ ()= dx ρ(x ) a[x()], ρ = dxρ(x). (7.8) ρ For rgodic mixing sysms, any smooh iniial dnsiy will nd o h asympoic naural masur in h limiρ(x, ) ρ (x). This allows us o ak any (b) avrag - 27oc22 ChaosBook.org vrsion4, Dc 3 22 avrag - 27oc22 ChaosBook.org vrsion4, Dc 3 22

3 CHAPTER 7. AVERAGING 353 smooh iniialρ(x) and dfin h xpcaion valua of an obsrvabl a as h asympoic im and spac avrag ovr h sa spac a = dx a[x]= lim dx dτ a[x()]. (7.9) W us h sam noaion as for h spac avrag (7.7) and disinguish h wo by h prsnc of h im variabl in h argumn: if h quaniya () bing avragd dpnds on im, hn i is a spac avrag; if i is h infini im limi, i is h xpcaion valua. Th xpcaion valu is a spac avrag of im avrags, wih vry x usd as a saring poin of a im avrag. Th advanag of avraging ovr spac is ha i smars h saring poins which wr problmaic for h im avrag (such as priodic poins). Whil asy o dfin, h xpcaion valua urns ou no o b paricularly racabl in pracic. Hr coms a simpl ida ha is h basis of all ha follows: Such avrags ar mor convninly sudid by invsigaing insad ofa h spac avrags of form β A = dx β A (x). (7.) In h prsn conx β is an auxiliary variabl of no physical significanc whos rol is o nabl us o rcovr h dsird spac avrag by diffrniaion, A = β A. β β= In mos applicaionsβ is a scalar, bu if h obsrvabl is a d-dimnsional vcor a(x) R d, hnβ R d ; if h obsrvabl is a [d d] nsor,β is also a rank-2 nsor, and so on. Hr w will mosly limi h considraions o scalarβ. If h im avrag limi a(x ) (7.4) xiss for almos all iniial x s and h sysm is rgodic and mixing (in h sns of sc..3.), w xpc h im avrag along almos all rajcoris o nd o h sam valu a, and h ingrad obsrvabl A o nd o a. Th spac avrag (7.) is an ingral ovr xponnials and hnc also grows xponnially wih im. So as w would xpc h spac avrag of xp(β A (x)) o grow xponnially wih im CHAPTER 7. AVERAGING 354 Now w undrsand on rason for why i is smarr o compu xp(β A ) rahr hana : h xpcaion valu of h obsrvabl (7.9) and h momns of h ingrad obsrvabl (7.) can b compud by valuaing h drivaivs of s(β) s β 2 s β 2 β= β= = lim A =a, ( = lim A A A A ) = lim (A a ) 2, (7.2) and so forh. W hav xplicily wrin ou h formulas for a scalar obsrvabl; h vcor cas is workd ou in xrcis 7.2 (w could hav usd full drivaiv ds/dβ in (7.2), bu for vcor obsrvabl w do nd parial drivaivs s/ β i ). If w can compu h funcion s(β), w hav h dsird xpcaion valu wihou having o sima any infini im limis from fini im daa. Suppos w could valua s(β) and is drivaivs. Wha ar such formulas good for? A ypical applicaion ariss in h problm of drmining ranspor cofficins from undrlying drminisic dynamics. xrcis 7.2 Exampl 7.4 Drminisic diffusion. (coninud from xampl 7.3) Considr a poin paricl scaring lasically off a d-dimnsional array of scarrs. If h scarrs ar sufficinly larg o block any infini lngh fr flighs, h paricl will diffus chaoically, and h ranspor cofficin of inrs is h diffusion consan x() 2 4D. In conras o D simad numrically from rajcoris x() for fini bu larg, h abov formulas yild h asympoic D wihou any xrapolaions o h limi. For xampl, for a i = v i and zro man drifv i =, in d dimnsions h diffusion consan is givn by h curvaur of s(β) a β =, scion 25. D= lim x() 2 = 2d 2d d 2 s, (7.3) β= β 2 i= i so if w can valua drivaivs of s(β), w can compu ranspor cofficins ha characriz drminisic diffusion. As w shall s in chapr 25, priodic orbi hory yilds an xac and xplici closd form xprssion for D. W urn o h problm of valuaing β A in sc. 7.2, bu firs w rviw som lmnary facs of saisics ha will b usful lar on. β A (cons) s(β), and is ra of growh o b givn by h limi fas rack: sc. 7.2, p. 355 s(β)= lim ln β A. (7.) avrag - 27oc22 ChaosBook.org vrsion4, Dc 3 22 avrag - 27oc22 ChaosBook.org vrsion4, Dc 3 22

