From Periodic Motion to Unbounded Chaos: Investigations of the Simple Pendulum

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1 Hom Sarch Collcions Journals Abou Conac us My OPscinc From Priodic Moion o Unboundd Chaos: nvsigaions of h Simpl Pndulum This conn has bn downloadd from OPscinc. Plas scroll down o s h full x. Viw h abl of conns for his issu, or go o h journal hompag for mor Download dails: P Addrss: This conn was downloadd on 02/03/2015 a 19:32 Plas no ha rms and condiions apply.

2 Physica Scripa. Vol. T9,510, 1985 From Priodic Moion o Unboundd Chaos: nvsigaions of h Simpl Pndulum* Lo P. Kadanoff Th Jams Franck and Enrico Frmi nsius, Th Univrsiy of Chicago, Chicago, L 60637, USA Rcivd Jun, 1984; accpd Jun 16, 984 Absrac Th simpls xampl of h ons of chaos in a Hamilonian sysm is providd by h sandard or ChirikovTaylor modl. As a nonlinariy paramr, k, is incrasd h long rm bhavior of h momnum, p, is xamind. A k = 0, p is consrvd. For k < k,, for all saringpoins,p is of boundd variaion. For som saring poins is bhavior is priodic, for ohrs quasipriodic, for ohrs chaoic. A som criical valu of k, unboundd chaoic variaion firs appars. A scaling analysis o dscrib his ons is dscribd. Th problm w will discuss in his lcur has a long hisory. Th basic work in h problm was don by Kolmogorov, Arnold and Mosr [l31; mor rcn work has bn don by J. Grn, B. V. Chirikov, D. Escand, F. Dovil, and R. MacKay. hav workd on his problm in collaboraion wih S. J. Shnkr; pars of his and rlad work wr don in collaboraion wih M. idom, A. Zisook, M. Fignbaum, D. Bnsimon and Subir Sarkar. Th phnomnology of Hamilonian sysms is qui diffrn from ha of dissipaiv sysms. n his lcur w shall analyz in dail h brakdown of a KAM curv and h ons of unboundd chaoic moion in a paricular map. Firs, l us giv hr physical sysms o moiva h sudy of his map. Firs considr a pndulum (Fig. 1) ml2nj = mgsin(2n) (1) in which w choos unis of 0 so ha 0 = 1 corrsponds o 360. L us ac upon his sysm priodically by modulaing g, h forc du o graviy (say, by wiggling h suppor of h pndulum) g = go + g, sin (u) (2) g quaions i. = k()sin(2nb) B=r Hr k() is a priodic funcion of lin wih frquncy w and r is h vlociy of h pndulum. can now us a rick du o PoincarB o ransform his diffrnial quaion ino a map. Obsrv h pndulum onc ach priod of h forc; l rj = r(j) and Bj = 8(j) whr j = (2n/~)~. Sinc h phas of h xrnal forc a im j is indpndn of j, on can ingra h quaions of moion (3) ovr his priod, xprssing h nw sa of h pndulum in rms of is sa on priod arlir: * From lcurs originally dlivrd a Eric in Th wriup coms from nos by R. d la Llav and J. Shna. (3) n gnral, F and G wil b som nonlinar funcions priodic in 8. Th simpls modl which sms o capur h physics of his sysm is h sandard map = rj (k/2n) sin (2~0,) =, +rj+] also known as h ChirikovTaylor modl [4]. (Fradkin and Hubrman [ 51 hav sudid his priodically modulad pndulum and hav indicad how q. (4) can b convrd o q. (5) in svral limiing siuaions.) Th scond sysm w will us o moiva his map is an acclraor modl (Fig. 2). Envision a paricl moving around a circular rack, acclrad ach im i nrs a small box; h acclraion is providd by an a.c. fild in