ASUMPTWTIKH ANALUSH MH GRAMMIKOU SUSTHMATOS DUO SUZEUGMENWN TALANTWTWN ME QRHSH TOU ALGORIJMOU CSP (COMPUTATIONAL SINGULAR PERTURBATION)

Transcription

1 EJNIKO METSOBIO POLUTEQNEIO SQOLH EFARMOSMENWN MAJHMATIKWN KAI FUSIKWN EPISTHMWN ASUMPTWTIKH ANALUSH MH GRAMMIKOU SUSTHMATOS DUO SUZEUGMENWN TALANTWTWN ME QRHSH TOU ALGORIJMOU CSP (COMPUTATIONAL SINGULAR PERTURBATION) Suggrafèac Kuriˆkoc P. Poulhmenˆkoc Epiblèpwn Kajhght c Dhm trioc A. GkoÔshc ParÐsi, 2013

2 Kuriˆkoc P. Poulhmenˆkoc DiplwmatoÔqoc Nauphgìc Mhqanolìgoc Mhqanikìc Copyright Kuriˆkoc Poulhmenˆkoc, 2013 Me epifôlaxh pantìc nìmimou dikai matoc. All rights reserved ApagoreÔetai h antigraf, apoj keush kai dianom thc paroôshc ergasðac, ex olokl rou tm matoc aut c, gia emporikì skopì. Epitrèpetai h anatôpwsh, apoj keush kai dianom gia skopì mh kerdoskopikì, ekpaideutik c ereunhtik c fôsewc, upì thn proôpìjesh na anafèretai h phg proèleushc kai na diathreðtai to parìn m numa. Erwt mata pou aforoôn th qr sh thc ergasðac gia kerdoskopikì skopì prèpei na apeujônontai sto suggrafèa. Oi apìyeic kai ta sumperˆsmata pou perièqontai sto èggrafo autì ekfrˆzoun to suggrafèa kai den prèpei na ermhneujeð ìti antiproswpeôoun tic epðshmec jèseic tou EjnikoÔ Metsìbiou PoluteqneÐou

3 3 EuqaristÐec Sto shmeðo autì ja jela na euqarist sw ìlouc ìsouc suntèlesan sthn pragmatopoðhsh aut c thc diplwmatik c, ton upoy fio didˆktora Panagi th Kourd, touc Sklˆbo SpÔro, Balsamˆ Miqˆlh, Andrèa Orgèta kai idiaðtera touc epiblèpontec kajhghtèc mou Dhm trio A. GkoÔsh kai Iwˆnnh T. GewrgÐou gia thn empistosônh pou mou èdeixan, gia thn polôtimh bo jeiˆ touc, thn kajod ghs touc kai telikˆ thn pragmatikˆ megˆlh upomon touc.

4 4 PerÐlhyh Sth fôsh upˆrqoun sust mata ta opoða qarakthrðzontai apì thn Ôparxh arg n kai gr gorwn qronoklimˆkwn. 'Otan oi gr gorec qronoklðmakec eðnai aposbetikèc, opìte exantloôntai polô sôntoma, tìte ta sust mata autˆ onomˆzontai dôskampta (stiff). H exˆntlhsh twn gr gorwn qronoklimˆkwn shmatodoteð thn anˆptuxh diˆforwn exisorrop sewn metaxô twn sunistws n tou sust matoc kai thn dunatìthta perigraf c thc exèlixhc tou fainìmenou apì èna aplopoihmèno argì (non-stiff) sôsthma. Paradosiakˆ, h katˆstrwsh tou aplopoihmènou sust matoc gðnontan me th bo jeia twn asumptwtik n mejìdwn, oi opoðec eðnai qr simec gia sqetikˆ aplˆ sust mata. H qr sh twn mejìdwn aut n bohjˆ sthn aplopoðhsh twn susthmˆtwn kai sth kalôterh katanìhsh twn fusik n diergasi n pou montelopoioôntai. TeleutaÐa, se antikatˆstash twn paradosiak n asumptwtik n mejìdwn, èqoun arqðsei na qrhsimopoioôntai algorijmikèc asumptwtikèc mèjodoi, oi opoðec proseggðzoun ton gr goro kai argì upoq ro tou efaptomenikoô q rou (tangent space) twn fˆsewn, stouc opoðouc droun oi gr gorec kai argèc, antðstoiqa, qronoklðmakec. 'Otan exantloôntai oi gr gorec qronoklðmakec, h lôsh exelðssetai ston argì upoq ro ìpou kuriarqoôn oi argèc qronoklðmakec, sômfwna me to aplopoihmèno sôsthma (AS). O q roc autìc kaleðtai arg analloðwth pollaplìthta (AAP) (Slow Invariant Manifold / SIM). Sth sugkekrimènh ergasða ja qrhsimopoihjeð h tropopoihmènh algorijmik mèjodoc CSP (Computational Singular Perturbation), h opoða prosdiorðzei me megˆlh akrðbeia thn AAP kai to AS, mèsw thc diadikasðac twn epanal yewn (refinements) ìpou epanaprosdiorðzontai ta dianôsmata bˆshc pou parˆgoun touc gr gorouc kai argoôc upoq rouc, ìpou droun antðstoiqa oi gr gorec kai oi argèc qronoklðmakec. To prìblhma pou ja mac apasqol sei eðnai èna mh-grammikì dôskampto sôsthma dôo suzeugmènwn talantwt n. 'Ena tètoio sôsthma mporeð na montelopoi sei diˆfora probl mata thc mhqanik c, talantwtik c fôsewc, me mh-grammik talantwtik sumperiforˆ. Sth paroôsa ergasða ja prosdioristeð o argìc kai o gr goroc upìqwroc me th bo jeia thc tropopoihmènhc mejìdou CSP kai ja kataskeuasteð h AAP. H akrðbeia thc AAP ja epibebaiwjeð me thn parakoloôjhsh thc poreðac diafìrwn lôsewn tou sust matoc.

5 5 Abstract In nature there are systems which are characterized by the existence of slow and fast time scales. When the fast time scales are of dissipative nature and consequently they are exhausted very soon, these systems are called stiff. The fact that the fast time scales are exhausted leads to the development of several equivalences among the various processes of the system and enables us to describe the evolution of the phenomenon by a simplified non stiff system. Traditionally the formulation of the simplified system was done with the asymptotic methods which are useful for relatively simple systems. The use of these methods helps in the simplification of the systems and in the better understanding of the natural processes which are modelized. Nowadays in replacement of the traditional methods, algorithmic methods are used, which are defining the fast and slow subspace of the tangent space of faces in which the fast and slow timescales act. When the fast timescales are exhausted the solution evolves in the slow subspace, where the slow timescales are dominant, according to the simplified system (SM). This space is called Slow Invariant Manifold (SIM) In this work the modified algorithmic method CSP (Computational Singular Perturbation) will be used, which can define the SIM and the SM with big accuracy through the procedure of refinements. This procedure redefines the base vectors which produce the fast and slow subspaces, where the fast and slow timescales evolve. The problem that we are going to deal with is a non-linear stiff system of two coupled oscillators. Such a system can model various engineering problems of oscillatory nature with non-linear oscillating behaviour. In the present work the slow and the fast subspaces will be defined with the use of the modified method CSP and the SIM will be described. The accuracy of the SIM will be guaranteed by the follow up of the evolution of the solutions of the system.

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7 Perieqìmena 1 Eisagwg 9 2 O Algìrijmoc CSP Perigraf tou algorðjmou Basikˆ shmeða thc mejìdou Tropopoihmènh mèjodoc CSP Perigraf thc AAP O Algìrijmoc IE Kataskeu thc AAP kai tou AS me qr sh tou pðnaka G r s (Tropopoihmènh mèjodoc CSP) IsodunamÐa twn dôo mejìdwn (CSP kai tropopoihmènh CSP) Upologismìc tou pðnaka G r s Upologismìc tou pðnaka R s r Qr sh tou pointer gia thn eklog twn gr gorwn metablht n SÔsthma dôo Suzeugmènwn Talantwt n Perigraf tou probl matoc Prosdiorismìc Paramètrwn Stajerèc elathrðwn k, k 1, k Mˆzec m 2, m Suqnìthtec ω 1, ω Aposbest rec GewmetrÐa Exwterik Fìrtish F Timèc paramètrwn p i SÔndesh twn tim n twn paramètrwn m 1, m 2, k 1, k 2 me ta sust mata pou montelopoioôntai Arqikèc sunj kec Majhmatik montelopoðhsh tou Probl matoc DiereÔnhsh 'Elegqoc idiotim n Iakwbian c Genikˆ Arqikì prìblhma PerÐptwsh PerÐptwsh PerÐptwsh PerÐptwsh Sumpèrasma SÔgkrish twn Peript sewn 1,2,

8 PERIEQŸOMENA Genikˆ Arqikì prìblhma PerÐptwsh PerÐptwsh PerÐptwsh Eklog PerÐptwshc Upologismìc thc AAP Genikˆ Q roc fˆsewn:(x 3, x 4, x 1 ) AAP F r1 = 0 kai F r2 = 0 (refinement) Q roc fˆsewn:(x 3, x 4, x 2 ) AAP F r1 = 0 kai F r2 = 0 (refinement) Sumpèrasma AAP kai akrib c lôsh Genikˆ Q roc fˆsewn:(x 3, x 4, x 1 ) AAP F r1 = 0, F r2 = 0(refinement) kai akrib c lôsh SÔgkrish AAP F r1 = 0 kai akriboôc lôshc SÔgkrish AAP F r2 = 0(refinement) kai akriboôc lôshc Q roc fˆsewn:(x 3, x 4, x 2 ) AAP F r1 = 0, F r2 = 0(refinement) kai akrib c lôsh SÔgkrish AAP F r1 = 0 kai akriboôc lôshc SÔgkrish AAP F r2 = 0(refinement) kai akriboôc lôshc Sumpèrasma Praktikèc efarmogèc Genikˆ SÔndesh susthmˆtwn me thn PerÐptwsh EpÐlogoc 85 Aþ POLLAPLOTHTES 87