4 CHAPTER 7. AVERAGING 355 CHAPTER 7. AVERAGING 356 Figur 7.2: Spac avraging pics oghr h im avrag compud along h orbi of figur 7. by a spac avrag ovr infinily rgb],,x many shor rajcory sgmns saring a all iniial poins a onc. rgb],,x 2 rgb],,ρ(x) rgb],, [ L ρ ] (x) rgb],,x2 rgb],,x Th simpls xampl is h β = cas, i.., h Prron-Frobnius opraor inroducd in sc Anohr xampl - dsignd o dlivr h Lyapunov xponn - will b h voluion opraor (7.4) discussd blow. Th acion of h voluion opraor on a funcionφ is givn by [ L φ ] (y)= dxδ ( y f (x) ) β A (x) φ(x). (7.7) 7.2 Evoluion opraors For i, h mysic voluion; No h righ only jusifid wha w call vil also jusifid. Wal Whiman, Lavs of Grass: Song of h Univrsal Th abov simpl shif of focus, from sudyinga o sudying xp ( β A ) is h ky o vryhing ha follows. ak h dpndnc on h flow xplici by rwriing his quaniy as β A = dx dyδ ( y f (x) ) β A (x). (7.4) Hrδ ( y f (x) ) is h Dirac dla funcion: for a drminisic flow an iniial poin x maps ino a uniqu poin y a im. Formally, all w hav don abov is o insr h idniy = dyδ ( y f (x) ), (7.5) ino (7.) o mak xplici h fac ha w ar avraging only ovr h rajcoris ha rmain in for all ims. Howvr, having mad his subsiuion w hav rplacd h sudy of individual rajcoris f (x) by sudying h voluion of h dnsiy of h oaliy of iniial condiions. Insad of rying o xrac a mporal avrag from an arbirarily long rajcory which xplors h sa spac rgodically, w can now prob h nir sa spac wih shor (and conrollabl) fini im pics of rajcoris originaing from vry poin in. As a mar of fac (and ha is why w wn o h roubl of dfining h gnraor (6.27) of infinisimal ransformaions of dnsiis) infinisimally shor im voluion inducd by h gnraor A of (6.27) suffics o drmin h spcrum and ignvalus ofl. W shall rfr o h krnl of h opraion (7.4) as h voluion opraor L (y, x)=δ ( y f (x) ) β A (x). (7.6) Th voluion opraor is diffrn for diffrn obsrvabls, as is dfiniion dpnds on h choic of h ingrad obsrvabl A in h xponnial. Is job is o dlivr h xpcaion valu of a, bu bfor showing ha i accomplishs ha, w nd o vrify h smigroup propry of voluion opraors. By is dfiniion, h ingral ovr h obsrvabl a is addiiv along h rajcory x() x( + 2) A +2 (x ) = = x() x( ) + dτ a[ f τ (x)] + x() + 2 x(+2) = A (x ) + A 2 ( f (x )). dτ a[ f τ (x)] xrcis 6.3 As A (x) is addiiv along h rajcory, h voluion opraor gnras a smigroup scion 6.5 L +2 (y, x)= dzl 2 (y, z)l (z, x), (7.8) as is asily chckd by subsiuion [ L 2 L a ] (y)= dxδ(y f 2 (x)) β A 2(x) [ L a ] (x)= [ L +2 a ] (y). This smigroup propry is h main rason why (7.4) is prfrabl o (7.9) as a saring poin for valuaion of dynamical avrags: i rcass avraging in form of opraors muliplicaiv along h flow. In rms of h voluion opraor, h spac avrag of h gnraing funcion (7.4) is givn by β A = dx dyφ(y)l (y, x)φ(x). whrφ(x) is h consan funcionφ(x)=. If h linar opraorl can b hough of as a marix, high powrs of a marix ar dominad by is fass growing marix lmns, and h limi (7.) s(β)= lim ln L. (7.9) avrag - 27oc22 ChaosBook.org vrsion4, Dc 3 22 avrag - 27oc22 ChaosBook.org vrsion4, Dc 3 22

5 CHAPTER 7. AVERAGING 357 yilds h lading ignvalu s (β), and, hrough i, all dsird xpcaion valus (7.2). In wha follows w shall larn how o xrac no only h lading ignvalu ofl, bu much of h dominan par of is spcrum. Clarly, w ar no inrsd ino h ignvalus ofl for any paricular fini im, bu hir bhavior as. Tha is achivd via a Laplac ransform, s sc Spcrum of an voluion opraor An xposiion of a subjc is of ncssiy squnial and on canno xplain vryhing a onc. As w shall acually nvr us ignfuncions of voluion opraors, w pospon hir discussion o sc For h im bing w ask h radr o accp uncriically h following skch: Schmaically, a linar opraor has a spcrum of ignvalus s α and ignfuncionsϕ α (x) [ L ϕ α ] (x)= s α ϕ α (x), α=,, 2,... (7.2) CHAPTER 7. AVERAGING Evoluion for infinisimal ims For infinisimal imδ, h voluion opraor (7.7) acs as ρ(y, δ) = = dx βaδ (x) δ(y f δ (x))ρ(x, ) dx βa(x)δ δ(y x δ v(x))ρ(x, ) = (+δβ a(y)) ρ(y, ) δ v x +δ v x ρ(y, ), (h dnominaor ariss from hδ linarizaion of h jacobian) giving h coninuiy quaion (6.25) a sourc rm ρ + x i (v i ρ)=β aρ. (7.22) Th voluion gnraor (6.27) ignfuncions now saisfy ordrd so ha R s α R s α+. For coninuous im flow ignvalus canno dpnd on im, hy ar ignvalus of h im-voluion gnraor (6.26) w always wri h ignvalus of an voluion opraor in xponniad form sα rahr han as muliplirsλ α W find i convnin o wri hm his way boh for h coninuous iml and h discr