
chithrasalabha prabhavam

araajakathva sidhaanthathil, oru nirnnayaka nonleeniyar sisttathinte oru avasthayile oru cheria maattam pilkkala avasthayil valia vyathyaasangalkk kaaranamakunna prarambha avasthakale sensitiivu aasrayikkunnathaanu butterfly ephakt.
ganithashaasthranjanum kalavastha shaasthranjanumaaya edward norttan lorensinte pravarthanangalumaayi eee padam adutha bandhamullathaan. aazhchakalkk munbu oru vidoora chithrashalabhatthinte chirakukal thattunnath polulla cheria aswasthathakal oru chuzhalikkaattinte vishadaamshangalude (roopeekaranathinte krithyamaaya samayam, krithyamaaya paatha) roopaka udaaharanathil ninnaan chithrasalabha prabhavam uruthirinjathennu addeham abhiprayappettu. laurence aadyam kodunkaattinu kaaranamakunna oru kadalkkuthirayaanu upayogichathenkilum 1972 aayappozhekkum chithrashalabhavum chuzhalikkattum upayogichu athine kooduthal kaavyathmakamaakkaan prerippikkappettu. thante kalavastha maathrkayude aadyakaala avastha daatta upayogichu aprasakthamaayi thonnikkunna reethiyil vrithaakrithiyilulla runs nireekshichappol addeham eee prabhavam kandethi. kalavastha maathrkaykku raundu cheyyaatha prarambha avastha daatta upayogichu rannukalude phalangal punarnirmmikkunnathil paraajayappedumennu addeham abhiprayappettu. prarambha saahacharyangalile valare cheria maattam ganyamaayi vyathyasthamaaya phalam srishtichu.
cheria kaaranagal kaalaavasthayil valia prathyaaghaathangal undaakkumenna ashayam franch ganithashaasthranjanum enchineeyarumaaya henry poincare neratthe amgeekarichirunnu. american ganithashaasthranjanum thathwachinthakanumaaya norbert winer eee sidhaanthathinu sambhavana nalki. laurencinte pravartthanangal bhoomiyude anthareekshathinte "asthiratha" enna ashayam oru alavilulla aditharayil sthaapikkukayum asthiratha enna ashayam nonleeniyar dynamicsinum nirnnayaka araajakathvathinum vidheyamaakunna valia tharam chalanathmaka samvidaanangalude gunangalumayi bandhippikkukayum cheythu.
oru cheria maattam valia prathyaaghaathangalkku kaaranamaakumenn karuthappedunna eathu sahacharyathinum oru vishalamaaya padamaayi kalavastha shaasthrathinte pashchaathalatthil butterfly ephakt ashayam upayogichu.
charithram.
dhii vokkeshan oof maan" (1800) enna pusthakatthil johann gottleab fitche parayunnu "athillathe ningalkk athinte sthaanathu ninnu oru manal polum neekkam cheyyaan kazhiyilla... alakkaan kazhiyaatha muzhuvan bhagangalilum enthengilum maattunnu.
araajakathva sidhaanthavum prarambha saahacharyangale sensitiivu ashrayikkunnathum niravadhi saahitya roopangalil vivarichittundu. 1890-ll poincare nadatthiya three-body prashnatthinte cases ithinu thelivaanu. atharam prathibhasangal saadhaaranamaayirikkaamennu addeham pinneedu nirddeshichu, udaaharanathinu, kaalaavasthaashaasthrathil.
1898-ll, negatteevu vakrathayude edangalile sanjaarapathangalude pothuvaaya vyathiyaanam jackwus hudmard rekhappeduthi. 1908ll piyar duhem ithinte pothuvaaya praadhaanyathekkurichu charcha cheythu.
1950-ll, alaan turing ingane abhiprayappettu "oru electroninte oru centimeterinte nooru kodiyilonnu oru nimishathil sthaanabhramsham oru varshathinusesham oru manushyan himapaathathil kollappedukayo rakshappedukayo cheiyunnathu thammilulla vyathyasam undaakkum."
oru chithrashalabhatthinte maranam pinneedu nadanna charitra sambhavangalil dooravyaapakamaaya tharamgam undaakkumenna ashayam 1952ll reey bradburyude cherukathayaaya "idimuzhakkathinte shabdam" enna kathayilaanu aadyamaayi prathyakshappettath. "idiminnalinte shabdam" samaya yaathrayude savisheshathayaanu.