9 Kefˆlaio 1 Eisagwg Sthn paroôsa ergasða ja asqolhjoôme me sust mata suzeugmènwn talantwt n. Lègontac sôsthma suzeugmènwn talantwt n ennooôme sthn pio apl tou morf dôo talantwtèc, ìpou o kˆje ènac èqei mˆza m kai stajerˆ elathrðou k, oi opoðoi sundèontai mèsw enìc trðtou talantwt me stajerˆ elathrðou k C ìpwc faðnetai sto Sq. 1. Sq ma 1. SÔsthma dôo suzeugmènwn talantwt n. Sth fôsh upˆrqoun pollˆ fainìmena ta opoða mporoôn na montelopoihjoôn me èna sôsthma dôo suzeugmènwn talantwt n [3], [4], [5], [15], ìpwc sto q ro thc anjrwpologðac to perpˆthma tou anjr pou, sto q ro thc mhqanik c ènac kinht rac pou eðnai edrasmènoc mèsw elastikoô sundèsmou se èna ploðo se èna autokðnhto, sto q ro thc nauphgik c to axonikì sôsthma (ˆxonac - propèla) enìc ploðou, h metallik kataskeu enìc ploðou h opoða apoteleðtai apì enisqutikˆ ta opoða uposthrðzontai apì enisqumènouc dokoôc. O trìpoc me ton opoðo ja kinhjoôn oi dôo suzeugmènoi talantwtèc exartˆtai apì th sqetik touc jèsh katˆ thn ènarxh tou fainìmenou [6], [7]. Upˆrqoun oi ex c peript seic: 1) An oi arqikèc metatopðseic eðnai Ðdiec katˆ mètro kai forˆ, oi dôo talantwtèc kinoôntai summetrikˆ petuqaðnontac tautìqrona ta mègista plˆth. Sq. 2(a) 2) An oi arqikèc metatopðseic eðnai Ðdiec katˆ mètro allˆ antðjetec katˆ forˆ, oi dôo talantwtèc kinoôntai antisummetrikˆ petuqaðnontac tautìqrona mègista allˆ antðjeta plˆth. Sq. 2(b) 9

10 KEFŸALAIO 1. EISAGWGŸH 10 (a) (b) Sq ma 2. KÐnhsh dôo talantwt n (a) me Ðdiec kai (b) antðjetec arqikèc metatopðseic. 3) An o ènac talantwt c xekinˆ me arqik metatìpish en o ˆlloc brðsketai sth jèsh isorropðac, tìte katˆ thn talˆntwsh ìtan o ènac petuqaðnei to mègisto plˆtoc o ˆlloc brðsketai sth jèsh isorropðac tou. Sq. 3(a) 4) An t ra xekin soun kai oi dôo me kˆpoia arqik allˆ diaforetik metatìpish, tìte katˆ thn talˆntwsh enallˆssontai gôrw apì autèc tic metatopðseic. Sq. 3(b) (a) (b) Sq ma 3. KÐnhsh dôo talantwt n me (a) mhdenik gia ton èna kai mh mhdenik gia ton ˆllo arqik metatìpish kai (b) diaforetikèc mh mhdenikèc arqikèc metatopðseic gia kˆje èna. Eˆn to prìblhma eðnai grammikì mporeð na brejeð h analutik èkfrash thc lôshc mèsw tou kajorismoô twn idiotim n kai idiodianusmˆtwn tou sust matoc. Eˆn to prìblhma eðnai mh grammikì h lôsh brðsketai mèsw arijmhtik c epðlushc tou sust matoc twn diaforik n exis sewn. Sun jwc autˆ ta probl mata qarakthrðzontai apì thn Ôparxh pollapl n qronoklimˆkwn, kˆpoiec apì tic opoðec eðnai gr gorec kai kˆpoiec ˆllec argèc. Eˆn oi gr gorec qronoklðmakec eðnai aposbetikèc, tìte exantloôntai polô pio gr gora apì tic argèc kai anaptôssetai ènac a- rijmìc exisorrop sewn metaxô twn diˆforwn diadikasi n tou probl matoc o arijmìc twn opoðwn eðnai Ðsoc me ton arijmì twn gr gorwn qronoklimˆkwn. Oi pio gnwstoð tôpoi aut n twn exisorrop sewn eðnai h QSSA (Quasi-Steady State Approximation) h opoða sqetðzetai me ta stoiqeða

11 KEFŸALAIO 1. EISAGWGŸH 11 tou dianôsmatoc pou perigrˆfei to prìblhma kai h PEA (Partial Equilibration Approximation) h opoða sqetðzetai me tic fusikèc diadikasðec pou montelopoioôntai. 'Ena tètoio prìblhma lègetai dôskampto [1], [2], [12], [16] kai h lôsh tou mporeð na upologisteð me th bo jeia thc QSSA, thc PEA, thc asumptwtik c anˆlushc kai eidikˆ thc mejìdou twn idiìmorfwn diataraq n. Me th bo jeia twn mejìdwn aut n, ekmetalleuìmenoi to gegonìc oti oi gr gorec aposbetikèc qronoklðmakec exantloôntai polô gr gora, mporoôme na prosdiorðsoume èna q ro lôsewn mikrìterhc diˆstashc o opoðoc onomˆzetai arg analloðwth pollaplìthta (AAP) (Slow Invariant Manifold / SIM), kaj c epðshc kai to aplopoihmèno argì sôsthma (AS) twn exis sewn pou prosdiorðzei thn kðnhsh thc lôshc sto q ro autì [1], [16]. Autìc eðnai ènac trìpoc me ton opoðo mporoôme na prospaj soume na lôsoume megˆla kai sônjeta sust mata aplopoi ntac ta mèsw thc elˆttwshc tou arijmoô twn metablht n. EpÐshc me tic mejìdouc autèc mporoôme na katano soume kalôtera th fusik tou probl matoc prosdiorðzontac tic diadikasðec pou summetèqoun sto sqhmatismì thc AAP kai tou AS kai tic metablhtèc pou sundèontai me tic argèc kai gr gorec qronoklðmakec. Prokeimènou na efarmìsoume tic parapˆnw paradosiakèc asumptwtikèc mejìdouc, to arqikì sôsthma exis sewn prèpei na grafeð se adiˆstath morf, na prosdioristeð o arijmìc twn gr - gorwn qronoklimˆkwn, na diaqwristoôn oi argèc apì tic gr gorec metablhtèc sto diˆnusma pou perigrˆfei to prìblhma kai na prosdioristeð h mikr parˆmetroc e h opoða eðnai o lìgoc thc pio arg c apì tic gr gorec proc thn pio gr gorh apì tic argèc qronoklðmakec kai eðnai endeiktik tou kenoô metaxô gr gorwn kai arg n qronoklimˆkwn tou sust matoc. Kˆti tètoio ìmwc gia megˆla kai sônjeta sust mata, ta opoða kurðwc mac endiafèroun, eðnai praktik c adônato. Sthn ergasða twn Georgiou, Schwartz [16] analôetai h diˆtaxh twn dôo talantwt n pou faðnetai sto Sq.4, h opoða montelopoieð mia dôskampth kataskeu pou apoteleðtai apì èna eôkampto mh grammikì talantwt m 1 kai ènan ˆkampto grammikì m. ParadeÐgmata tètoiwn kataskeu n eðnai to sôsthma enìc peristrefìmenou ˆxona pˆnw se mia biskoelastik èdrash to sôsthma èlika-ˆxonac enìc ploðou. 'Otan h kðnhsh tou sust matoc exartˆtai apì mia argˆ metaballìmenh sunist sa kai apì mia gr gora metaballìmenh sunist sa h opoða exantleðtai polô sôntoma o upologismìc thc lôshc autoô tou sust matoc gðnetai me th bo jeia thc asumptwtik c anˆlushc kai pio sugkekrimèna mèsw thc anˆlushc idiìmorfwn diataraq n. Sq ma 4. Mh grammikìc talantwt c mˆzac m 1 sundedemènoc me èna grammikì talantwt mˆzac m. Me th bo jeia thc anˆlushc idiìmorfwn diataraq n prosdiorðzoume èna sôsthma exis sewn h lôsh tou opoðou katal gei se mia elktik ametˆblhth pollaplìthta, gnwst wc AAP. Aut