iml=l cass, and w shall assum ha spcrum of L is discr. L is a linar opraor acing on a dnsiy of iniial condiionsρ(x), x, so h limi will b dominad by s = s(β), h lading ignvalu ofl, [ L ρ β ] (y) := dxδ ( y f (x) ) β A (x) ρ β (x)= s(β) ρ β (y), (7.2) whrρ β (x) is h corrsponding ignfuncion. Forβ=h voluion opraor (7.6) is h Prron-Frobnius opraor (6.), wihρ (x) h naural masur. From now on w hav o b carful o disinguish h wo kinds of linar opraors. In chapr 5 w hav characrizd h voluion of h local linar nighborhood of a sa spac rajcory by ignvalus and ignvalus of h linarizd flow Jacobian marics. Evoluion opraors dscribd in his chapr ar global, and hy ac on dnsiis of orbis, no on individual rajcoris. As w shall s, n of h wondrs of chaoic dynamics is ha h mor unsabl individual rajcoris, h nicr ar h corrsponding global dnsiy funcions. (s(β) A)ρ(x,β)=β a(x)ρ(x,β). (7.23) Diffrniaing wih rspc oβ s (β)ρ(x,β) + s(β) β ρ(x,β)+ x (v(x) β ρ(x,β) ) = a(x)ρ(x,β)+β a(x) β ρ(x,β) In h vanishing auxiliary paramr limiβ, w hav s()=,ρ(x, )=ρ (x) s ()ρ (x)+ (v i (x) β ) x ρ(x, ) = a(x)ρ (x). i By ingraing, h scond rm vanishs by Gauss horm s ()= dx a(x)ρ (x)=a, vrifying quaion (7.8): spaial avrag of h obsrvabl a is givn by h drivaiv of h lading ignvalu s (). fas rack: sc. 8, p. 37 avrag - 27oc22 ChaosBook.org vrsion4, Dc 3 22 avrag - 27oc22 ChaosBook.org vrsion4, Dc 3 22

6 CHAPTER 7. AVERAGING 359 CHAPTER 7. AVERAGING Rsolvn ofl Hr w limi ourslvs o a brif rmark abou h noion of h spcrum of a linar opraor. Th Prron-Frobnius opraor L acs muliplicaivly in im, so i is rasonabl o suppos ha hr xis consans >, s such ha L s for all. Wha dos ha man? Th opraor norm is dfind in h sam spiri in which on dfins marix norms: W ar assuming ha no valu ofl ρ(x) grows fasr han xponnially for any choic of funcion ρ(x), so ha h fass possibl growh can b boundd by s, a rasonabl xpcaion in h ligh of h simpls xampl sudid so far, h scap ra (.3). If ha is so, muliplying L by s w consruc a nw opraor s L = (A s) which dcays xponnially for larg, (A s). W say ha s L is an lmn of a boundd smigroup wih gnraora s I. Givn his bound, i follows by h Laplac ransform d s L = s A, R s> s, (7.24) ha h rsolvn opraor (s A) is boundd s A d s s =. (7.25) s s If on is inrsd in h spcrum ofl, as w will b, h rsolvn opraor is a naural objc o sudy; i has no im dpndnc, and i is boundd. I is calld rsolvn bcaus i sparas h spcrum of L ino individual consiuns, on for ach spcral lin. From (7.9), i is clar ha h lading ignvalu s (β) corrsponds o h pol in (7.25); as w shall s in chapr 8, h rs of h spcrum is similarly rsolvd ino furhr pols of h Laplac ransform. Th main lsson of his brif asid is ha for coninuous im flows, h Laplac ransform is h ool ha brings down h gnraor in (6.29) ino h rsolvn form (7.24) and nabls us o sudy is spcrum. 7.3 Avraging in opn sysms Ifis a compac rgion or s of rgions o which h dynamics is confind for all ims, (7.9) is a snsibl dfiniion of h xpcaion valu. Howvr, if h rajcoris can xiwihou vr rurning, dyδ(y f (x ))= for > xi, x, Figur 7.3: A picwis-linar rpllr (6.): All rajcoris ha land in h gap bwn h f and f branchs scap (Λ = 4,Λ = 2)..5 w migh b in roubl. In paricular, a rpllr is a dynamical sysm for which h rajcory f (x ) vnually lavs h rgion, unlss h iniial poin x is on h rpllr, so h idniy dyδ(y f (x ))=, >, iff x non wandring s (7.26) migh apply only o a fracal subs of iniial poins of zro Lbsgu masur (non wandring s is dfind in sc. 2..). Clarly, for opn sysms w nd o modify h dfiniion of h xpcaion valu o rsric i o h dynamics on h non wandring s, h s of rajcoris which ar confind for all ims. Dno by a sa spac rgion ha ncloss all inrsing iniial poins, say h 3-disk Poincaré scion consrucd from h disk boundaris and all possibl incidnc angls, and dno by h volum of. Th volum of sa spac conaining all rajcoris, which sar ou wihin h sa spac rgion and rcur wihin ha rgion a im, is givn by () = dxdyδ ( y f (x) ) γ. (7.27) As w hav alrady sn in sc..4.3, his volum is xpcd o dcras xponnially, wih h scap raγ. Th ingral ovr x aks car of all possibl iniial poins; h ingral ovr y chcks whhr hir rajcoris ar sill wihin by h im. For xampl, any rajcory ha falls off h pinball abl in scion 22. figur. is gon for good. If w xpand an iniial disribuion ρ(x) in (7.2), h ignfuncion basis ρ(x)= α a α ϕ α (x), w can also undrsand h ra of convrgnc of fini-im simas o h asympoic scap ra. For an opn sysm h fracion of rappd rajcoris dcays as scion 7.3 Γ () = dx[ L ρ ] (x) dxρ(x) = α f(x).5 sα a dxϕ α(x) α dxρ(x) = s( (cons.)+o( (s s) ) ). (7.28) x avrag - 27oc22 ChaosBook.org vrsion4, Dc 3 22 avrag - 27oc22 ChaosBook.org vrsion4, Dc 3 22

