Discontinuity, Nonlinearity, and Complexity

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1 Dsonnuy Nonneay an Compey hps://hsenfpubshng.om/jounas/dnc-defau.asp Unvesa Pnpes of Pefe Chaos Segey Kamenshhov Physa Depamen Mosow Sae Unvesy of M.V.omonosov Mosow 999 Russa Submsson Info Communae by Reeve Aepe Avaabe onne Keywos Chaos Despon eavy Nonnea speson Aao Uneanes eaon Absa The pupose of hs wo s o noue s ompehensve efnon of pefe haos o fn ou s bas popees n ems of phase ansons an o gve onneons fo uneanes yng n he base of pefe haos onep. The one as noneemns espon was noue base on wo fomaze neessay an suffen onons: fne esouon of phase spae an nsaby of phase spae aeoes. The popees of Komogoov sysem nung phase mng une ou o be onsequenes of hao sae bu no s ompehensve an suffen onons. Despon eavy was efne as a manaoy popey of pefe haos he same aeas of phase spae may show egua o hao popees epenng on espon of spae - me auay. Aso was foun ou ha fo hao sae wh unfom ffuson nonnea speson aw s a manaoy popey. In s un nonnea speson neessay eas o spae me nsaby of pobaby ensy an appeaane of pobaby aves n phase spae so ae phase spae aaos whee paes ensy gows up. The ase of hao sae wh fe bounay an onsan ffuson was onsee n hs pape. I was pove ha Foue eomposon aows evng eaons beween oonae momenum an me - enegy efnon uneanes. The haos ffuson fao s he ony paamee mng pou of oesponng uneanes whh was pove n hs pape. &H Senf Pubshng C. A ghs eseve.. Pefe haos an eavy Sevea senaos of ubuene anson have been popose sne 883 yea when ubuene onep was noue hough epemens of Engsh engnee Osbone Reynos. He has noe ynam phase anson n qu seam haaeze by unsabe voe appeaane an noue wo m saes of moon: amna an ubuen. Sne sevea senaos of ubuene anson have been eveope. Among hem anau Hopf nsaby mehansm [] oenz aao mehansm [] senao of Ponae Feygebaum [3] an senao of Komogoov - Ano Mose [4]. Eah of oune mehansm has s nvua aea of appaon an bas assumpons. Fo hs eason none of hem s unvesa moeove unambguous onneons beween hem ae no sae ye. Sne nouon of ubuene onep s popees wee nvesgae an geneaze. Fo now oneps of ynam m saes hemseves wee geneaze an ansfome no saes of egua Coesponng auho. Ema aess: amphys@gma.om

2 moon an pefe haos sae. Theefoe eemne moon oespons o amna seam whe pefe haos o ubuen moon sae. e us onse seon m sae - he onep of pefe haos. One s efne as uneemne espon n gven phase spae esouon. Unpeaby of moon s onsequene of wo onons eazaon: a fne esouon of geneaze phase spae; b nsaby of phase spae aeoes. Conep of geneaze phase spae may be epane hough sysem moe onssng of M paes whh have nepenen phase aeoes. If moon of eah pae s eemne n N mensona phase spae hen geneaze phase s M N mensona an oesponng veo w be sysem haaes veo n Hbe spae. If onneons ae noue menson of geneaze spae w be equa o P=M N- whee s numbe of onneon equaons. Then esouon fneness n a eas one eon of geneaze phase spae hen eas o uneany n na ynam sysem sae. Fomay hs onon may be epesene n he foowng way: P mn Hee mn s eemen of esbng geneaze phase spae whe s haaes veo poeon oesponng o eon of Hbe phase spae. If we assume ha mnma uneany s soop mn hen eemenay e voume of geneaze phase spae s epesse n he foowng way: P P mn. mn e s onse seon onon of pefe haos sae une suggeson ha fs one s sasfe. If na any wo sysem pas paes have nsabe aeoes vegng n phase spae eemne ynam espon of he moon omes mpossbe an pefe haos sae s eahe. Insaby equemen may be epesse hough sum of posve yapunov faos fo eah menson of geneaze phase spae: K h h Uneemne haaes aeoy s bas popey of pefe haos sysem whh eas o wo onsequenes. Fs one egas auo oeaon funon of ynam vaue f. Hee sysem evouon s efne by haaes geneaze funon - evese mappng s no snge vaue n genea ase. Aong o Eq. an Eq. g = m f an g = m f ae nepenen funons f an f ae abay ynam funons hen auo oeaon haaes funon R f sasfes Eq.3: m R f = 3 Ths eaon efes ae popey of mng aong o emnoogy noue by G.M. Zasavsy [5]. In fa eazaon of Eq.3 eas o eeuon of Susy eon fo ego sysem: T m R f 4 T T Hee s eay me beween sa an he en of sysem evouon obsevaon. Aong o 4 sysem beomes ego fo. Fo physa sysems hs onon an be foowng epesson: mn ns ns 5 h Hee s fne me esouon whe mn ns s nsaby nemen fo ha may be epesse hough negae yapunov fao Eq.. Sasfaon of h haos onon aows eevng foowng equaons fo any ynam funon n fame of ego espon: T f f f f 6 T