12 KEFŸALAIO 1. EISAGWGŸH 12 eðnai ènac upoq roc tou q rou twn fˆsewn pou perilambˆnei tic argèc sunist sec tou sust - matoc. 'Enac ˆlloc trìpoc asumptwtik c anˆlushc tou probl matoc eðnai me th qr sh thc algorijmik c mejìdou CSP(Computational Singular Perturbation) [1], [2], [8], [12], h opoða lìgw thc algorijmik c fôshc thc mporeð na prosdiorðsei thn AAP kai to AS gia megˆla kai sônjeta majhmatikˆ montèla, parèqontˆc mac megalôterh akrðbeia apì tic klasikèc mejìdouc pou anafèrjhkan pio pˆnw. Autì epitugqˆnetai mèsw tou orismoô twn dianusmˆtwn pou proseggðzoun ton argì kai gr goro upoq ro kai ton epanaprosdiorismì (refinement) twn dianusmˆtwn aut n. H akrðbeia thc mejìdou exartˆtai apì ton arijmì twn epanal yewn (refinement), kaj c epðshc kai apì to mègejoc tou kenoô e pou parousiˆzetai metaxô twn arg n kai gr gorwn qronoklimˆkwn, to opoðo orðzetai wc h apìstash metaxô thc pio arg c apì tic gr gorec qronoklðmakec kai twn arg n qronoklimˆkwn oi opoðec tautðzontai. Sthn ergasða twn Gousis, Valorani [12] anaptôsetai h tropopoihmènh mèjodoc CSP. H mèjodoc aut efarmìzetai se dôskampta sust mata ta opoða qarakthrðzontai apì thn Ôparxh arg n kai gr gorwn aposbetik n qronoklimˆkwn oi opoðec exantloôntai gr gora, af nontac tic argèc na prosdiorðsoun thn exèlixh tou fainìmenou. Tìte ìpwc mporoôme na doôme stic ergasðec twn Kourdis, Steuer, Gousis, Valorani [2], [8], [12] h exˆntlhsh ttwn gr gorwn qronoklimˆkwn mac dðnei th dunatìthta na prosdiorðsoume algorijmikˆ me th bo jeia thc mejìdou CSP thn AAP, kaj c epðshc kai to AS to opoðo perigrˆfei thn kðnhsh thc lôshc pˆnw sthn AAP. 'Etsi aplopoioôme to dôskampto sôsthma afoô èqoume mikrìtero arijmì agn stwn kai douleôoume se èna q ro qwrðc gr gorec qronoklðmakec. EpÐshc to AS mac dðnei th dunatìthta na prosdiorðsoume th lôsh tou arqikoô montèlou, prˆgma to opoðo den eðnai pˆnta dunatì me thn asumptwtik anˆlush, idðwc ìtan èqoume megˆla kai sônjeta montèla. Tèloc o prosdiorismìc twn fusik n diadikasi n, ìpwc eðnai h fôsh twn exisorrop sewn oi opoðec mac perigrˆfoun thn AAP, kaj c epðshc o prosdiorismìc twn kurðarqwn exisorrop sewn pou mac odhgeð sth diamìrfwsh tou AS, mac bohjoôn sthn kalôterh fusik katanìhsh kai ermhneða tou probl matoc. H tropopoihmènh mèjodoc CSP eðnai pio akrib c se sqèsh me thn mèjodo CSP ston prosdiorismì thc AAP kai tou AS idðwc stic peript seic ìpou h AAP èqei polô mikr diˆstash, dhlad ìpou èqoume shmantik aplopoðhsh tou sust matoc. H diaforˆ se sqèsh me thn arqik mèjodo CSP eðnai oti prokeimènou na xekin soume prèpei na prosdiorðsoume tic gr gorec metablhtèc, prˆgma to opoðo mporeð na gðnei me th bo jeia tou CSP pointer. St n ergasða tou Asbestˆ [1] upologðzetai gia to sôsthma twn dôo talantwt n pou faðnetai sto Sq.4 h akrib c lôsh kai h AAP me th bo jeia thc asumptwtik c anˆlushc kai thc mejìdou CSP bˆsh tou gegonìtoc ìti to sôsthma qarakthrðzetai apì thn Ôparxh arg n kai gr gorwn aposbetik n qronoklimˆkwn oi opoðec exantloôntai polô sôntoma. ParathroÔme pwc h AAP pou prokôptei apì thn asumptwtik anˆlush den tautðzetai apìluta me thn AAP pou prosdiorðzetai me th mèjodo CSP. Katìpin sugkrðnontai oi treic lôseic ìpou faðnetai pwc h akrib c lôsh pèftei epˆnw sthn AAP pou prosdiorðsthke me th mèjodo CSP, prˆgma to opoðo shmaðnei pwc h mèjodoc aut proseggðzei thn AAP me megalôterh akrðbeia. Autì ofeðletai sto ìti sth mèjodo CSP gðnontai epanal yeic (refinements), ìpou kˆje forˆ upologðzontai me megalôterh akrðbeia o argìc kai gr goroc upoq roc, se antðjesh me thn asumptwtik anˆlush ìpou h akrðbeia sthn prosèggish thc AAP exartˆtai mìno apì thn tˆxh tou anaptôgmatoc thc sunˆrthshc pou perigrˆfei thn AAP. O dunamikìc autìc epanaprosdiorismìc twn dianusmˆtwn bˆshc twn dôo upoq rwn, ìpou droun antðstoiqa oi argèc kai oi gr gorec

13 KEFŸALAIO 1. EISAGWGŸH 13 qronoklðmakec kˆnei th mèjodo CSP pio akrib ston upologismì thc AAP. Sthn paroôsa ergasða prìkeitai na asqolhjoôme me èna dôskampto sôsthma dôo suzeugmènwn talantwt n to opoðo faðnetai sto Sq. 5. Oi katakìrufoi talantwtèc qarakthrðzontai antðstoiqa apì tic stajerèc k 1, k 2 tou elathrðou apì tic aposbèseic c 1, c 2, kai apì tic mˆzec m 1, m 2, en o talantwt c pou sundèei touc dôo katakìrufouc talantwtèc eðnai mh grammikìc kai qarakthrðzetai apì th stajerˆ k tou elathrðou. Gia na mporèsoume na efarmìsoume th mèjodo CSP prèpei na èqoume èna sôsthma pou na qarakthrðzetai apì ènan arijmì idiotim n me pragmatikì mèroc arnhtikì kai kat' apìluth tim polô megalôtero apì to fantastikì, prˆgma to opoðo mac exasfalðzei thn Ôparxh arg n kai gr gorwn aposbetik n qronoklimˆkwn oi opoðec exantloôntai polô pio sôntoma apì tic argèc. Prokeimènou na fèroume to sôsthmˆ mac se aut th morf prosdiorðzoume tic katˆllhlec timèc twn paramètrwn k 1, k 2, c 1, c 2, m 1, m 2. Autì ja mac odhg sei sthn perðptwsh enìc sust matoc ìpou upˆrqei megˆlh diaforˆ stic paramètrouc c 1, c 2 (c 1 > c 2 ) h opoða odhgeð sthn anˆptuxh dôo gr gorwn aposbetik n qronoklimˆkwn oi opoðec exantloôntai polô sôntoma. H melèth tou sust matoc ja gðnei me thn tropopoihmènh mèjodo CSP. To sôsthma autì twn dôo talantwt n mporeð na ekfrˆsei th majhmatik montelopoðhsh twn parakˆtw diatˆxewn: 1) Enìc kinht ra pou eðnai edrasmènoc mèsw elastikoô sundèsmou se èna ploðo se èna autokðnhto. 2) Tou sust matoc anˆrthsh-amˆxwma enìc autokin tou. 3) Tou axonikoô sust matoc (ˆxonac - propèla) enìc ploðou. 4) Tou diploô toiq matoc enìc ploðou. 5) Tou sust matoc ploðo-plwt exèdra antl sewc petrelaðou ìtan eðnai sundedemèna. 6) Tou sust matoc ploðo(montelopoieðtai wc elastik dokìc)-jˆlassa. 7) Thc metallik c kataskeu c enìc ploðou h opoða apoteleðtai apì enisqutikˆ ta opoða uposthrðzontai apì enisqumènouc dokoôc. Ta apotelèsmata aut c thc melèthc ja mac bohj soun na prosdiorðsoume thn kðnhsh kˆje antikeimènou kai kat' epèktash kradasmoôc (peript seic 1,2,3,5) paramorf seis-tˆseic (perðptwsh 4,6,7). Sto Kefˆlaio 2 gðnetai mia perigraf thc mejìdou CSP kai thc tropopoihmènhc mejìdou CSP ìpwc aut èqei diatupwjeð apì touc Lam kai GkoÔsh kai apodeiknôetai h isodunamða twn dôo mejìdwn. Sto Kefˆlaio 3 perigrˆfetai to sôsthma twn suzeugmènwn talantwt n pou ja melethjeð kai gðnetai h majhmatik montelopoðhsh tou probl matoc en sto kefˆlaio 4 gðnetai h diereônhsh twn paramètrwn prokeimènou na katal xoume se èna sôsthma pou na qarakthrðzetai apì thn Ôparxh arg n kai gr gorwn aposbetik n qronoklimˆkwn oi opoðec exantloôntai polô gr gora. AkoloujeÐ sto kefˆlaio 5 o upologismìc thc AAP me thn tropopoihmènh mèjodo CSP en sto kefˆlaio 6 gðnetai o upologismìc thc akriboôc lôshc me th bo jeia tou k dika LSODE kai sôgkrish thc lôshc aut c me thn AAP ìpou blèpoume pwc h akrib c lôsh pèftei pˆnw sthn AAP. 'Etsi h paroôsa ergasða mac dðnei me th bo jeia thc mejìdou CSP thn AAP tou sust matoc tou Sq. 5 to opoðo montelopoieð ta fusikˆ fainìmena talantwtik c fôsewc pou anafèrjhkan pio pˆnw. H AAP aut eðnai o argìc, mikrìterhc diˆstashc, upoq roc tou q rou twn fˆsewn prˆgma to opoðo mac epitrèpei na prosdiorðsoume poiec metablhtèc kajorðzoun thn exèlixh tou fainìmenou sto qrìno kai mac bohjˆei na aplopoi soume th morf thc lôshc tou sust matoc kai na thn ekfrˆsoume bˆsh twn arg n metablht n. EpÐshc mèsw tou prosdiorismoô thc fusik c

14 KEFŸALAIO 1. EISAGWGŸH 14 shmasðac twn exisorrop sewn kai twn fusik n diadikasi n oi opoðec odhgoôn stic sugkekrimènec exisorrop seic mporoôme na katano soume kalôtera th fôsh tou probl matoc kai na sundèsoume tic metablhtèc me tic argèc kai gr gorec qronoklðmakec kaj c epðshc kai me tic diˆforec fusikèc diadikasðec.