7 CHAPTER 7. AVERAGING 36 CHAPTER 7. AVERAGING 362 Th consan dpnds on h iniial dnsiy ρ(x) and h gomry of sa spac cuoff rgion, bu h scap raγ= s is an inrinsic propry of h rplling s. W s, a las hurisically, ha h lading ignvalu ofl dominasγ () and yilds h scap ra, a masurabl propry of a givn rpllr. δx x() x( ) δx Th non wandring s can b vry difficul o dscrib; bu for any fini im w can consruc a normalizd masur from h fini-im covring volum (7.27), by rdfining h spac avrag (7.) as β A = dx β A (x) dx β A (x)+γ. (7.29) () Figur 7.4: A long-im numrical calculaion of h lading Lyapunov xponn rquirs rscaling h disanc in ordr o kp h narby rajcory sparaion wihin h linarizd flow rang. 7.4 Lyapunov xponns x( ) 2 δ x 2 in ordr o compnsa for h xponnial dcras of h numbr of surviving rajcoris in an opn sysm wih h xponnially growing facor γ. Wha dos his man? Onc w hav compudγw can rplnish h dnsiy los o scaping rajcoris, by pumping in γ of nw rajcoris in such a way ha h ovrall masur is corrcly normalizd a all ims, =. Exampl 7.5 Escap ra for a picwis-linar rpllr: (coninuaion of xampl 6.) Wha is gaind by rformulaing h dynamics in rms of opraors? W sar by considring a simpl xampl in which h opraor is a [2 2] marix. Assum h xpanding -dimnsional map f (x) of figur 7.3, a picwis-linar 2 branch rpllr (6.). Assum a picwis consan dnsiy (6.2). Thr is no nd o dfinρ(x) in h gap bwn and, as any poin ha lands in h gap scaps. Th physical moivaion for sudying his kind of mapping is h pinball gam: f is h simpls modl for h pinball scap, figur.8, wih f and f modlling is wo srips of survivors. As can b asily chckd using (6.9), h Prron-Frobnius opraor acs on his picwis consan funcion as a [2 2] ransfr marix (6.3) xrcis 6. xrcis 6.5 ( ρ ρ ) ( Λ Lρ= Λ )( Λ ρ Λ ρ ), srching bohρ andρ ovr h whol uni inrvalλ, and dcrasing h dnsiy a vry iraion. In his xampl h dnsiy is consan afr on iraion, solhas only on non-zro ignvalu s = / Λ +/ Λ, wih consan dnsiy ignvcor ρ =ρ. Th quaniis / Λ, / Λ ar, rspcivly, h sizs of h, inrvals, so h xac scap ra (.3) h log of h fracion of survivors a ach iraion for his linar rpllr is givn by h sol ignvalu of L: γ= s = ln(/ Λ +/ Λ ). (7.3) Voila! Hr is h raional for inroducing opraors in on im sp w hav solvd h problm of valuaing scap ras a infini im. (coninud in xampl 23.5) (J. ahisn and P. Cvianović) L us apply h nwly acquird ools o h fundamnal diagnosics in his subjc: Is a givn sysm chaoic? And if so, how chaoic? If all poins in a nigh- xampl 2.3 borhood of a rajcory convrg oward h sam rajcory, h aracor is a fixd poin or a limi cycl. Howvr, if h aracor is srang, any wo rajcoris scion.3. x()= f (x ) and x()+δx()= f (x +δx ) (7.3) ha sar ou vry clos o ach ohr spara xponnially wih im, and in a fini im hir sparaion aains h siz of h accssibl sa spac. This snsiiviy o iniial condiions can b quanifid as δx() λ δx (7.32) whrλ, h man ra of sparaion of rajcoris of h sysm, is calld h Lyapunov xponn Lyapunov xponn as a im avrag W can sar ou wih a smallδx and ry o simaλfrom (7.32), bu now ha w hav quanifid h noion of linar sabiliy in chapr 4 and dfind h dynamical im avrags in sc. 7.., w can do br. Th problm wih masuring h growh ra of h disanc bwn wo poins is ha as h poins spara, h masurmn is lss and lss a local masurmn. In h sudy of xprimnal im sris his migh b h only opion, bu if w hav quaions of moion, a br way is o masur h growh ra of vcors ransvrs o a givn orbi. Th man growh ra of h disanc δx() / δx bwn nighboring rajcoris (7.32) is givn by h Lyapunov xponn, which for long (bu no oo long) im can b simad as λ ln δx() / δx (7.33) avrag - 27oc22 ChaosBook.org vrsion4, Dc 3 22 avrag - 27oc22 ChaosBook.org vrsion4, Dc 3 22