3 3 In gven eaon Г an T ae phase spae voume oupe by phase aeoy ung obsevaon me an obsevaon me sef. Fo negae yapunov fao gven popey aows o oune onsequene of Eq.. h h h 7 h Hee h s ynam enopy of Komogoov Sna ha may be epesse hough enopy of sysem n gven phase spae epesenaon [5]: S Г h 8 Г Quany S n Г s Gbbs enopy of hao sysem wh aoun of fne phase spae esouon an onon Eq.5. Sasfaon of haos onons an eas o manaoy gowh of Gbbs enopy even n ase when oesponen eemns espon s onsevave. Consequenes Eq.3 Eq.6 an Eq.7 fo eaons Eq. an Eq. n fa oespon o efnon of Komogoov sysem [6] sae K sysem une onon ha mn ns. Howeve we have o noe ha K sysem equemens ae neessay bu no suffen fo pefe haos sae PCS obsevaon. I may be usefu o sae anohe quaave popey of PCS espon eavy. As was shown PCS s m sae of ynam sysem haaeze by popees oune beow: P 9 mn K h h Sasfaon nequay epens on he espon paamees mn an h. Aong o Eq.9 an Eq. magnue of hese paamees may ea o oppose m saes. They ae pefe haos sae PCS an egua sae RS. e s onse eampe of physa sysem. Then fneness of mn s pove by quanum uneany eaons. In genea ase mn s funon of me esouon: mn = f mn. Fne magnue of mn aows o eave one ono paamee - negae yapunov fao. Theefoe egua sae of sysem w be epesene by goup of Eq. an Eq.: P mn h h K Seon eaon onans me as paamee. In suh a way geneay anson beween wo m saes may ou a any nsan of me. If evouon of physa sysem n gven geneaze phase spae s epesene by onsequene of egua saes an oesponng ansons an be efne as quasegua sae of moon QRS. Tanson beween wo egua aeoes m yes s eaze hough hao saes. Aong o emnoogy of G.M.Zasavsy [7] n phase spae suh ype of moon s epesene by sohas sea wh saby sans. Tme eay of wo onsequen ansons R R an R R aso ae bfuaons n genea s funon of me paamee an mn : mn. 3 e s onse phase aeoy n hee geneaze phase spaes an suh ha 3 mn mn mn. Then he same phase aeoy 3 epesene hough an w have ffeen faons of egua sae saby sans an ansona sae pefe haos. Phenomenon of espon eavy s epane by Fg. a an Fg. b whee wo mensona phase spaes ae suppose o have unfom esouon. Eah sysem ynam sae s epesene as pon nse oesponng e whh ms phase spae uneany. Tansons beween enumeae saes ae symboay esgnae as sagh ne we on ae no aoun phase ways of oesponng bfuaons. In gven fgue he same segmens 3 an 5 of phase aeoes ae efne as hao moon - Fg. a o quasegua moon - Fg. b - wh fne fe me quas egua aeoes symboay shown n Fg. b nse age es. In genea uaon of sysem esene.e. fe me = 8 n any maosop ynam sae s abay. Regua moon appeaane may ea o spae - me