15 KEFŸALAIO 1. EISAGWGŸH 15

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17 Kefˆlaio 2 O Algìrijmoc CSP 2.1 Perigraf tou algorðjmou Me th upologistik mèjodo CSP prospajoôme na proseggðsoume th lôsh dôskamptwn problhmˆtwn ta opoða prosomoiˆzontai apì majhmatikˆ montèla pou qarakthrðzontai apì thn Ôparxh arg n kai gr gorwn qronoklimˆkwn. Th lôsh enìc dôskamptou probl matoc mporoôme na thn proseggðsoume kataskeuˆzontac èna majhmatikì montèlo to opoðo ja eðnai mikrìterhc diˆstashc kai ja eðnai apallagmèno apì tic gr gorec qronoklðmakec tou arqikoô montèlou. Ousiastikˆ autì pou zhteðtai eðnai h kataskeu enìc q rou mikrìterhc diˆstashc, dhlad miac pollaplìthtac, h opoða elkôei thn troqiˆ. Aut h pollaplìthta onomˆzetai Arg AnalloÐwth Pollaplìthta (AAP) (Slow Invariant Manifold / SIM) tou dôskamptou probl matoc. Upˆrqoun dôo kathgorðec mejìdwn oi opoðec mac bohjoôn na upologðsoume thn AAP. Sthn pr th kathgorða an koun autèc pou prosdiorðzoun thn AAP wc gewmetrikì tìpo shmeðwn, en sth deôterh an koun autèc pou prosdiorðzoun thn AAP ìpou droun oi argèc qronoklðmakec, allˆ kai touc gr gorouc q rouc ìpou droun oi gr gorec qronoklðmakec prˆgma to opoðo mac dðnei th dunatìthta na prosdiorðsoume kai to argì aplopoihmèno sôsthma AS pou kajorðzei thn kðnhsh thc lôshc pˆnw sthn AAP. H upologistik mèjodoc CSP h opoða anaptôqjhke apì touc Lam kai GkoÔsh an kei sthn deôterh kathgorða. Dedomènou enìc N-diˆstatou probl matoc, h CSP prosdiorðzei ta sônola twn gr gorwn kai arg n dianusmˆtwn bˆshc, a r kai a s kaj c kai ta antðstoiqa sônola twn duik n touc, b r kai b s. Ta a r perigrˆfoun to gr goro M-diˆstato upoq ro, en apì ta a s perigrˆfoun ton argì (N M)-diˆstato upoq ro AAP (M < N) o opoðoc eðnai topikˆ orjog nioc wc proc ton gr goro upoq ro. H mèjodoc CSP perilambˆnei mia epanalhptik diadikasða, h opoða prosdiorðzei ta dianôsmata bˆshc pou parˆgoun ton gr goro kai argì upoq ro, stouc opoðouc droun oi gr gorec kai argèc qronoklðmakec antðstoiqa. H diadikasða xekinˆei me mia tuqaða arqik ektðmhsh twn gr gorwn dianusmˆtwn bˆshc a r kaj c kai twn duik n touc b r. Metˆ apì kˆje epanˆlhyh lambˆnontai kalôterec proseggðseic twn gr gorwn kai twn sumplhrwmatik n touc arg n upoq rwn. H b r refinement kai h a r refinement eðnai dôo eid n anexˆrthtec epanal yeic ìpou h pr th belti nei thn akrðbeia thc perigraf c thc AAP kai thc lôshc tou AS me to na proseggðzei kalôtera ton argì upoq ro en h deôterh exaleðfei thn akamyða tou aplopoihmènou argoô sust matoc proseggðzontac kalôtera ton gr goro upoq ro. Metˆ apì kˆje epanˆlhyh refinement ta gr gora plˆth gðnontai O(e) mikrìtera ìpou e eðnai mia posìthta pou isoôtai me to lìgo thc pio arg c apì thc gr gorec qronoklðmakec proc thn 17

18 KEFŸALAIO 2. O ALGŸORIJMOS CSP 18 qarakthristik qronoklðmaka (pio gr gorh apì tic argèc). AkoloÔjwc h akrðbeia prosèggishc thc lôshc apì to AS belti netai katˆ thn Ðdia tim. 2.2 Basikˆ shmeða thc mejìdou 'Estw èna fusikì fainìmeno to opoðo perigrˆfetai apì to sôsthma twn N sun jwn diaforik n exis sewn thc morf c dy dt = g(y) (2.2.1) ìpou y eðnai to N-diˆstato diˆnusma twn exarthmènwn metablht n kai g mia algebrik exðswsh tou y. Upojètoume ìti h exðswsh (2.2.1) eðnai dôskampth, dhlad kˆpoiec apì tic qronoklðmakec tou fainìmenou eðnai polô pio gr gorec apì tic qronoklðmakec pou mac endiafèroun. Upˆrqoun dôo upoq roi tou q rou twn fˆsewn. O M-diˆstatoc gr goroc ìpou droun oi gr gorec qronoklðmakec kai o (N M)-diˆstatoc argìc ìpou droun oi argèc qronoklðmakec. Se kˆje shmeðo tou q rou twn fˆsewn to diˆnusma g analôetai se dôo sunistamènec oi opoðec eðnai oi probolèc tou antðstoiqa ston M-diˆstato gr goro kai ston (N M)-diˆstato argì upoq ro. Oi upoq roi autoð parˆgontai antðstoiqa apì ta M dianôsmata st lh N diastˆsewn ston NxM pðnaka a r kai apì ta (N M) dianôsmata st lh N diastˆsewn ston Nx (N M) pðnaka a s. [ a r a r = r a s r ] [ a r, a s = s a s s ] (2.2.2) JewroÔme t ra èna deôtero sônolo dianusmˆtwn gramm c ta opoða eðnai duikˆ twn parapˆnw kai perilambˆnontai stouc pðnakec b r diastˆsewc MxN kai b s diastˆsewc (N M)xN b r = [ b r r, b r s ], b s = [ b s r, b s s ] (2.2.3) ìpou ta a r r, b r r eðnai diastˆsewc MxM, ta as r, b s r eðnai diastˆsewc (N M)xM, ta ar s, b r s eðnai diastˆsewc Mx(N M) ta a s s, b s s eðnai diastˆsewc (N M)x(N M) An probˆloume to g ston argì kai gr goro upoq ro èqoume apì thn exðswsh (2.2.1) ìpou [ dy a dt = a r(b r g) + a s (b s r g) = r a s r ] f r + [ a r s a s s ] f s (2.2.4) f r = b r g = [ b r r, b r s f s = b s g = [ b s r, b s s ] [ g r g s ] [ g r g s ] = b r rg r + b r sg s (2.2.5) ] = b s rg r + b s sg s (2.2.6) Opìte h AAP perigrˆfetai apì thn exðswsh: kai to AS apì thn exðswsh: b r rg r + b r sg s = 0 (2.2.7)

19 KEFŸALAIO 2. O ALGŸORIJMOS CSP 19 [ dy a dt = r s a s s ] (b s rg r + b s sg s ) (2.2.8) 'Opwc anafèrjhke prohgoumènwc, upˆrqoun dôo eðdh CSP refinement. To pr to metabˆllei to b r kai a s ( b r refinement) af nontac ta b s kai a r anephrèasta kai belti nei thn akrðbeia sthn perigraf thc AAP kai thc lôshc tou AS. To deôtero metabˆllei to a r kai b s ( a r refinement) af nontac ta b r kai a s anephrèasta kai egguˆtai thn mh-duskamyða tou aplopoihmènou probl matoc. Oi dôo autèc epanalhptikèc diadikasðec leitourgoôn anexˆrthta, sunep c opoiosd pote a- rijmìc apì autˆ mporeð na efarmosteð. Oi sqèseic pou dðnoun ton algìrijmo twn dôo refinement eðnai: b r refinement (2.2.9) ( db r ) b r (k 1 + 1, m 1 ) = T r (k 1, m 1 ) r(k 1, m 1 ) + b r (k 1, m 1 )J, dt a r (k 1 + 1, m 1 ) = a r (k 1, m 1 ), b s (k 1 + 1, m 1 ) = b s (k 1, m 1 ), a s (k 1 + 1, m 1 ) = [I a r (k 1 + 1, m 1 )b r (k 1 + 1, m 1 )]a s (k 1, m 1 ), a r refinement (2.2.10) b r (k 2, m 2 + 1) = ( b r (k 2, m 2 ), ) a r (k 2, m 2 + 1) = dar (k 2, m 2 ) + Ja r (k 2, m 2 ) T r dt r(k 2, m 2 ), b s (k 2, m 2 + 1) = b s (k 2, m 2 )[I a r (k 2, m 2 + 1)b r (k 2, m 2 + 1)], a s (k 2, m 2 + 1) = a s (k 2, m 2 ), ìpou [( db r T r (k i, m i ) r(k i, m i ) = dt ) 1 + b r (k i, m i )J a r (k i, m i )] (2.2.11) 2.3 Tropopoihmènh mèjodoc CSP Perigraf thc AAP 'Estw ìti èqoume mia AAP diastˆsewc (N M) sto N-diˆstato q ro twn fˆsewn kai y eðnai to N-diˆstato diˆnusma twn exarthmènwn metablht n. 'Estw oti h AAP parametropoieðtai apì s j = (j = 1,..., N M) omalèc sunart seic tou y, tètoiec ste: s j = s j (y) = s j (y 1,..., y N ), j = 1, N M ( ) To diˆnusma jèsewc pˆnw sthn AAP mporeð na upologisteð apì:

20 KEFŸALAIO 2. O ALGŸORIJMOS CSP 20 ìpou s = ( s 1,..., s N M) T. y i = y i (s) = y i (s 1,..., s N M ), i = 1, N, ( ) H dianusmatik sunˆrthsh s = s(y) eðnai mia apeikìnish tou dianôsmatoc jèsewc apì ton N-diˆstato fˆsewn sth (N M)-diˆstath AAP. ParagwgÐzontac thn ( ) wc proc ton qrìno èqoume: dy dt = Y ds s dt = g(y) ( ) ìpou ds dt = S dy y dt = S yg(y) ( ) kai Y s = y 1 s 1. y N s 1 y 1 s N M. y N s N M, S y = s 1 y 1 s 1 y N. s N M y 1. yn M y N ( ) Oi Y s, S s eðnai Nx(N M) kai (N M)xN pðnakec antðstoiqa pou ikanopoioôn th sqèsh: I s s S y Y s = I s s ( ) eðnai o (N M)x(N M) monadiaðoc pðnakac. Apì tic ( ) kai ( ) èqoume to N-diˆstato sôsthma algebrik n exis sewn: [I N N Y s S y ]g(y) = 0 ( ) ìpou mìno M sunist sec eðnai grammikˆ anexˆrthtec kai mporoôn na perigrˆyoun thn AAP. H lôsh pˆnw sthn AAP ja dðnetai apì to N-diˆstato sôsthma: dy dt = Y ss y g(y) ( ) 'Opou mìno (N M) sunist sec eðnai grammikˆ anexˆrthtec.

21 KEFŸALAIO 2. O ALGŸORIJMOS CSP O Algìrijmoc IE Ja anafèroume ed ta basikˆ stoiqeða miac ˆllhc upologistik c mejìdou thc AAP, thc IE, prokeimènou sth sunèqeia na parajèsoume thn tropopoihmènh mèjodo CSP. 'Estw ìti oi omalèc sunart seic s j eðnai oi N M teleutaðec sunist sec tou y. s = (y M+1,..., y N ) T ( ) OrÐzoume to M-diˆstato diˆnusma z wc ex c: z = (y 1 (s),... y M (s)) T = z(s) ( ) Tìte h sqèsh ( ) grˆfetai wc: g r (z, s) G r s (z, s)gs (z, s) = 0 ( ) Ta g r = ( g 1,..., g M) T kai g s = ( g M+1,..., g N) T eðnai ta M-diˆstata kai (N M)-diˆstata dianôsmata pou apoteloôntai antistoðqwc apì ta pr ta M kai ta teleutaða (N M) stoiqeða, tou dianusmatikoô pedðou g. O Mx(N M) pðnakac G r s perilambˆnei tic merikèc parag gouc twn M sunistws n tou y sto z wc proc tic upìloipec (N M) sunist sec tou y sto s G r s(z, s) = z s = z 1 z 1 s 1 s N M.. z M s 1 z M s N M ( ) H sqèsh ( ) onomˆzetai analloðwth exðswsh kai apoteleðtai apì M exis seic me M agn stouc. SÔmfwna me ton algìrijmo IE h (n + 1) epanˆlhyh z n+1 upologðzetai gia dosmèno s kai arqik ektðmhsh z 0 apì thn peplegmènh sunˆrthsh g r (z n+1, s) G r s (z n, s)g s (z n+1, s) = 0 ( ) apì mia analutik sunˆrthsh thc morf c z n+1 = H(z n, s) ( ) Wstìso, èwc autì to shmeðo, o algìrijmoc IE den prosfèrei kanèna kanìna wc proc ton diaqwrismì tou dianôsmatoc jèsewc y se s kai z, dhlad se argèc kai gr gorec sunist sec, antðstoiqa, oôte proteðnei thn katˆllhlh morf thc sunˆrthshc H(z n, s) oôtwc ste na epitugqˆnetai sôgklish.

22 KEFŸALAIO 2. O ALGŸORIJMOS CSP Kataskeu thc AAP kai tou AS me qr sh tou pðnaka G r s (Tropopoihmènh mèjodoc CSP) H adiˆstath exðswsh ( ) grˆfetai wc: Sunep c orðzontai oi pðnakec: A r = [I r r G r s] [ I r r G r sr s r R s r [ g r (z, s) g s (z, s) ] = 0 ( ) ] [ G r s, A s = I s s ], ( ) B r = [I r r, G r s], B s = [R s r, I s s R s rg r s], ( ) ìpou o G r s orðzetai apì thn ( ) kai o Rs r ja oristeð argìtera ìtan ja tejoôn oi sunj kec gia thn mh akamyða tou aplopoihmènou sust matoc. Oi diastˆseic twn pinˆkwn A r, A s, B r, B s eðnai antistoðqwc NxM, Nx(N M), MxN, (N M)xN kai ikanopoioôn tic sqèseic orjogwniìthtac [ ] [ ] B r B r [A r A s ] = [A r A s ] = I N N ( ) B s B s Apì tic sqèseic ( ) kai ( ) blèpoume pwc to arqikì prìblhma (2.2.1) mporeð na grafeð sth morf : [ dy I dt = A rf r + A s F s r r G r = sr s r R s r ] [ G F r r s + I s s ] F s ( ) ìpou [ ] g F r = B r g = [I r r, G r r s] = g r G r sg s $ ( ) g s [ ] g F s = B s g = [R s r, I s s R s rg r r s] = R s rg r + (I s s R s rg r s) g s ( ) g s 'Otan h lôsh brðsketai pˆnw sthn AAP ikanopoieðtai h exðswsh ( ). Dhlad èqoume apì ( ) F r = g r G r sg s = 0 ( ) Sunep c mìno o argìc ìroc diathreðtai sthn exðswsh ( ) kai mporoôme na grˆyoume: [ dy G r dt = s I s s ] (R s rg r + (I s s R s rg r s) g s ) ( ) Oi exis seic ( ) kai ( ) perigrˆfoun antðstoiqa thn AAP kai to AS.

23 KEFŸALAIO 2. O ALGŸORIJMOS CSP IsodunamÐa twn dôo mejìdwn (CSP kai tropopoihmènh CSP) Prokeimènou na apodeðxoume thn isodunamða twn dôo mejìdwn eðnai aparaðthto na deðxoume pwc oi dôo morfèc (2.2.4) kai ( ) thc arqik c exðswshc tautðzontai ìtan: G r s = a r s(a s s) 1 = (b r r) 1 b r s ( ) R s r = a s sb s r = a s rb r r ( ) dedomènou oti ta (b r r) 1, (a s s) 1 upˆrqoun. Apì tic parapˆnw sqèseic mporoôme na deðxoume oti oi exis seic pou perigrˆfoun thn AAP kai to AS sth mèjodo CSP ( ), ( ) kai sthn tropopoihmènh mèjodo CSP (2.2.7), (2.2.8) epðshc tautðzontai. dy dt = [ G r s I s s F r = g r G r sg s = 0 ] ( ) (R s rg r + (I s s R s rg r s) g s ) ( ) b r rg r + b r sg s = 0 ( ) [ ] dy a dt = r s (b s rg r + b s sg s ) ( ) a s s Apì thn exðswsh ( ) èqoume gia ta dianôsmata bˆshc twn dôo mejìdwn: ìpou B r = (b r r) 1 b r, A r = a r b r r + a s N s r ( ) B s = M s rb r + a s sb s, A s = a s (a s s) 1 ( ) N s r = (a s s) 1 [R s r + a s rb r r], M s r = [R s r + a s rb r r](b r r) 1 ( ) Blèpoume pwc ta dianôsmata B r, A s parˆgoun ton Ðdio upoq ro me ta dianôsmata b r, a s, ìpwc autìc prokôptei metˆ apì to CSP b r refinement (2.2.9). AntÐjeta ta dianôsmata B s, A r den parˆgoun ton Ðdio upoq ro me ta dianôsmata b s, a r ìpwc autìc prokôptei metˆ apì to CSP a r refinement (2.2.10) Antikajist ntac ton pðnaka R s r ìpwc prosdiorðsthke apì thn exðswsh ( ) stic exis seic ( ) kai ( ) èqoume: [ ] (a B r = [I r r, G r s], A r = a r b r r r + s G r sa s s)b s r ( ) B s = [0 s r, a s r(b r rg r s + b r s)] + a s sb s, A s = 0 s r [ G r s I s s ] ( ) Blèpoume pwc ta dianôsmata B r, A s, B s, A r den parˆgoun ton Ðdio upoq ro me ta dianôsmata b r, a s, b s, a r H exðswsh ( ) mporeð me th qr sh thc exis sewc metatrop c ( ) na mac d sei ènan akrib prosdiorismì thc AAP ìpwc kai to CSP b r refinement, ìqi ìmwc kai tou AS ìpwc autì prosdiorðzetai apì thn CSP. Autì mporeð na gðnei ìtan kai oi dôo exis seic metatrop c ( ) kai ( ) qrhsimopoioôntai, opìte ta dianôsmata twn dôo mejìdwn sundèontai me tic sqèseic:

24 KEFŸALAIO 2. O ALGŸORIJMOS CSP 24 [ B r B s ] = [A r A s ] = [a r a s ] [ (b r r) 1 0 r s 0 s r a s s [ b r r 0 r s 0 s r (a s s) 1 ] [ b r ] b s ] ( ) ( ) Blèpoume pwc ta dianôsmata A r, A s, B r, B s eðnai Ðdia me ta CSP dianôsmata bˆshc a r, a s, b r, b s. Ta kainoôrgia autˆ dianôsmata orðzoun thn tropopoihmènh mèjodo CSP h opoða mporeð na efarmosteð mìno ìtan to diˆnusma jèsewc y diameristeð katˆllhla stic s kai z sunist sec. kai R s r Gia ton prosdiorismì thc AAP kai tou AS eðnai aparaðthtoc o upologismìc twn pinˆkwn G r s Upologismìc tou pðnaka G r s O pðnakac G r s mporeð na upologisteð apì thn parag gish thc adiˆstathc exðswshc Br g = 0 wc proc to qrìno ìpou L r rf r + L r sf s = 0 ( ) L r r = ( dbr dt L r s = ( dbr dt + B r J)A r ( ) + B r J)A s ( ) apì ta opoða èqoume: db r dt + B r J = L r rb r + L r sb s ( ) Pˆnw sthn AAP èqoume F r = B r g = 0. Dedomènou oti F s 0 h exðswsh ( ) mac dðnei L r s = 0 r s. Opìte h exðswsh ( ) gðnetai: db r dt + B r J = L r rb r ( ) Jètoume: [ J r J= rj r s J s rj s s ] ( ) Opìte apì thn exðswsh ( ) kai thn ( ) èqoume: h parapˆnw exðswsh mac dðnei: [ ] 0 r r dgr s + [J r r G r dt sj s r, J r s G r sj s s] = L r r [I r r, G r s] ( )

25 KEFŸALAIO 2. O ALGŸORIJMOS CSP 25 J r r G r sj s r = L r r ( ) dg r s dt + G r sj s s J r s = L r rg r s ( ) Opìte mporoôme na pˆroume ton pðnaka G r s apì thn parakˆtw exðswsh: dg r s dt + G r sj s s J r s = [J r r G r sj s r] G r s ( ) h opoða mporeð na lujeð epanalhptikˆ wc ex c: [ ] G r s (n + 1) = (J r r G r s (n) J s r) 1 G r s (n) J s s J r s + dgr s (n) dt ( ) Upologismìc tou pðnaka R s r H exèlixh twn arg n plat n F s sto qrìno dðdetai apì thn exðswsh: df s dt = L s rf r + L s sf s ( ) ìpou L s r = ( dbs dt + B s J)A r ( ) L s s = ( dbs dt + B s J)A s ( ) Efìson oi gr gorec qronoklðmakec den epidroôn sthn exèlixh twn arg n plat n F s ja èqoume L s r = 0 s r. Opìte anˆloga me thn exðswsh ( ) prokôptei h akìloujh exðswsh gia to A r : - da r dt + JA r = A r rl r r ( ) Opìte apì ( ) kai ( ) mporoôme na pˆroume gia ton pðnaka R s r exðswsh: thn parakˆtw dr s r dt + J s r(i r r G r sr s r) J s sr s r = R s rl r r ( ) h opoða mporeð na lujeð epanalhptikˆ wc ex c: ìpou R s r(j + 1) = [J s sr s r(j) J s r(i r r G r s(n)r s r(j)) drs r(j) ](L r dt r(n)) 1 ( ) L r r(n) = J r r G r s(n)j s r ( )

26 KEFŸALAIO 2. O ALGŸORIJMOS CSP Qr sh tou pointer gia thn eklog twn gr gorwn metablht n O pointer mac deðqnei poioð apì touc arqikoôc ˆxonec twn metablht n pèftoun ston gr goro upoq ro. Oi M gr gorec metablhtèc epishmaðnontai apì tic megalôterec timèc tou pointer o opoðoc prosdiorðzetai apì thn akìloujh exðswsh gia th mèjodo CSP: q r = diag[ a rb r M ] = 1 M Gia thn tropopoihmènh mèjodo CSP èqoume: Q r = diag[ A rb r M ] = 1 M [ diag(a r r b r r) diag(a s rb r s) Lìgw twn sqèsewn orjogwniìthtac ( ) ja èqoume: ìpou 1 < i < M kai 1 < j < N M ] [ diag(i r r G r sr s r) diag(r s rg r s) ] ( ) ( ) 1 M [ (a r rb r r) i i + (a s rb r s) j j ] = 1 ( ) Timèc tou a i k bk i pou eðnai kontˆ sth monˆda prosdiorðzoun th i gr gorh metablht pou sundèetai me thn k qronoklðmaka.

27 Kefˆlaio 3 SÔsthma dôo Suzeugmènwn Talantwt n 3.1 Perigraf tou probl matoc H diˆtaxh tou probl matoc pou ja mac apasqol sei faðnetai sto Sq. 5 Sel. 14. Perilambˆnei dôo katakìrufouc talantwtèc (1) kai (2) oi opoðoi sundèontai me èna trðto keklimèno talantwt o opoðoc brðsketai upì gwnða, prˆgma to opoðo kˆnei to sôsthma mh grammikì. O katakìrufoc talantwt c (1) èqei mˆza m 1, stajerˆ elathrðou k 1, suntelest apìsbeshc c 1 kai suqnìthta ω 1. H metatìpis tou kai h taqôthtˆ tou eðnai antðstoiqa u 1 kai u 1. O katakìrufoc talantwt c (2) èqei mˆza m 2, stajerˆ elathrðou k 2, suntelest apìsbeshc c 2 kai suqnìthta ω 2. H metatìpis tou kai h taqôthtˆ tou eðnai antðstoiqa u 2 kai u 2. O keklimènoc talantwt c èqei stajerˆ elathrðou k, suntelest apìsbeshc c, m koc L kai brðsketai upì gwnða φ 0, h opoða gðnetai φ katˆ thn kðnhsh tou sust matoc. H orizìntia apìstash metaxô twn dôo talantwt n eðnai D kai h katakìrufh apìstash metaxô twn dôo maz n eðnai L. JewroÔme ìti h exwterik dônamh tou sust matoc eðnai F 1, en F eðnai h dônamh pou anaptôssetai ston keklimèno talantwt. 3.2 Prosdiorismìc Paramètrwn Stajerèc elathrðwn k, k 1, k 2 Jèloume k < k 1, k 2 kai k 1 > k 2 prokeimènou na montelopoi soume sust mata ìpwc autˆ pou anafèrontai ston prìlogo kai na èqoume enallag sthn kðnhsh twn dôo talantwt n, dhlad stˆsh tou enìc talantwt ìtan o ˆlloc talant netai me to mègisto plˆtoc. JewroÔme tic timèc k = 1, k 1 = 10, k 2 = 5 27

28 KEFŸALAIO 3. SŸUSTHMA DŸUO SUZEUGMŸENWN TALANTWTŸWN Mˆzec m 2, m 1 Jèloume m 2 > m 1 prokeimènou na montelopoi soume sust mata ìpwc autˆ pou anafèrontai ston prìlogo kai na èqoume enallag sthn kðnhsh twn dôo talantwt n, dhlad stˆsh tou enìc talantwt ìtan o ˆlloc talant netai me to mègisto plˆtoc. JewroÔme gia tic mˆzec tic timèc m 1 = 0.2, m 2 = Suqnìthtec ω 1, ω 2 Apì tic parapˆnw timèc twn m 2, m 1, kai k 1, k 2 prokôptoun oi mh suzeugmènec suqnìthtec: ω 1 = ω 2 = k1 = 50 = 7.07 m 1 k2 5 = m 2 3 = 1.29 Efìson ω 2 << ω 1, èqoume èna eôkampto-ˆkampto sôsthma Aposbest rec JewroÔme gia touc parˆgontec apìsbeshc tic timèc ζ 1, ζ 2 = 0.01 kai upologðzoume touc suntelestèc apìsbeshc c 1, c 2 apì tic exis seic: GewmetrÐa c 1 = 2ω 1 ζ 1 m 1 = 0.03 c 2 = 2ω 2 ζ 2 m 2 = 0.08 Sqetikˆ me th gewmetrða ìpwc aut faðnetai sto Sq. 5 jewroôme ta akìlouja: D = 1 5, L = Exwterik Fìrtish F 1 H exwterik fìrtish èqei hmitonoeid morf kai dðdetai apì exðswsh: F 1 = Asin(Ωt) Sthn perðptws mac jewroôme pwc den èqoume exwterik fìrtish.