8 CHAPTER 7. AVERAGING 363 CHAPTER 7. AVERAGING 364 Figur 7.5: Th symmric marix J= ( J )T J maps a swarm of iniial poins in an infinisimal sphrical nighborhood of x ino a cigar-shapd nighborhood fini im lar, wih smiaxs drmind by h local srching/shrinking Λ, bu local individual rajcory roaions by h complx phas of J ignord. x x + δx f ( ) x()+ Jδx Figur 7.6: A numrical sima of h lading Lyapunov xponn for h Rösslr flow (2.7) from h dominan xpanding ignvalu formula (7.35). Th lading Lyapunov xponn λ.9 is posiiv, so numrics suppors h hypohsis ha h Rösslr aracor is chaoic. Th big unxplaind jump illusras prils of Lyapunov xponns numrics. (J. ahisn) (For noaional brviy w shall ofn supprss h dpndnc of quaniis such asλ=λ(x ),δx()=δx(x, ) on h iniial poin x ). On can ak (7.33) as is, ak a small iniial sparaionδx, rack disanc bwn wo narby rajcoris unil δx( ) gs significanly biggr, hn rcord λ = ln( δx( ) / δx ), rscal δx( ) by facor δx / δx( ), and coninu add infinium, as in figur 7.4, wih h lading Lyapunov xponn givn by λ= lim i λ i. (7.34) i Howvr, w can do br. Givn h quaions of moion, for infinisimal δx w know hδx i ()/δx j () raio xacly, as his is by dfiniion h Jacobian marix (4.38) lim δx() δx i () δx j () = x i() x j () = J i j (x ), so h lading Lyapunov xponn can b compud from h linar approximaion (4.23) λ(x )= lim ln J (x )δx δx = lim 2 ln(ˆn T( J )T J ˆn ). (7.35) In his formula h scal of h iniial sparaion drops ou, only is orinaion givn by h iniial orinaion uni vcor ˆn=δx / δx mars. Th ignvalus of J ar ihr ral or com in complx conjuga pairs. As J is in gnral no symmric and no diagonalizabl, i is mor convnin o work wih h symmric and diagonalizabl marix = ( J )T J, wih ral posiiv ignvalus{ Λ 2... Λ d 2 }, and a compl orhonormal s of ignvcors of {u,...,u d }. Expanding h iniial orinaion ˆn= (ˆn u i )u i in h u i = Λ i 2 u i ignbasis, w hav ˆn T ˆn= d (ˆn u i ) 2 Λ i 2 = (ˆn u ) 2 2µ( +O( 2(µ µ2) ) ), (7.36) i= whr ln Λ i (x, ) =µ i, wih ral pars of characrisic xponns (4.8) ordrd byµ µ 2 µ 3. For long ims h largs Lyapunov xponn dominas xponnially (7.35), providd h orinaion ˆn of h iniial sparaion was no chosn prpndicular o h dominan xpanding ign-dircion u. Th Lyapunov xponn is h im avrag λ(x ) { = lim ln ˆn u +ln Λ (x, ) +O( 2(λ λ2) ) } = lim ln Λ (x, ), (7.37) whrλ (x, ) is h lading ignvalu of J (x ). By choosing h iniial displacmn such ha ˆn is normal o h firs (i-) ign-dircions w can dfin no only h lading, bu all Lyapunov xponns as wll: λ i (x )= lim ln Λ i(x, ), i=, 2,,d. (7.38) Th lading Lyapunov xponn now follows from h Jacobian marix by numrical ingraion of (4.9). Th quaions can b ingrad accuraly for a fini im, hnc h infini im limi of (7.35) can b only simad from plos of 2 ln(ˆnt ˆn) as funcion of im, such as figur 7.6 for h Rösslr flow (2.7). As h local xpansion and conracion ras vary along h flow, h mporal dpndnc xhibis small and larg humps. Th suddn fall o a low lvl is causd by a clos passag o a folding poin of h aracor, an illusraion of why numrical valuaion of h Lyapunov xponns, and proving h vry xisnc of a srang aracor is a difficul problm. Th approximaly monoon par of h curv can b usd (a your own pril) o sima h lading Lyapunov xponn by a sraigh lin fi. As w can alrady s, w ar couring difficulis if w ry o calcula h Lyapunov xponn by using h dfiniion (7.37) dircly. Firs of all, h sa spac is dns wih aypical rajcoris; for xampl, if x happns o li on a priodic orbi p,λwould b simply ln Λ p /T p, a local propry of cycl p, no a global propry of h dynamical sysm. Furhrmor, vn if x happns o b a gnric sa spac poin, i is sill no obvious ha ln Λ(x, ) / should b convrging o anyhing in paricular. In a Hamilonian sysm wih coxising llipic islands and chaoic rgions, a chaoic rajcory gs capurd in h nighborhood of an llipic island vry so ofn and can say hr for arbirarily long im; as hr h orbi is narly sabl, during such pisod ln Λ(x, ) / can dip arbirarily clos o +. For sa spac volum non-prsrving flows h rajcory can avrag - 27oc22 ChaosBook.org vrsion4, Dc 3 22 avrag - 27oc22 ChaosBook.org vrsion4, Dc 3 22