4 4 sabzaon of sysem. If sabzaon ous fo sae hen. In ohe ase uen sabzaon s empoay an quas apue s eaze [7]. In hs ase egua aeoy s sabe ung fne me engh. Afe hs me quas egua ous omes unsabe efoms an may fnay sappea. Fg.. a phase spae epesenaon. Chao phase aeoy ; b phase spae epesenaon. Quas apues n segmens 3 an 5 egua moon aeas wh fne fe me quas egua saby sans. Hoow es upaes sae pons n phase spae epesenaon. Inease of geneaze phase spae esouon may ea o appeaane of new quas egua aeas o ovea spae - me sabzaon of aeoy. In fs ase some poon of paes n es epesenaon of oasene esouon uns ou o ansfom no ouses wh fne o nfne fe me. One s efne by oa me of sysem obsevaon nfne fe me w oespon n hs ase sabe esene of egua aea ung a obsevaon me. As we an see spae me eavy aows eevng quaavey ffeen hao egua popees fo he same pa of gven ynam sysem.. Nonneay as manaoy popey of pefe haos In equaon 3 evng we use popey of nepenene fo abay ynam funons f an f f mn ns. e s assume ha onsee sysem onsss of M subsysems paes haaeze by oesponng pobaby enses M = M. Then f f fo pefe haos sysem we have geneaze Eq.3: m C Hee C s oeaon funon. Eq. may be ae oeaon eay o sysem memoy oss. One of appoahes appe fo haaezaon of ansona popees n gven fame s base on Foe - Pan - Komogoov moe [8]. One aows obanng bas equaon of anspo fom Chapman - Komogoov Eq.3. P Inegaon s mae fo phase voume oupe by sysem phase aeoy. Uppe nees of haaes veo oespon o onsequen me momens 3 : 3. Funon 3 s onona pobaby ensy wh fe na onon. e s ea bas assumpons mae fo evaon of Foe - Pan Komogoov equaon [8]. 3

5 5.. Gven onon means ha pobaby of bfuaon oesn epen on absoue magnue of na me pon: ns mn. Ths maon s sasfe f Eq. Eq. an Eq.5 fo haos ae va. Eq.5 s eaze neessay f we spea abou fome nsaby;. - fna onona pobaby ensy oesn epen on he na oonae veo. In ems of haaes geneaze funon hs onon s va as we fo he easons gven n Pon ; 3.. Fo fne phase spae e an me aoun hs epesson an be eaze fo mn an mn ; 4. Ina sbuon ensy s efne by Da ea funon:.e. na oonae an be efne auaey n fame of phase spae fne esouon Da ea funon oespons o eangua funon; 5. '' ' b a. Hee fo esene of seon evave of Da funon s neessay fo o sasfy foowng onon: mn n fame of ean esouon phase spae. Coeffens a an b ae efne by eaons 4 an 5: a 4 b 5 On bass of eaon 5 seon anspo oeffen an be noue: mn m 6 Gven assumpons aow o fomuae nown no paame fom of Foe Pan Komogoov equaon FPK equaon: 7 I an be shown ha n Eq.6 an Eq.7 me s hen paamee [8]. e s epesen enegy of sysem mass un: mn 8 Aong o Eq.7 seon anspo fao an be epesse n he mofe fom of Eq.9 - supesps ae ome. mn mn mn 9 In Eq.8 ae geneay nepenen agumens fo enegy epesson. Inee beause of phase aeoy mng Eq.3 spef enegy an oonae may no have muua oesponene. Then fo onona pobaby ensy we have mofe equaon: '' ' b a