h box a h im h paricl nrs; h qs. (5) dscrib h sa of h paricl as i nrs h box for h j + 1s im in rms of is sa as i nrd h im bfor. On can rxprss q. (5) in h form j+l 28, + j+ = (k/2n) sin (2~0,) (6) which as a discr vrsion of q. (1) prhaps maks h conncion o h pndulum problm mor clar. Finally, considr a solid sa modl of a ondimnsional array of aoms adsorbd on a priodic subsra (Fig. 3). Th jh aom fls a forc from h springs conncing i o is wo nighbors, and from h gradin of h ponial nrgy a is posiion on h priodic subsra. f w choos 0, = xj/a o b h posiion of h jh aom xi dividd by h subsra laic consan a, hn kspring[(j Bj+l) + (j j1)] = ksubsrasin (2nj) (7)!\ h8% Sysm is wiggld up cind down Fzg. 1, Our firs modl. A pndulum is acclrad up and down. (4) (5)

3 6 Lo P. Kadanoff puricl moving wih spd rj phas of paricl occlrod paricl a via oscillaing nry is Bj lcric fild Fig. 2. Th scond modl. A simplifid acclraor. dscribs a saic configuraion of aoms. This is of h form (6). Th brakdown of h KAM surfac has a physical maning hr as a pinning of h dnsiy wav of adsorbd aoms; his problm has bn sudid by Copprsmih al. [6], Aubry [7], and ohrs. Th sandard map q. (5) has no consrvd nrgy; as nod abov, i can rprsn an xrnally forcd pndulum which xchangs nrgy wih is nvironmn. dos, howvr, oby Liouvill s horm Th map posssss a las hr qualiaivly diffrn kinds of orbis: priodic, chaoic, and KAM curvs. To dscrib hs orbis shall draw wo kinds of picurs: Figur 4 plos Oj vs j o show h avrag vlociy and Figur 5 plos (rj,oj) in phas spac o show h naur of h ypical orbis. Th op orbi is priodic; afr hr iraions 81 has incrasd by wo whil rj has rurnd o is iniial valu; sinc 8 and 8 1 ar idnifid, his is a rurn o h iniial sa. Th orbi progrsss q = 2 unis in is priod p = 3, so is avrag spd is p/q = 213. Th boom orbi is chaoic. Chaoic orbis appar o fill aras which w shall argu ar boundd by h KAM curvs. Th middl curv is an xampl of such a KAM curv. All h poins on h orbi fall on a smooh priodicgy climbing curv; i has a wlldfind avrag spd of = (45 1)/2. For larg valus of k, h chaoic rgion bcoms unboundd in h vrical dircion along r (Fig. 6). For many (bu no all) iniial condiions h moion in his rgim appars diffusiv, wih a diffusion consan Chaoic rajcory TERATON NUMBER Fig. 4. Valu of 0j vrsus j for h hr kinds of orbis. On his scal, 0 sms o simply incras linarly. indica ha D grows from zro a a criical valu (now known o b givn by k, = ) lik (k k,)2.56. On h ohr hand, for larg valus of k, D k2. Th lar fac can b undrsood; wri j+ni rj+n rj = (k) sin (2n,) 1=j j+l = j +rj+l For larg k, h succssiv Bj s can b considrd virually indpndn random numbrs mod on, sinc h rj chang by a numbr of ordr k afr ach iraion. Thus (rj+,, rj)2 = (k/27~)~(z sin 2n8,) = ( k/2~)~ z(sin 2n = ( k/2~)~n/2 = D/2n (1 1) and D = ( k/2~)~. hy is h chaos boundd for k d k,? L s sar a k = 0, whr h orbis of q. (5) li on h sraigh horizonal lins r =. For raional, h orbis along r = ar priodic; for irraional any orbi fills h nir lin dnsly. Som of hs which dpnds on k (Fig. 6). Numrical sudis du o Chirikov (9) \ U subs ra a Fig. 3. A solid sa xampl. Paricls on a wavy surfac Fig. 5. Thr kinds of orbis in h r 0 plan.