ennirunnaalum, kooduthal krithyamaayi paranjaal, anthareekshathinte muzhuvan kaattum badhikkunna oru cheria praaniyude chirakinte krithyamaaya ashayavum padaprayogavum-oru kuttikalude pusthakatthil prasidheekarichu, athu 1962 ll, laurence prasidheekarikkunnathinna oru varsham munbu, aagolathalathil angeyattam vijaykaravum prashasthavumaayirunnu
1961ll, oru kurukkuvazhiyaayi muun oottathinte madhyathil ninnu oru kalavastha pravachanam veendum cheiyunnathinaayi laurence oru samkhyaa computer model pravarthippikkukayirnu. poornnamaaya krithyatha 0.506127 moolyam nalkunnathinupakaram print aouttil ninnu addeham prarambha vyavastha 0.506 nalki. thikachum vyathyasthamaaya oru kaalaavasthayaayirunnu phalam.
laurence ezhuthi
1963-ll laurence eee phalathekkurichulla oru saidhaanthika patanam "nirnnayakamaaya nonperiadic phloo" enna peril uyarnna udharicha, seminal pepparil prasidheekarichu (kanakkukoottalukal oru royal mcbee eljipi-30 computeril nadatthi). mattevideyengilum addeham paranju
sahapravarthakaril ninnulla nirddeshangalethudarnnu, pilkkala prasamgangalilum prabandhangalilum laurence kooduthal kaavyathmaka chithrashalabhangal upayogichu. laurencinte abhipraayathil, 1972-ll american association phoor dhii advaansment oof sayansinte 139-aamath yogathil avatharippikkaanirunna oru prasamgathinu oru thalakkett nalkunnathil paraajayappettappol, philipp merilees "brazeelile chithrashalabhatthinte chirakukalude flap texacil oru chuzhalikkaatt undaakkunnundo?"oru thalakkett. chirakukal thattunna oru chithrashalabhatthinte eee aashayathinte aavishkaarathil sthiratha nilanilkkunnundenkil, chithrashalabhatthinte sthaanam, anantharafalangal, anantharafalangalude sthaanam ennivayil vyapakamaaya vyathyaasamund.
oru chithrashalabhatthinte chirakukal anthareekshathil cheria mattangal srishtikkumenna ashayathe eee vaachakam soochippikkunnu, athu aathyanthikamayi oru chuzhalikkaattinte paathaye mattiyekkaam allengil kaalathamasam varuthukayo vegathilaakkukayo mattoru sthalathu oru chuzhalikkaatt undaakunnath thadayukayo cheyyaam. chithrashalabhangal chuzhalikkaattine shakthippeduthukayo nerittu srishtikkukayo cheiyunnilla, ennaal chithrashalabhatthinte chirakukalude flap chuzhalikkaattin kaaranamaakumenn soochippikkaan uddeshichullathaanu eee padam chirakukalude flap parasparabandhithamaya sankeernamaaya webinte prarambha avasthakalude bhagamaanu enna ardhathil; oru koottam avasthakal oru chuzhalikkaattilekku nayikkunnu, athesamayam mattu koottam avasthakal angane cheiyunnilla. flapping wing sisttathinte prarambha avasthayile oru cheria maattathe prathinidheekarikkunnu, ithu sambhavangalude valia thothilulla maattangalilekku vyaapikkunnu (thaarathamyam cheyyuka domino prabhavam). chithrashalabhangal chirakukal thattiyittillengil, samvidaanathinte paatha valare vyathyasthamaayirunnirikkam-ennaal chithrashalabhangal chirakukal thattathe oru koottam saahacharyangal oru chuzhalikkaattilekku nayikkunna settayirikkaanum saadhyathayund.
kaalavastha polulla oru samvidaanathinte prarambha avasthakal orikkalum poornnamaaya krithyathayode arian kazhiyaathathinal chithrasalabha prabhavam pravachanathinu vyakthamaaya velluvili uyarthunnu. eee prashnam samagramaya pravachanatthinte vikasanathinu prachodanamaayi, athil aswasthamaaya prarambha saahacharyangalil ninnu niravadhi pravachanangal nadathunnu.
neratthe vishwasichathupole kalavastha vyavastha prarambha saahacharyangalodu samvedanakshamathayullathallena chila shaasthranjar vaadichu. kalavastha pravachana pishakinte pradhana sambhavana model pishakaanennum prarambha avasthakalodulla samvedanakshamatha thaarathamyena cheria punk vahikkunnuvennum david oorel vaadikkunnu. laurence samavaakyangal valare lalithamaanennum viscos efakttukale prathinidheekarikkunna padangal adangiyittillennum steefan wolfrum rekhappeduthunnu; eee padangal cheria aswasthathakale illaathaakkumennu addeham vishwasikkunnu. kooduthal disipative padangalum nonleeniyarittiyum ulppedunna saamaanyavalkkaricha lorenz modalukal upayogichulla sameepakaala padanangal soochippikkunnathu kuzhappam aarambhikkunnathin oru valia choodaakkal paaraameettar aavasyamaanennu.