6 6 A he same me evave of pobaby an be epesene usng Chapman Komogoov Eq. n he foowng way: m In hs equaon s ansona pobaby ensy. Subsuon of Eq. no Eq. gves eene FPK equaon EFPK [8]: Vaaon of suh ha mn aows epesenng equaon 9 n asympo fom fo an eevng abnoma anspo equaon: ' 3 Roo eaon of equaon boh pas eas o aw of abnoma ffuson [9]: D 4 In hs eaon D s anomaous ffuson fao. Taonay abnoma ffuson aw s epane afay noung faa FPK equaon FFPK [9]. e s onse unfom sae fo aveage haaes enegy of hao sysem: f. Eq.9 aows eevng oesponen fom of anspo oeffen: mn f. In hs ase Foue eomposon of one mensona oa EFPK Eq. may be epesene n he foowng way: ep ep P 5 Hee s oesponng mofe anspo oeffen fo menson. Ampues of Foue eomposon ae oune hough Eq.6 an Eq.7: ep 6 T ep 7 Seon Foue eomposon gves eaons Eq.8 an Eq.9 wh equvaen opeao s enes ' : ep ' 8 K K ep ' 9 Inegas ms ae efne aong o Koenov heoem: mn mn K. Subsuon of Eq.8 an Eq.9 no equaon Eq.5 gves wave pae fom: ep ' ep ' 3 Genea abaness of negaon ms fnay aows epesenng aw: 3

7 7 As foows fom oune epesson nonnea speson aw of Eq.3 s manaoy popey of unfom hao sae. Aoaon of ea pa eas o Eq.3: Im Re Im Re Re 3 Re an Posveness of physay measue quanes Re aows eevng foowng popey of ompe wave numbe: Im. Hee posveness of spef enegy an onsequeny anspo oeffen ae aen no aoun. Fs Foue eomposon of pobaby ensy hen an be gven by Eq.33: epim ep Re 33 Hee Im as posve spae nemen shows esene of spae nsaby fo pobaby ensy ampue. e s onse he magnay eaon fo boh pas of Eq.3: Im Im Re 34 Posveness of me nemen shows me nsaby of pobaby ensy: ep Re ep 35 As we see spae me nsaby of pobaby ensy s efne by manaoy nonnea speson aw of Eq.3 of hao sysem. Gven nsaby eas o appeaane of pobaby aves n phase spae - phase spae aaos whee paes ensy gows up. Ths poess onnues up o he momen when spef enegy an anspo fao aheves spae nhomogeney:. Sne ha oa EFPK equaon has o be onsee n genea fom of Eq.. 3. Uneany eaon of phase sae I was menone above ha wo possbe ypes of phase aeoes ae possbe n fame of haaes veo espon: beon an muvaue mappng. Eah ype s haaeze by spef enegy n fom of an oesponngy. Gven vson aows noung quaave popees of ynam sysem basng on anspo paamee mn. We sha esgnae phase saes as beon saes of onsan aveage enegy.e. enegy whou ep me epenene. Then muvaue mappng oespons o ansona moon wh phase aeoy mng. Appeaane of ansona sae s efne by fs eun of haaes veo. Phase ansons ae esbe by EFPK Eq.. In ems of ffuson faos gven ypes of moon ae aso esgnae as noma an abnoma ffuson [9]. e s onse ase of unfom phase sae wh fe bounay: ons ons. Ths phenomenon appeas une onon of phase spae me saby of pobaby avy as was shown n Seon II. Despon of oesponng sysem sae an be eaze n fame of noma ffuson FPK Eq.7 fo fe me of phase sae: f f nea ffuson equaon:. Fo seee menson we an epesen Eq.7 as unfom f f 36 Souon an be seahe n fom of Foue epanson sees Eq.37 Eq.38 whh sasfes bounay onon an na sae:. N sn 37