4 2.01, From Priodic Moion o Unboundd Quos: nvsigaions of h Simpl Pndulum 7, Fig. 6. Unboundd Chaos for k >k,. Hr 500 iraions of a singl saring poin ar plod whn k = 1.8. Th cnral curv is h bs KAM rajcory o brak up and is shown hr o display h diffusion across i. lar orbis will form h KAM rajcoris as w incras k; h m raional orbis will dsroy narby irraional orbis o form chaoic rgions. Fig. 8. Sparaion of chaoic rgions by KAM rajcoris. Th x's and For k > 0 bu small, hr ar los of KAM rajcoris; in dos rprsn wo diffrn chaoic rgions, sparad by many KAM bwn any wo, lis a chaoic rgion (in bwn any wo rajcoris. n (b) kl is largr han in (a). As ~ incrass, KAM rairraionals lis a raional). assr ha h chaoic rgions jcoris disappar and chaoic rgions mrg. ar confind by h prsisnc of horizonal KAM rajcoris (as in Fig. 5). Considr h chaoic rgion, for xampl, condaild mchanism for h brak up according o h lins Of ahing h fixd Poin r = 0, 6 = 0 Th union of h of a small nighborhood of h origin (Fig. 7) undr succssiv iraions of h map ill h gnral form a vry conord Opn s. Howvr, i mus always includ h origin, and canno inrsc a KAM surfac (sinc poins on a KAM surfac ar imags only of ohr poins on h Surfac). Thus no Poin in h nighborhood can vr lav h rgion boundd by horizonal KAM surfacs. As w coninu o incras k, mor and mor KAM curvs brak up, and h chaoic rgions mrg (Fig. 8) unil a = k h las rmaining horizonal KAM curvs disappar and unboundd vrical chaoic diffusion nsus. Empirically, h las surfacs o go hav w = (4: 1)/2, h invrs of h Grk's goldn man [8101 (up o an ingr). Tha is, a kc (Fig. 9) only isolad horizonal curvs ar lf amid h chaos, and hs hav bcom vry crinkld. For k > k, hs curvs also brak up; gaps form in hm, changing h curv ino a Canor s (Fig. O). Th rmaindr of his lcur is dvod o dscribing h h rsarch carrid on by Sco J. Shnkr and myslf [ll. How can w sudy his in dail? G~~~~~ has obsrvd ha hs las KAM curvs can b hough of as a limi of h paricular priodic cycls wih, =pn/q, and = Pn+l = F,, h Fibonacci numbrs. (F, saisfis F~ = F~ = 1, F ~ = + F, ~ + F~]). n Fig. 11, for xampl, w s wo priodic cycls wih w = 213 abov h KAM surfac, and wo blow wih = 3/5. Ths priodic cycls alrnaly convrg o h KAM surfac wih w = ( from abov and blow. ndd, h solid lin in Fig. 11 acually is composd of wo cycls wih = 4181 = Fls, as w can s in h xpandd scal of Fig. 12. Hr no how magnificanly smooh h KAM surfac appars on his lngh scal. why do w choo_s prcisly his squnc of priodic orbis o look a = (d5 1)/2? prhaps hr ar br.. CL KAM 0 is imag X x x x x x 6 1 Fig. 7. mag of iniial rgion including origin. No ha h imag can nvr cross a KAM curv.. *..,.,. * * '.. 4 X =8 Fig. 9. A k,, only a fw whparad KAM rajcoris ar lf so ha hr ar larg chaoic rgions.