1963 leeii thante prabandhatthil laurence vivaricha (munbu poincare nireekshicha) tharatthilulla prarambha avasthakale sensitiivu aasrayikkunnathinulla paryaayamaayi "butterfly ephakt" palappozhum vishadeekarikkappedunnum, butterfly roopakathe aadyam 1969 ll addeham prasidheekaricha srishtikalil prayogichu, athu ashayathe oru padi koodi munnott kondupoyi. anthareekshathile cheria chalanagal valia samvidaanangale engane badhikkunnu ennathinu laurence oru ganitha maathruka nirddeshichu. aaa modelle samvidaanangal bhaviyil oru prathyeka ghattam vare maathrame pravachikkaan kazhiyoo ennum athinappuram, prarambha saahacharyangalile pishak kuraykkunnathu pravachanaatheetha vardhippikkillennum (pishak poojyamallaatthidatholam) addeham kandethi. pravachanaatheethathayude kaaryatthil oru nirnnayaka samvidaanathe nirnnaayakamallaathathil ninnu "nireekshanaparamaayi verthirichariyaan kazhiyilla" ennu ithu theliyichu. eee prabandhatthinte sameepakaala punarpareekshakal soochippikkunnathu, kwaandam physics vaagdaanam cheiyunna velluvilikalumaayi thaarathamyappeduthumbol, nammude prapancham nirnnaayakamaanenna aashayathin ithu oru pradhana velluvili nalki ennaanu.
1993ll prasidheekaricha "araajakathvathinte saaramsham" enna pusthakatthil lorenz chithrashalabhatthinte phalathe nirvachichathu ingane "oru chalanathmaka samvidaanathinte avasthayil oru cheria maattam varuthiyaal thudarnnulla avasthakal maattamillathe thudarnnuvarunna avasthakalil ninnu valareyadhikam vyathyaasappedum."eee savisheshatha prarambha avasthakale (ezz. dii. aii. sii) parihaarangalude sensitiivu aashrithathvathinu thulyamaanu. athe pusthakatthil, lorenz skiinginte pravartthanam prayogikkukayum prarambha sthaanangalilekkulla samaya vyathyasamulla paathakalude samvedanakshamatha velippeduthunnathinaayi oru aadarshaparamaaya skiing model vikasippikkukayum cheythu. ezz. dii. aii. sii aarambhikkunnathin munbu oru pravachanatheethamaaya chakravaalam nirnnayikkappedunnu.
sidhaanthavum ganithashaasthra nirvachanavum.
aavarthanam, oru sisttam athinte prarambha avasthakalilekku ekadeshamaayi madanguka, prarambha avasthakale sensitiivu aasrayikkal enniva araajakathva chalanathinulla rund pradhana gadakangalaanu. aarambikkunna anthareeksha saahacharyangal poornnamaayum krithyamaayi alakkaan kazhiyaathathinal kaalavastha polulla sankeernamaaya samvidaanangal nirmmikkunnathinte praayogika anantharafalangal avaykku oru nischitha samayaparidhi (kaalaavasthayude kaaryatthil ekadesham orazhcha) marikadannu pravachikkaan prayaasamaan.
kaalakramena exponential nirakkil pooyintukal ekapaksheeyamaayi parasparam adaykkukayaanengil oru chalanathmaka samvidaanam prarambha avasthakale sensitiivu aashrayikkunnath pradarshippikkunnu. nirvachanam topological allah, marichu adisthaanaparamaayi alavukolullathaanu. sensitiivu aashrithathvathe lorenz inipparayunna reethiyil nirvachichu
"oru bhramanapathatthinte swabhaava savisheshatha (i.iii.oru pariharam) eathenkilum ghattathil athinaduthekku kadannupokunna mattu bhramanapathangal samayam munnott pookumbool athinaduthaayi nilanilkkunnillenkil."
m" ennathu mapp formula _ 1-nte stettu space aanenkil, "m"-leeii eathenkilum x-num eathenkilum δ> 0-num, "m"-ll y undengil, formula _ 3 polulla dooravum angane ullathumaaya prarambha avasthakalodulla sensitiivu aashrithathvam formula _ 1 pradarshippikkunnu.