29 KEFŸALAIO 3. SŸUSTHMA DŸUO SUZEUGMŸENWN TALANTWTŸWN Timèc paramètrwn p i SÔmfwna me ìsa èqoun anaferjeð stic paragrˆfouc èwc apodðdontai stic paramètrouc p i oi timèc pou faðnontai stouc PÐnakec 1 kai 2. Timèc paramètrwn p 1 = k p 2 = k 1 p 3 = k 2 p 4 = m 1 p 5 = m Pinakac 1. Timèc twn paramètrwn p 1 p 5 tou probl matoc Timèc paramètrwn p 6 = ζ 1 p 7 = ζ 2 p 8 = D p 9 = L p 10 = A Pinakac 2. Timèc twn paramètrwn p 6 p 10 tou probl matoc Apì tic timèc twn Pinˆkwn 1 kai 2 prokôptoun akoloôjwc oi timèc twn paramètrwn p i (i = 11 15) oi opoðec eðnai sunduasmìc twn paramètrwn p i (i = 1 10). Oi timèc autèc twn paramètrwn p i (i = 1 15) ja qrhsimopoihjoôn sth majhmatik montelopoðhsh. p 11 = p 12 = p2 k1 = = ω 1 = 50 = 7.07 p 4 m 1 p3 k2 5 = = ω 2 = p 5 m 2 3 = 1.29 p 13 = 2p 11 p 6 p 4 = 2ω 1 ζ 1 m 1 = c 1 = 0.03 p 14 = 2p 12 p 7 p 5 = 2ω 2 ζ 2 m 2 = c 2 = 0.08 p 15 = p p 2 9 = D 2 + DL 2 = L = SÔndesh twn tim n twn paramètrwn m 1, m 2, k 1, k 2 me ta sust mata pou montelopoioôntai Oi parˆmetroi m 1, m 2, k 1, k 2 ìpwc prosdiorðsthkan stic prohgoômenec paragrˆfouc sqetðzontai me tic diatˆxeic pou anafèrontai ston prìlogo kai mporoôn na montelopoihjoôn apì to sôsthma twn dôo suzeugmènwn talantwt n wc akoloôjwc: 1) Kinht rac pou eðnai edrasmènoc mèsw elastikoô sundèsmou se èna ploðo se èna autokðnhto. Oi m 1, k 1 mporoôn na eðnai h mˆza kai h stajerˆ elathrðou tou kinht ra, en oi m 2, k 2 h mˆza kai h stajerˆ elathrðou tou ploðou tou autokin tou. 2) SÔsthma anˆrthsh-amˆxwma enìc autokin tou. Oi m 1, k 1 mporoôn na eðnai h mˆza kai h stajerˆ elathrðou thc anˆrthshc, en oi m 2, k 2 h mˆza kai h stajerˆ elathrðou tou autokin tou. 3) Axonikì sôsthma (ˆxonac - propèla) enìc ploðou. Oi m 1, k 1 mporoôn na eðnai h mˆza kai h stajerˆ elathrðou tou axonikoô sust matoc, en oi m 2, k 2 h mˆza kai h stajerˆ elathrðou thc propèlac.

30 KEFŸALAIO 3. SŸUSTHMA DŸUO SUZEUGMŸENWN TALANTWTŸWN 30 4) SÔsthma ploðo-plwt exèdra antl sewc petrelaðou ìtan eðnai sundedemèna. Oi m 1, k 1 mporoôn na eðnai h mˆza kai h stajerˆ elathrðou tou ploðou, en oi m 2, k 2 h mˆza kai h stajerˆ elathrðou thc plwt c exèdrac. 5) SÔsthma ploðo(montelopoieðtai wc elastik dokìc)-jˆlassa. Oi m 1, k 1 mporoôn na eðnai h mˆza kai h stajerˆ elathrðou tou ploðou wc elastik dokìc, en oi m 2, k 2 h mˆza kai h stajerˆ elathrðou thc jˆlassac. 6) Metallik kataskeu enìc ploðou h opoða apoteleðtai apì enisqutikˆ ta opoða uposthrðzontai apì enisqumènouc dokoôc. Oi m 1, k 1 mporoôn na eðnai h mˆza kai h stajerˆ elathrðou thc enisqumènhc dokoô, en oi m 2, k 2 h mˆza kai h stajerˆ elathrðou twn enisqutik n. 3.4 Arqikèc sunj kec Jètoume se kðnhsh to sôsthma twn dôo talantwt n metatopðzontac th mˆza m 1 apì th jèsh isorropðac thc se mia kainoôrgia jèsh kai af nontˆc th na kinhjeð me mhdenik arqik taqôthta. Oi timèc twn metablht n faðnontai ston PÐnaka 3 ìpou jewroôme ìti h mˆza m 1 èqei arqik metatìpish 1. Timèc metablht n x 1 = u 1 x 2 = u 1 x 3 = u 2 x 4 = u Pinakac 3. Timèc metablht n tou probl matoc 3.5 Majhmatik montelopoðhsh tou Probl matoc Oi dôo mˆzec sundèontai me grammikì elat rio upì gwnða φ. H klðsh aut dhmiourgeð th mh grammikìthta. H majhmatik montelopoðhsh tou sust matoc twn dôo suzeugmènwn talantwt n èqei wc akoloôjwc: ExÐswsh kðnhshc tou talantwt 1: ExÐswsh kðnhshc tou talantwt 2: m 1 ü 1 + k 1 u 1 + c 1 u 1 = F (u, u) cos φ + F 1 (3.5.1) m 2 ü 2 + k 2 u 2 + c 2 u 2 = F (u, u) cos φ (3.5.2) An φ eðnai h klðsh tou orizìntiou talantwt èqoume: cos φ = (D + u 2 u 1 )/(L + u) (3.5.3) An L eðnai arqikˆ to m koc tou keklimènou talantwt èqoume: L 2 = D 2 + ( L) 2 (3.5.4) An L + u eðnai to m koc tou keklimènou talantwt metˆ thn ekkðnhsh thc talˆntwshc èqoume: (L + u) 2 = ( L) 2 + (D + u 2 u 1 ) 2 (3.5.5)

31 KEFŸALAIO 3. SŸUSTHMA DŸUO SUZEUGMŸENWN TALANTWTŸWN 31 Opìte apì (3.5.3) kai (3.5.5) èqoume: cos φ = D + u 2 u 1 (D + u 2 u 1 ) 2 + ( L) 2 (3.5.6) Apì (3.5.5) èqoume: L + u = (D + u 2 u 1 ) 2 + ( L) 2 u = (D + u 2 u 1 ) 2 + ( L) 2 L (3.5.7) JewroÔme ìti h apìsbesh c tou keklimènou talantwt kai h exwterik dônamh F 1 eðnai 0. Opìte h dônamh F pou anaptôssetai ston keklimèno talantwt ja eðnai: Apì (3.5.6), (3.5.7), (3.5.8), èqoume: F (u, u) = ku + c u = ku (3.5.8) F (u, u) cos φ = (k (D + u 2 u 1 ) 2 + ( L) 2 D + (u 2 u 1 ) kl) (D + u 2 u 1 ) 2 + ( L) 2 F (u, u) cos φ = k(d + u 2 u 1 ) kl(d + u 2 u 1 ) (D + u 2 u 1 ) 2 + ( L) 2 F (u, u) cos φ = k(d + u 2 u 1 )(1 L (D + u 2 u 1 ) 2 + ( L) 2 ) (3.5.9) OrÐzoume tic metablhtèc x 1, x 2, x 3, x 4 tou probl matoc c akoloôjwc: Metatìpish tou talantwt 1: x 1 = u 1 TaqÔthta tou talantwt 1: x 2 = u 1 Metatìpish tou talantwt 2: x 3 = u 2 TaqÔthta tou talantwt 2: x 4 = u 2 Opìte èqoume: Apì (3.5.1), (3.5.9) èqoume: ẋ 1 = u 1 = x 2 (3.5.10) ẋ 2 = ü 1 = F (u, u) cos φ + F 1 k 1 u 1 c 1 u 1 = F (u, u) cos φ + F 1 k 1 x 1 c 1 x 2 m 1 m 1 ẋ 2 = k L (D + x 3 x 1 )(1 ) + F 1 k 1 x 1 c 1 x 2 m 1 (D + x 3 x 1 ) 2 + ( L) 2 m 1 m 1 m 1 (3.5.11)

32 KEFŸALAIO 3. SŸUSTHMA DŸUO SUZEUGMŸENWN TALANTWTŸWN 32 Apì (3.5.2), (3.5.9) èqoume: ẋ 3 = u 2 = x 4 (3.5.12) Jètoume: ẋ 4 = ü 2 = F (u, u) cos φ k 2u 2 c 2 u 2 m 2 = F (u, u) cos φ k 2x 3 c 2 x 4 ẋ 4 = k m 2 (D + x 3 x 1 )(1 m 2 L ) k 2 (D + x 3 x 1 ) 2 + ( L) 2 x 3 c 2 x 4 (3.5.13) m 2 m 2 z 0 = (p 8 + x 3 x 1 ) 2 + p 2 9 z 1 = 1 p 15 z0 z 2 = p 8 + x 3 x 1 Opìte gia tic timèc twn paramètrwn p 1 p 15 kai gia ta z 0, z 1, z 2 èqoume: q 1 = ẋ 1 = u 1 = x 2 (3.5.14) ( ) ( ) ( ) p1 p2 p13 q 2 = ẋ 2 = ü 1 = z 2 z 1 x 1 x 2 (3.5.15) p 4 p 5 p 4 q 3 = ẋ 3 = u 2 = x 4 (3.5.16) ( ) ( ) ( ) p1 p3 p14 q 4 = ẋ 4 = ü 2 = z 2 z 1 x 3 x 4 (3.5.17) 'Estw y to N-diˆstato diˆnusma twn exarthmènwn metablht n. p 5 p 4 p 5 y= x 1 x 2 x 3 x 4 (3.5.18) An g eðnai mia algebrik exðswsh tou y tìte apì (3.5.14), (3.5.15), (3.5.16), (3.5.17), (3.5.18) èqoume: dy dt = g(y) = q 1 q 2 q 3 q 4 (3.5.19) Autì eðnai to sôsthma twn tessˆrwn diaforik n exis sewn to opoðo perigrˆfei thn exèlixh tou fainìmenou.