5 8 Lo P. Kadanoff r '.*,... 9: *..' L '.,.L.. * *'.*..7, *.. * rgion KAM c ,! ' ,,,, Fig. 12. bu roughly his squnc provids an ordrly progrssion, convrging rapidly and uniformly in is propris o hos of h surfac. On can hink of h n + 1s cycl as "almos" bing h nh cycl followd by h n 1s cycl. Sinc h convrgnc is alrnaing a propr wighing of h prvious wo cycls is a naural choic for h nx; for = (d5 1)/2 h propr wighing is qual. A his poin i will b usful o inroduc h Mosr rprsnaion. Rmmbr ha is an avrag spd for 6 :Bj = Bo + j + O(1) on a priodic orbi or a KAM surfac. f w dfin a im i = o + j. Mosr has shown for small k ha Oj = (j) whr () can b wrin () = + u() (12) wih u( + 1) = ~ (). On h KAM curv for k < k,, u() is a smooh funcion of priod on, and w may dfin a Fourir ransform m u() = A, sin (27"). =1 For cycls =p/q, u() is dfind only on a discr s of poins. Nonhlss, on may dfin a discr Fourir ransform Clarly, A,(,) wih, = F,,/Fn vanishs if w 2 F,. Howvr, (Grn assrs) h valus Aw(n) +Aw() as nz. n Fig. 13 w s u() and wa,() for =d5 1/2, k = 0.5. ( plo u() only o = 0.5;~ is odd abou his poin u(1/2 + ) = u(1/2 ).) Th obvious smoohnss of U is rflcd on A(w), which is xponnially small as w gs larg. No ha alrady A5 is largr han Ad; 4 is no a Fibonacci numbr. n Fig. 14, w s u and wa, fork = 0.9. Nw bumps hav appard in U, and h Fourir ransform rflcs his wih prominn srucur a h low Fibonacci numbrs. (Noic ha hs paks do no rflc numrical ffcs from our us of Fibonacci lngh orbis in approximaing h KAM surfac. Ths frquncis ar naurally arising in h spcrum of h surfac.) Finally, in Fig. 15 w look a k = k,. Figur 15(a(l)) is a graph of u() and a small par of h phas rajcory is shown xpandd in Fig. 15(a(2)). n sriking conras o Fig. 12, i is vry bumpy vn on his small lngh scal. A longr orbi han in Fig. 15(a(2)) would fill h gaps bwn hs dos wih mor bumps lik hs. Figur 15(b) shows h spcrum o*aw a k,. On should ignor h low frquncis (long wavlnghs) as hy dpnd upon h dails C 0.03 v (a' i p a Fig. 11. =3 Fig

6 From Priodic Moion o Unboundd Chaos: nvsigaions of h Simpl Pndulum 9 k = k, = " ' '. " ooL ',.!,,, ' Fig q l&l (j of h map and ar no univrsal. On mus also ignor h vry high frquncis (shor wavlnghs) as h fini lngh of h orbi numrically inroducs rrors. Th scal rgion of h spcrum is xpandd in Fig. 15(c). Th big lins a h Fibonacci numbrs hav sld down: lqna(qn)l is clarly consan as n + =. Also h smallr paks ar sling down, giving a bauiful slfsimilariy. L us look in mor dail a o() nar = 0 (Fig. 16). ( = 0 and = 1/2 ar spcial symmry poins of our map (3). Th funcion () a k, is slfsimilar, scaling [12] and lik [ 131 o() = xo sign () O(T) 7 = n l/ln (LPK [13]) whr 0(7) is a univrsal funcion. Ovrall, h funcion dvlops an infini slop a zro, wih O(l/Fn) = consan.