9 CHAPTER 7. AVERAGING 365 CHAPTER 7. AVERAGING 366 ravrs locally conracing rgions, and ln Λ(x, ) / can occasionally go ngaiv; vn wors, on nvr knows whhr h asympoic aracor is priodic or chaoic, so any fini sima ofλmigh b dad wrong. xrcis Evoluion opraor valuaion of Lyapunov xponns A soluion o hs problms was proposd in sc rplac im avraging along a singl orbi by acion of a muliplicaiv voluion opraor on h nir sa spac, and xrac avrag of h Lyapunov xponn from is lading ignvalu. from fini lngh cycls. If h chaoic moion fills h whol sa spac, w ar indd compuing h asympoic Lyapunov xponn. If h chaoic moion is ransin, lading vnually o som long araciv cycl, our Lyapunov xponn, compud on a non wandring s, will characriz h chaoic ransin; his is acually wha any xprimn would masur, as vn a vry small amoun of xrnal nois suffics o dsabiliz a long sabl cycl wih a minu immdia basin of aracion. Th main ida - wha is h Lyapunov obsrvabl - can b illusrad by h dynamics of a -dimnsional map. Exampl 7.6 Lyapunov xponn, discr im -dimnsional dynamics. Du o h chain rul (4.47) for h drivaiv of an irad map, h sabiliy of a -dimnsional mapping is muliplicaiv along h flow, so h ingral (7.) of h obsrvabl a(x) = ln f (x), h local rajcory divrgnc ra, valuad along h rajcory of x, is addiiv: A n (x )=ln f n (x ) n = ln f (x k ). (7.39) k= For a -dimnsional iraiv mapping, h Lyapunov xponn is hn h xpcaion valu (7.9) givn by a spaial ingral (7.8) wighd by h naural masur λ= ln f (x) = dxρ (x) ln f (x). (7.4) Th associad (discr im) voluion opraor (7.6) is L(y, x)=δ(y f (x)) β ln f (x). (7.4) from (7.2), h drivaiv of h lading ignvalu s (β) of h voluion opraor (7.4). xampl 2.2 Th only qusion is: How? (By chapr 2 you will know.) Résumé Th xpcaion valua of an obsrvabl a(x) ingrad, A (x)= dτ a(x(τ)), and im avragd, A /, ovr h rajcory x x() is givn by h drivaiv a = s β β= of h lading ignvalu s(β) of h voluion opraorl. By compuing h lading ignfuncion of h Prron-Frobnius opraor (6.), on obains h xpcaion valu (6.2) of any obsrvabl a(x). Thus w can consruc a spcific, hand-ailord voluion opraor L for ach and vry obsrvabl. Th good nws is ha, by h im w arriv a chapr 2, h scaf- chapr 2 folding will b rmovd, bohl s and hir ignfuncions will b gon, and only h xplici and xac priodic orbi formulas for xpcaion valus of obsrvabls will rmain. Th nx qusion is: How do w valua h ignvalus ofl? In xampl 7.5, w saw a picwis-linar xampl whr hs opraors rduc o fini marics L, bu for gnric smooh flows, hy ar infini-dimnsional linar opraors, and finding smar ways of compuing hir ignvalus rquirs som hough. In chapr w undrook h firs sp, and rplacd h ad hoc pariioning (6.4) by h inrinsic, opologically invarian pariioning. In chapr 5 w applid his informaion o our firs applicaion of h voluion opraor formalism, valuaion of h opological nropy, and h growh ra of h numbr of opologically disinc orbis. In chaprs 8 and 9, his small vicory will b rfashiond ino a sysmaic mhod for compuing ignvalus of voluion opraors in rms of priodic orbis. Hr w hav rsricd our considraions o -d maps, as for highr-dimnsional flows only h Jacobian marics ar muliplicaiv, no h individual ignvalus. Consrucion of h voluion opraor for valuaion of h Lyapunov spcra for a d-dimnsional flow rquirs mor clvrnss han warrand a his sag in h narraiv: an xnsion of h voluion quaions o a flow in h angn spac. All ha rmains is o drmin h valu of h Lyapunov xponn λ= ln f (x) = s(β) β = s () (7.42) β= Commnary Rmark 7. Prssur. Th quaniy xp(β A ) is calld a pariion funcion by Rull [9.]. ahmaicians dcora i wih considrably mor Grk and Gohic lrs han is don in his rais. Rull [7.] and Bown [7.2] had givn nam prssur P(a) o s(β) (whr a is h obsrvabl inroducd in sc. 7..), dfind by h larg sysm limi (7.). As w shall also apply h hory o compuaing h physical gas prssur xrd on h walls of a conainr by a bouncing paricl, w rfr o s(β) as simply h lading ignvalu of h voluion opraor inroducd in sc avrag - 27oc22 ChaosBook.org vrsion4, Dc 3 22 avrag - 27oc22 ChaosBook.org vrsion4, Dc 3 22