chila positteev paaraamettarukalkk "a". nirvachanathinu oru ayalpakkathil ninnulla ellah pooyintukalum bays pooyint "x" ll ninnu verthirikkendathilla, pakshe ithinu oru positteev liapunov eksponant aavasyamaanu. positteev liapunov exponentinu purame, araajakathva samvidaanangalkkullile mattoru pradhana savisheshathayaanu boundedness.
prarambha avasthakale sensitiivu aayi aashrayikkunnath prakadippikkunna ettavum lalithamaaya ganitha chattakkood logistic maappinte oru prathyeka paaraameettartaiseshan nalkunnu
ettavum araajakathvamulla bhoopadangalil ninnu vyathyasthamayi, oru clossed phoom parihaaramunda
ivide prarambha avastha paaraameettar formula _ 7 formula _ 8 nalkunnu. yukthisahamaya formula _ 7-nuu, parimithamaaya ennam aavarthana formula _ 10 oru periodic seekwansilekku mapp cheiyunnu. ennaal mikkavarum ellah soothravaakyangalum yukthirahithamaan, yukthirahithamaya soothravaakyathinu, soothravaakyam 10 orikkalum aavarthikkilla-ithu kaalaanusrithamalla. eee parihara samavaakyam kuzhappathinte rund pradhana savisheshathakal vyakthamaayi prakadamaakkunnu-stretching folding ghadakam 2 "n" stretchinginte exponential valarcha kaanikkunnu, ithu prarambha avasthakale (butterfly ephakt) sensitiivu aashrayathin kaaranamaakunnu, athesamayam squared saine function formula _ 10 neey paridhikkullil madakkunnu [0,1].
bhauthika samvidaanangalil.
kaalaavasthayil.
kaalaavasthayude kaaryatthil chithrashalabhatthinte prabhavam ettavum parichithamaanu; udaaharanathinu, saadhaarana kalavastha pravachana modalukalil ithu eluppathil pradarshippikkaan kazhiyum. kalavastha shaasthranjaraaya jaims annanum vilyam konnoliyum vishadeekarikkunnathu kaalavastha pravachana reethikalude vikasanathil araajakathvam pradhaanamaanennu; maathrkakal prarambha saahacharyangalodu samvedanakshamathayullavayanu. avar munnariyipp cherkkunnu "theerchayaayum, oru anjjaatha chithrashalabhatthinte chirakukal thattunnath kalavastha pravachanangale nerittu baadikkilla, kaaranam ithrayum cheria aswasthatha ganyamaaya valippathilekku valaraan valareyadhikam samayamedukkum, koodaathe namukk vishamikkenda niravadhi pettennulla anishchithathvangalundu. athinaal kaalavastha pravachanatthil eee prathibhaasatthinte nerittulla swaadheenam palappozhum alppam thettaanu."prarambha saahacharyangale sensitiivu aayi ashrayikkunnathum valia doorangalil oru samghaditha rakthachamkramanam srishtikkunnathinulla oru cheria aswasthathayude kazhivum ulppede rund tharatthilulla chithrasalabha phalangal krithyamaayi onnalla. rund tharatthilulla chithrashalabhangalude phalavum moonnaamathe tharatthilulla chithrashalabhangalude phalavum thaarathamyam cheythathayi rekhappeduthiyittundu. sameepakaala padanangalil, oru cheria aswasthathayude falamaayundaakunna hraswavum ennaal kaaryamaayathumaaya exponential valarchayude savisheshathayaaya chithrasalabha prabhavam srishtikkunnathil asthiratha oru punk vahikkunnuvennu kalavastha, kalavastha ithara rekheeya maathrkakal kaanikkunnuvenn report cheyyappettu.
laurence modalukalkkullil sahavarthithvamulla araajakathvavum araajakathvamillaathathumaaya akarshanangal velippeduthunnathiloode, "kaalavastha araajakathvamaanu" enna parambaraagatha kaazhchappaadin vipareethamaayi "kaalaavasthaykku araajakathvavum kramavumundu" enna oru puthukkiya veekshanam shen sahapravarthakarum nirddeshichu. thalphalamaayi, prarambha avasthakale (ezz. dii. aii. sii) aashrayikkunnath ellaayppozhum prathyakshappedunnilla. athaayath, rund bhramanapathangal (i.iii.pariharangal) kuzhappamulla akarshakamaayi maarunnu; rund bhramanapathangal ore pooyint aakarshakathilekku neengumbol athu drishyamaakilla. eratta pendulam chalanathinulla melpparanja aanimeshan oru saamyam nalkunnu. valia konukalil oonjaalinte chalanam palappozhum kuzhappathilaanu. thaarathamyappeduthumbol, cheria konukalil chalanagal noonn-kottik aanu.