(l/f,)%o (16) Th xponn xo is characrisic of h bhavior of 0 in his ransiion. A similar xponn yo can b dvlopd govrning h bhavior of r(l/f,) r(o). Nar = 1/2, wo mor xponns x1 and y1 govrning 6 and r can b dfind. Finally, wo mor xponns R, and R, can b dfind as follows. To ach Fibonacci raio approximaing h goldn man hr ar wo priodic orbis on sabl (llipic) and on unsabl (hyprbolic). (n h figurs h circls dno llipic, h crosss hyprbolic orbis.) Th drivaiv of h qims irad map for a p/4 cycl has wo ignvalus h and ' (wih X pur imaginary for llipic orbis). Th rsidu is dfind o b 1/4 (2 ' '); is bhavior for = F,,/F, as n + a (k, k) Q 1 is dscribd by h xponns R, and R,. Ths xponns ar abulad in Tabl. (Ths ar old numbrs now ach is known o a las on mor dcimal poin.) Th firs column shows h numrical valus of hs r k = kc =, ,,, [ Fig k = k, k =kc q = 4181 (b) xponns for h brakdown of h las KAM curv in h sandard map (5). Th las column shows h brakdown of h las curv in a diffrn ara prsrving map of h plan; h agrmn in h xponns shown has prsisd using h

7 10 Lo P. Kadanoff () / 1: 0.10 Tabl. Ponially univrsal obsrvabls = [1,1,1. [3,1,4,1. 1 [2,2,2. [1,1,1. x, x, 1, yo y, R, R, ::E// 0.Ooj Fig. 16. Singular bhavior a k = k,. Each bump in his plo of () vs marks an infiniy in h slop. br accuracy availabl oday. Th scond column shows h brakdown of anohr KAM curv, whos winding numbr has coninud fracion which nds in [l, 1, 1.] [bu bgins wih h digis of n]. Again, h xponns agr. Howvr, choosing = 45 1 = [2,2,2... ] and varying k unil his KAM curv braks down, givs disincly diffrn valus for R, and R,. Th grar prcision availabl oday indicas ha his diffrnc is ral, and xnds o h ohr xponns as wll. is hough ha h curv wih winding numbr nding in ons sparas chaoic rgions a is brakdown, whil (,,... ) is surroundd by surviving curvs (.g. [2,2,2..., 2, 1,1,1,...) a is brakdown is nvironmn is qui diffrn as i gos. Thus w s univrsal bhavior; h singulariis a k, ar indpndn of h xac map, or (wihin limis) h xac. s scal invarian bhavior (1 5) wih srucurs which rcur on all scals as + 0, 1/2, and which occurs again and again as h frquncy gos hrough Fibonacci numbrs. s bhavior which is conncd o Fibonacci numbrs. nd an xplanaion for h scaling and univrsaliy bhavior w hav sn. This can b obaind from a rnormalizaion group ramn [14161 which xplains som bu no all of h faurs nod hr. Rfrncs 1. Kolmogorov, A. N., Dokl. Akad. SSSR 98,527 (1954) (Russian). 2. Arnol d,v.., zv. Akad. Nauk 25,21 (1961). 3. Mosr, J., Nachr. Akad. iss. Goingn Mah. Phys. K1. a, 1 (1962). 4. Chirikov, B., Phys. Rv. 52,265 (1979). 5. Fradkin, E. and Hubrman, B., Urbana prprin. 6. Cooprsmih, S. N. al., Phys. Rv. L. 46,459 (1981). 7. Aubry,S., in Solions in Condnsd Mar Physics (ds. A. R. Bishop and T. Schnidr), Springr Vrlag, Nw York (1978). 8. Grn, J. M., J. Mah. Phys. 9,760 (1968). 9. Grn, J. M., J. Mah. Phys. 20,1183 (1979). 10. Grn, J. M. and Prcival,., Physica 3D (1981). 11. Shnkr, Sco J. and Kadanoff, Lo P., J. Sa. Phys. 27, 631 (1 9 82). 12. idom, B., J. Chm. Phys. 43,3989 (1983). 13. Kadanoff, Lo P., Phys. Rv. L. 47,1641 (1981). 14. Escand, D. and Dovil, F.,. Sa. Phys. 26,257 (1981). 15. MacKay, R., Princon Thsis (1983). 16. Kadanoff, Lo P., in Mling, Localizaion and Chaos (ds. R. K. Kalia and P. Vashisa), p Norh Holland, Nw York (1982).