10 CHAPTER 7. AVERAGING 367 EXERCISES 368 Th convxiy propris such as P(a) P( a ) will b pry obvious consquncs of h dfiniion (7.). In h cas ha L is h Prron-Frobnius opraor (6.), h ignvalus{s (β), s (β), } ar calld h Rull-Pollico rsonancs [7.3, 7.4, 7.5], wih h lading on, s(β)= s (β) bing h on of main physical inrs. In ordr o aid h radr in digsing h mahmaics liraur, w shall ry o poin ou h noaional corrspondncs whnvr appropria. Th rigorous formalism is rpl wih lims, sups, infs,ω-ss which ar no rally ssnial o undrsanding of h hory, and ar avoidd in his book. Rmark 7.2 icrocanonical nsmbl. In saisical mchanics h spac avrag (7.7) prformd ovr h Hamilonian sysm consan nrgy surfac invarian masur ρ(x)dx=dqdpδ(h(q, p) E) of volumω(e)= dqdpδ(h(q, p) E) a() = dqdpδ(h(q, p) E)a(q, p, ) (7.43) ω(e) is calld h microcanonical nsmbl avrag. Rmark 7.3 Lyapunov xponns. Th uliplicaiv Ergodic Thorm of Osldc [7.6] sas ha h limis ( ) xis for almos all poins x and all angn vcors ˆn. Thr ar a mos d disinc valus ofλas w l ˆn rang ovr h angn spac. Ths ar h Lyapunov xponns [7.8]λ i (x ). W ar doubful of h uiliy of Lyapunov xponns as mans of prdicing any obsrvabls of physical significanc, bu ha is h minoriy posiion - in h liraur on ncounrs many provocaiv spculaions, spcially in h conx of foundaions of saisical mchanics ( hydrodynamic mods) and h xisnc of a Lyapunov spcrum in h hrmodynamic limi of spaiomporal chaoic sysms. Thr ar volums of liraur on numrical compuaion of h Lyapunov xponns, s for xampl rfs. [7.4, 7.5, 7.7]. For arly numrical mhods o compu Lyapunov vcors, s rfs. [7.6, 7.7]. Th drawback of h Gram-Schmid mhod is ha h vcors so consrucd ar orhogonal by fia, whras h sabl/unsabl ignvcors of h Jacobian marix ar in gnral no orhogonal. Hnc h Gram-Schmid vcors ar no covarian, i.., h linarizd dynamics dos no ranspor hm ino h ignvcors of h Jacobian marix compud furhr downsram. For compuaion of covarian Lyapunov vcors, s rfs. [7.8, 7.2]. Rmark 7.4 Sa spac discrizaion. Rf. [7.2] discusss numrical discrizaons of sa spac, and consrucion of Prron-Frobnius opraors as sochasic marics, or dircd wighd graphs, as coars-graind modls of h global dynamics, wih ranspor ras bwn sa spac pariions compud using his marix of ransiion probabiliis; a rigorous discussion of som of h formr faurs is includd in rf. [7.22]. Exrciss 7.. How unsabl is h Hénon aracor? (a) Evalua numrically h Lyapunov xponn λ by iraing som, ims or so h Hénon map [ ] [ ] x ax y = 2 + y bx for a=.4, b=.3. (b) Would you dscrib h rsul as a srang aracor? Why? (c) How robus is h Lyapunov xponn for h Hénon aracor? Evalua numrically h Lyapunov xponn by iraing h Hénon map for a = , b =.3. How much do you now rus your rsul for par (a) of his xrcis? (d) R-xamin his compuaion by ploing h iras, and rasing h plod poins vry iras or so. Kp a i unil h srang aracor vanishs lik h smil of h Chsir ca. Wha rplacs i? Do a fw numrical xprimns o sima h lngh of ypical ransin bfor h dynamics sls ino his long-im aracor. () Us your Nwon sarch rouin o confirm xisnc of his aracor. Compu is Lyapunov xponn, compar wih your numrical rsul from abov. Wha is h iinrary of h aracor. (f) Would you dscrib h rsul as a srang aracor? Do you sill hav confidnc in claims such as h on mad for h par (b) of his xrcis? 7.2. Expcaion valu of a vcor obsrvabl. Chck and xnd h xpcaion valu formulas (7.2) by valuaing h drivaivs of s(β) up o 4-h ordr for h spac avrag xp(β A ) wih a i a vcor quaniy: (a) (b) s β i β= 2 s β i β j β= = lim A i =ai, (7.44) ( ) = lim A i A j A i A j = lim (A i a i )(A j a j ). No ha h formalism is smar: i auomaically yilds h varianc from h man, rahr han simply h 2nd momn a 2. (c) compu h hird drivaiv of s(β). (d) compu h fourh drivaiv assuming ha h man in (7.44) vanishs,a i =. Th 4-h ordr momn formula x 4 () K()= x2 () 3 (7.45) 2 ha you hav drivd is known as kurosis: i masurs a dviaion from wha h 4-h ordr momn would b wr h disribuion a pur Gaussian (s (25.22) for a concr xampl). If h obsrvabl is a vcor, h kurosis K() is givn by i j [ Ai A