8 8 sn 38 Subsuon of 37 no 36 gves Eq.39 Eq.4 fo Foue oeffens: sn N 39 4 Coesponng vaues of anspo fao ae epesene by Eq.4: 4 Aong o Eq.4 oeffens sasfes foowng onon: ons.consequeny fo we have: ep. Tang no aoun Eq.9 fo aveage spef enegy we have go foowng epesson fo see enegy speum: mn 4 e s esgnae mn hen fo enegy evave we have Eq.43 gven beow Une onons of fne phase spae an me esouon Eq. Eq.5 fo hao sysem we an mofy gven eaon no fom of Eq.44: 3 44 Fo ynam espon wh pefe auay na pobaby ensy s epesene as Da funon Seon II Iem 4: 45 In vny of poeon of haaes veo s beon. Nomazaon onon fo hen an be epesene n he foowng way: / 46 Da funona s epesene hee hough me agumen. Ine oespons o zeos of funon. In onsee ase we have ony one vaue of agumen oesponng o zeo -. Then Eq.46 an be mofe n he foowng way: / / / m T 47

9 9 As we an see n vny of spae - me beon aows noung pobaby ensy oesponene: sgn / 48 Fne spae - me esouon aows subsuon of Dea funon by s see aenave eangua puse. Whou oosng of geneay we may assume ha : C mn mn Aong o nomazaon onon oeffens C an C an be epesse n he foowng way: C C. mn mn Fo gven veo puse eaon onneng haaes wh of speum an puse wh an be wen n he foowng way: 5 Subsuon of Eq.44 no Eq.5 gves Eq One aows eevng onneon beween enegy an me esouon Eq.5. 5 Hee wh aoane o Eq.37 wave numbe Epesson fo auay funon s epesene beow: Then eaon 5 an be mofe n foowng way: s noue. C 4 mn mn mn Hee s mnma anspo fao fo menson. In fame of ffuson epesenaon Eq.54 an mn be epesene n gven fom owe nees ae ome: 55 Hee D s mnma ffuson fao fo menson of phase sae. e s eeve onneon beween spae an me uneanes. Sasfaon of egoy onon fo hao sae aows gves aby o mofy Eq.9: T T 56 T Uppe unesoe hee means me aveagng. Spae me nepenene of phase sae eas o spae nepenene n. Fo abaness of negaon me hs means ha eaon 56 an be smpfe n he foowng way: 57 D mn

10 Fne ffeena fo enegy hen an be epesse hough momenum: p p. Subsuon of gven eaon n Eq.55 aows eevng ffeena equaon fo momenum: Momenum s epesse n fne fom: p onneon beween p an : p p 4 D 58. Subsuon of hs epesson n Eq.58 gves p 59 4 ess s fom of eaon 59 aows unfom epesenng of Eq.59 an Eq.55 gven beow. D D p 6 6 Eq.6 an Eq.6 show onneons beween uneanes of oonae momenum an me - enegy efnon oesponngy. I may be usefu o noe ha any of gven uneanes may be eemne as oesponng sana evaons: p. REFERENCES p. anau.d an vshs E.M. 7 Hyoynams Fsma Russa: Mosow oenz E. 98 Deemns nonpeo moon Sange aaos: Mosow. 3. Fegenbaum M.J. 979 The unvesa me popees of nonnea ansfomaons Jouna of Sasa Physs Mose J. 96 On nvaan uves of aea pesevng mappngs on an annuus Nah. Aa. Wss. Goengen Mah. Phys Zasavsy G.M. Sageev R.Z. 988 Inouon o nonnea physs: fom he penuum o ubuene an haos Naua: Mosow Zasavsy G.M. Sageev R.Z. 988 Inouon o nonnea physs: fom he penuum o ubuene an haos Naua: Mosow Zasavsy G.M. 7 The physs of haos n Hamonan sysems Impea Coege Pess: onon Kamenshhov S.A. 3 Eene founaons of sohas peon Communaons n nonnea sene an numea smuaon CNSNS-D--496 une evew subme on Aug.. Ogna pape n Av - hp://av.og/abs/ Zasavsy G.M. 7 The physs of haos n Hamonan sysems Impea Coege Pess: onon 5-5. D