oru sisttam (udha.jii.eratta pendulam sisttam) prarambha avasthakale maathram aashrayichirikkunna onniladhikam bounded akarshakangal adangiyirikkunnu. valathuvashathulla chithratthil kayaking upayogichaanu multistability chithreekarichathu (i.iii.chithram 1) sakthamaaya pravaahangaludeyum nischalamaaya pradeshathinteyum roopam yathaakramam asthirathayum praadeshika sthirathayum soochippikkunnu. thalphalamaayi, rund kayaakkukal sakthamaaya pravaahangaliloode neengumbol, avayude pathakal sdic pradarshippikkunnu. maruvashathu, rund kayaakkukal nischalamaaya oru pradeshathekku neengumbol, avar kudungunnu, saadhaarana sdic kanikkunnilla (oru arajakamaaya shanikatha sambhavikkaamenkilum). esdiisinteyo noow esdiisinteyo atharam savisheshathakal rund tharam pariharangal nirddeshikkukayum multistabilityude swabhaavam vyakthamaakkukayum cheiyunnu.
valia thothilulla prakriyakalude moduleshanumaayi bandhapetta samaya vyathyasamulla multistability kanakkiledukkunnathiloode (udha.jii., seasonal fozhsing), cherukida prakriyakalude mothatthilulla prathikaranam (udha.jii., samvahanam), melpparanja parishkaricha kazhcha inipparayunna reethiyil parishkarichirikkunnu
"anthareekshathil araajakathvavum kramavum undu; udaaharanathinu, uyarnnuvarunna samghaditha samvidaanangalum (chuzhalikkaattukal polullava) aavarthichulla seasonukalil ninnu vyathyasthamaaya samayavum ithil ulppedunnu."
kwaandam mechanicsil.
prarambha avasthakale (butterfly ephakt) sensitiivu aasrayikkunnathinulla saadhyatha ardha classical, kwaandam physixil sakthamaaya feeldukalile aattangalum anisotropic keppler prashnavum ulppede niravadhi kesukalil padichittundu. shudhamaaya kwaandam chikilsakalil prarambha avasthakale angeyattam (exponential) aashrayikkunnath pratheekshikkunnillenna chila rachayithakkal vaadichittundu; ennirunnaalum, classical chalanatthil prakadamaaya prarambha avasthakalekkurichulla sensitiivu aashrithathvam martin gutsviller joonn biyum vikasippicha ardha classical chikilsakalil ulppeduthiyittundu. delos sahapravarthakarum. kramarahithamaya matricks sidhaanthavum kwaandam computerukalumaayulla simuleshanukalum kwaandam mechanicsile butterfly efakttinte chila pathippukal nilavilillenn theliyikkunnu.
kwaandam sisttangalil chithrashalabhatthinte prabhavam nireekshikkaamenn mattu rachayithakkal abhipraayappedunnu. isbisec pii. karkusevsky mattullavarum. alpam vyathyasthamaaya hamiltonians ulla kwaandam sisttangalude samaya parinamam pariganikkuka. kwaandam sisttangalude hamiltoniansile cheria mattangalodulla samvedanakshamathayude thoth avar anweshikkunnu. david poulin thudangiyavar. fidility decay alakkunnathinayi oru kwaandam algoritam avatharippichu, ithu "alpam vyathyasthamaaya dynamicsinu vidheyamaakumbol samana prarambha avasthakal vyathichalikkunna nirakk alakkunnu". fidility decay (poornnamaayum classical) butterfly efakttinte ettavum adutha kwaandam analog aayi avar kanakkakkunnu. classical butterfly ephakt oru nischitha hamiltonian sisttathile oru vasthuvinte sthaanathum/allengil vegathayilum oru cheria maattatthinte phalathe pariganikkumbol, kwaandam butterfly ephakt oru nischitha prarambha sthaanavum vegathayum upayogichu hamiltonian sisttathile oru cheria maattatthinte phalathe pariganikkunnu. eee kwaandam butterfly prabhavam pareekshanaathmakamaayi prakadamaakkiyittundu. prarambha avasthakalodulla sisttam sensittivittiyude kwaandam, ardha classical chikilsakale kwaandam kaas ennu vilikkunnu.