i A j A j + 2 ( Ai A j A j A i Ai A ( ia i A i ) Pinball scap ra from numrical simulaion. Esima h scap ra for R : a=6 3-disk pinball by shooing, randomly iniiad pinballs ino h 3-disk sysm and ploing h logarihm of h numbr of rappd orbis as funcion of im. For comparison, a numrical simulaion of rf. [8.3] yilds γ = Rösslr aracor Lyapunov xponns. (a) Evalua numrically h xpanding Lyapunov xponnλ of h Rösslr aracor (2.7). (b) Plo your own vrsion of figur 7.6. Do no worry if i looks diffrn, as long as you undrsand why your plo looks h way i dos. (Rmmbr h nonuniform conracion/xpansion of figur 4.3.) (c) Giv your bs sima ofλ. Th liraur givs surprisingly inaccura simas - s whhr you can do br. (d) Esima h conracing Lyapunov xponnλ c. Evn hough i is much smallr hanλ, a glanc a h sabiliy marix (4.4) suggss ha you can probably g i by ingraing h infinisimal volum along a long-im rajcory, as in (4.42). avrag - 27oc22 ChaosBook.org vrsion4, Dc 3 22 rfsavr - sp27 ChaosBook.org vrsion4, Dc 3 22

11 Rfrncs 369 Rfrncs [7.] D. Rull, Bull. Amr. ah. Soc. 78, 988 (972). [7.2] R. Bown, Equilibrium sas and h rgodic hory of Anosov diffomorphisms, Springr Lc. Nos on ah. 47 (975). [7.3] D. Rull, Th hrmodynamical formalism for xpanding maps, J. Diff. Go. 25, 7 (987). [7.4]. Pollico, On h ra of mixing of Axiom A flows, Invn. ah. 8, 43 (985). [7.5] D. Rull, J. Diff. Go. 25, 99 (987). [7.6] V. I. Osldc, Trans. oscow ah. Soc. 9, 97 (968). [7.7]. Pollico, Lcurs on Ergodic Thory and Psin Thory in Compac anifolds, (Cambridg Univ. Prss, Cambridg 993). [7.8] A.. Lyapunov, Gnral problm of sabiliy of moion, Ann.ah. Sudis 7 (949) (Princon Univ. Prss). [7.9] Ya.B. Psin, Uspkhi a. Nauk 32, 55 (977), [Russian ah. Survys 32, 55 (977)] [7.] Ya.B. Psin, Dynamical sysms wih gnralizd hyprbolic aracors: hyprbolic, rgodic and opological propris, Ergodic Thory and Dynamical Sysms, 2, 23 (992). [7.] Ya.B. Psin, Func. Anal. Applic. 8, 263 (974). Rfrncs 37 [7.9] C. Skokos, Th Lyapunov Characrisic Exponns and hir compuaion, arxiv: [7.2] A. Polii, A. Torcini and S. Lpri, J. Phys. IV 8, 263 (998). [7.2]. Dllniz, O. Jung, W.S. Koon, F. Lkin,.W. Lo, J.E. arsdn, K. Padbrg, R. Pris, S.D. Ross, and B. Thir, Transpor in Dynamical Asronomy and ulibody Problms, Inrna. J. Bifur. Chaos 5, 699 (25); koon/paprs [7.22] G. Froyland, Compur-assisd bounds for h ra of dcay of corrlaions, Commun.ah.Phys. 89, 237 (997); C. Livrani, Rigorous numrical invsigaion of h saisical propris of picwis xpanding maps. A fasibiliy sudy, Nonlinariy 4, 463 (2). [7.23] P. Cvianović, Chaos for cycliss, in E. oss, d., Nois and chaos in nonlinar dynamical sysms (Cambridg Univ. Prss, Cambridg 989). [7.24] P. Cvianović, Th powr of chaos, in J.H. Kim and J. Sringr, ds., Applid Chaos, (John Wily & Sons, Nw York 992). [7.25] P. Cvianović, d., Priodic Orbi Thory - hm issu, CHAOS 2, -58 (992). [7.26] P. Cvianović, Dynamical avraging in rms of priodic orbis, Physica D 83, 9 (995). [7.27] A. Trvisan and F. Pancoi, Priodic orbis, Lyapunov vcors, and singular vcors in h Lornz sysm, J. Amos. Sci. 55, 39 (998). [7.2] A. Kaok, Lyapunov xponns, nropy and priodic orbis for diffomorphisms, Publ. ah. IHES 5, 37 (98). [7.3] D. Bssis, G. Paladin, G. Turchi and S. Vaini, Gnralizd Dimnsions, Enropis and Lyapunov Exponns from h Prssur Funcion for Srang Ss, J. Sa. Phys. 5, 9 (988). [7.4] A. Wolf, J.B. Swif, al., Drmining Lyapunov Exponns from a Tim Sris, Physica D 6, 285 (985). [7.5] J.-P. Eckmann, S.O. Kamphors, al., Lyapunov xponns from im sris, Phys. Rv. A 34, 497 (986). [7.6] I. Shimada and T. Nagashima, Prog. Thor. Phys. 6, 65 (979). [7.7] G. Bnin, L. Galgani, A. Giorgilli and J.-. Srlcyn, Lyapunov characrisic xponns for smooh dynamical sysms and for Hamilonian sysms: a mhod for compuing all of hm. Par :Thory, ccanica 5, 9 (98); Par 2: Numrical Applicaion, ccanica 5, 2 (98). [7.8] F. Ginlli, P. Poggi, A. Turchi, H. Chaé, R. Livi and A. Polii, Characrizing dynamics wih covarian Lyapunov vcors, Phys Rv L 99, 36 (27); arxiv:76.5. rfsavr - sp27 ChaosBook.org vrsion4, Dc 3 22 rfsavr - sp27 ChaosBook.org vrsion4, Dc 3 22