33 Kefˆlaio 4 DiereÔnhsh 4.1 'Elegqoc idiotim n Iakwbian c Genikˆ Me ton k dika LSODE kai gia tic timèc twn paramètrwn p 1 p 10 ìpwc autèc orðsthkan sthn parˆgrafo upologðzoume tic idiotimèc λ k = λ kr ± iλ ki thc Iakwbian c (J = grad(g)) tou sust matoc, ìpou h g orðsthke sthn parˆgrafo 3.5. g(y) = q 1 q 2 q 3 q 4 J= q 1 q 1 q 1 q 1 x 1 x 2 x 3 x 4 q 2 q 2 q 2 q 2 x 1 x 2 x 3 x 4 q 3 q 3 q 3 q 3 x 1 x 2 x 3 x 4 q 4 q 4 q 4 q 4 x 1 x 2 x 3 x 4 Oi gr gorec qronoklðmakec enìc dôskamptou probl matoc, oi opoðec eðnai upeôjunec gia tic exisorrop seic pou anaptôssontai metaxô twn diˆforwn diadikasi n tou probl matoc, sqetðzontai me tic idiotimèc thc Iakwbian c tou sust matoc to pragmatikì mèroc twn opoðwn eðnai arnhtikì kai kat' apìluth tim polô megalôtero apì to fantastikì. Sto prìblhmˆ mac ja prospaj soume na epanaprosdiorðsoume timèc twn paramètrwn p 1 p 10 ètsi ste na petôqoume idiotimèc thc Iakwbian c thc morf c pou perigrˆyame parapˆnw prokeimènou na èqoume èna dôskampto prìblhma to opoðo na qarakthrðzetai apì thn Ôparxh arg n kai gr gorwn aposbetik n qronoklimˆkwn oi opoðec na teðnoun gr gora sto mhdèn. Katìpin ja efarmìsoume thn tropopoihmènh mèjodo CSP h opoða ekmetalleuìmenh thn exˆntlhsh twn gr gorwn qronoklimˆkwn ja mac odhg sei ston prosdiorismì thc AAP kai katˆ sunèpeia sth lôsh enìc probl matoc mikrìterhc diˆstashc. 33

34 KEFŸALAIO 4. DIEREŸUNHSH Arqikì prìblhma Sto arqikì prìblhma oi timèc twn paramètrwn p 1 p 10 eðnai autèc pou anafèrontai sthn parˆgrafo kai oi idiotimèc pou paðrnoume faðnontai sthn pr th seirˆ tou PÐnaka 20. Blèpoume pwc oi idiotimèc pou paðrnoume den eðnai thc morf c pou jèloume, ìpwc aut prosdiorðsthke sthn parˆgrafo Prokeimènou na pˆroume idiotimèc thc zhtoômenhc morf c ja proqwr soume se metabolèc twn paramètrwn p 1 p 10 (Peript seic 1-4) StoÔc PÐnakec 4, 5, 6 parousiˆzontai oi timèc twn paramètrwn p 1 p 15 Timèc paramètrwn (Arqikì prìblhma) p 1 = k p 2 = k 1 p 3 = k 2 p 4 = m 1 p 5 = m Pinakac 4. Timèc twn paramètrwn p 1 p 5 tou arqikoô probl matoc Timèc paramètrwn (Arqikì prìblhma) p 6 = ζ 1 p 7 = ζ 2 p 8 = D p 9 = L p 10 = A Pinakac 5. Timèc twn paramètrwn p 5 p 10 tou arqikoô probl matoc Timèc paramètrwn (Arqikì prìblhma) p 11 = ω 1 p 12 = ω 2 p 13 = c 1 p 14 = c 2 p 15 = L Pinakac 6. Timèc twn paramètrwn p 10 p 15 tou arqikoô probl matoc PerÐptwsh 1 Arqikˆ auxˆnoume kai elatt noume kˆje parˆmetro katˆ treic (3) tˆxeic megèjouc ( 10 3 ) kai ( 10 3 ) kai paðrnoume tic idiotimèc pou faðnontai ston PÐnaka 20. Blèpoume pwc mìno h aôxhsh twn p 6 = ζ 1, p 7 = ζ 2 mac dðnei idiotimèc thc morf c pou jèloume. Eklègoume gia thn p 6 = ζ 1 thn tim 3.00 apì 0.01 kai kratˆme Ðdiec tic ˆllec paramètrouc prˆgma to opoðo mac dðnei gia thn apìsbesh c 1 thn tim p 13 = c 1 = Ston PÐnaka 21 blèpoume ìti oi idiotimèc èqoun th morf pou jèloume, me to pragmatikì mèroc twn dôo pr twn na eðnai arnhtikì kai kat' apìluth tim polô megalôtero apì to fantastikì, en oi epìmenec dôo idiotimèc sqhmatðzoun èna migadikì zeugˆri. StoÔc PÐnakec 7, 8, 9 parousiˆzontai oi timèc twn paramètrwn p 1 p 15 Timèc paramètrwn (PerÐptwsh 1) p 1 = k p 2 = k 1 p 3 = k 2 p 4 = m 1 p 5 = m Pinakac 7. Timèc twn paramètrwn p 1 p 5 thc PerÐptwshc 1 Timèc paramètrwn (PerÐptwsh 1) p 6 = ζ 1 p 7 = ζ 2 p 8 = D p 9 = L p 10 = A Pinakac 8. Timèc twn paramètrwn p 5 p 10 thc PerÐptwshc 1

35 KEFŸALAIO 4. DIEREŸUNHSH 35 Timèc paramètrwn (PerÐptwsh 1) p 11 = ω 1 p 12 = ω 2 p 13 = c 1 p 14 = c 2 p 15 = L Pinakac 9. Timèc twn paramètrwn p 10 p 15 thc PerÐptwshc PerÐptwsh 2 Gia thn tim thc paramètrou p 6 = ζ 1 auxˆnoume kai elatt noume tic ˆllec paramètrouc katˆ dôo (2) tˆxeic megèjouc ( 10 2 ) kai ( 10 2 ) kai paðrnoume tic idiotimèc pou faðnontai ston PÐnaka 22. Blèpoume pwc kurðwc h aôxhsh thc p 2 = k 1 h elˆttwsh thc p 4 = m 1 mac dðnei idiotimèc thc morf c pou jèloume. Kratˆme gia thn p 6 = ζ 1 thn tim 3.00, eklègoume gia thn p 2 = k 1 thn tim apì en kratˆme Ðdiec tic ˆllec paramètrouc. Autì mac dðnei gia thn apìsbesh c 1 kai gia th suqnìthta w 1 tic timèc p 13 = c 1 = kai p 11 = w 1 = Oi idiotimèc pou paðrnoume faðnontai ston PÐnaka 23 ìpou to pragmatikì mèroc twn dôo pr twn idiotim n eðnai arnhtikì kai kat' apìluth tim polô megalôtero apì to fantastikì, en oi epìmenec dôo idiotimèc sqhmatðzoun èna migadikì zeugˆri. Se sôgkrish me thn PerÐptwsh 1 h diaforˆ metaxô pragmatikoô kai fantastikoô mèrouc eðnai saf c megalôterh gia kˆje idiotim, kaj c epðshc megalôterh eðnai kai h diaforˆ metaxô twn pragmatik n mer n twn antðstoiqwn idiotim n. StoÔc PÐnakec 10, 11, 12 parousiˆzontai oi timèc twn paramètrwn p 1 p 15 Timèc paramètrwn (PerÐptwsh 2) p 1 = k p 2 = k 1 p 3 = k 2 p 4 = m 1 p 5 = m Pinakac 10. Timèc twn paramètrwn p 1 p 5 thc PerÐptwshc 2 Timèc paramètrwn (PerÐptwsh 2) p 6 = ζ 1 p 7 = ζ 2 p 8 = D p 9 = L p 10 = A Pinakac 11. Timèc twn paramètrwn p 5 p 10 thc PerÐptwshc 2 Timèc paramètrwn (PerÐptwsh 2) p 11 = ω 1 p 12 = ω 2 p 13 = c 1 p 14 = c 2 p 15 = L Pinakac 12. Timèc twn paramètrwn p 10 p 15 thc PerÐptwshc PerÐptwsh 3 Gia tic timèc twn paramètrwn p 2 = k 1 kai p 6 = ζ 1 auxˆnoume kai elatt noume tic ˆllec paramètrouc katˆ dôo (2) tˆxeic megèjouc ( 10 2 ) kai ( 10 2 ) kai paðrnoume tic idiotimèc pou faðnontai ston PÐnaka 24. Blèpoume pwc kurðwc h elˆttwsh thc p 4 = m 1 mac dðnei idiotimèc thc morf c pou jèloume. Kratˆme gia thn p 6 = ζ 1 thn tim 3.00, gia thn p 2 = k 1 thn tim kai eklègoume gia thn p 4 = m 1 thn tim 0.02 apì 0.20 en kratˆme Ðdiec tic ˆllec paramètrouc. Autì mac dðnei gia thn apìsbesh c 1 kai gia th suqnìthta w 1 tic timèc p 13 = c 1 = 8.48 kai p 11 = w 1 =

36 KEFŸALAIO 4. DIEREŸUNHSH 36 Oi idiotimèc pou paðrnoume faðnontai ston PÐnaka 25 ìpou to pragmatikì mèroc twn dôo pr twn idiotim n eðnai arnhtikì kai kat' apìluth tim polô megalôtero apì to fantastikì, en oi epìmenec dôo idiotimèc sqhmatðzoun èna migadikì zeugˆri. Se sôgkrish me thn PerÐptwsh 2 h diaforˆ metaxô pragmatikoô kai fantastikoô mèrouc eðnai saf c megalôterh gia kˆje idiotim, kaj c epðshc megalôterh eðnai kai h diaforˆ metaxô twn pragmatik n mer n twn antðstoiqwn idiotim n. Pio sugkekrimèna oi diaforèc eðnai megalôterec apì tic antðstoiqec diaforèc metaxô twn Peript sewn 1 kai 2. StoÔc PÐnakec 13, 14, 15 parousiˆzontai oi timèc twn paramètrwn p 1 p 15 Timèc paramètrwn (PerÐptwsh 3) p 1 = k p 2 = k 1 p 3 = k 2 p 4 = m 1 p 5 = m Pinakac 13. Timèc twn paramètrwn p 1 p 5 thc PerÐptwshc 3 Timèc paramètrwn (PerÐptwsh 3) p 6 = ζ 1 p 7 = ζ 2 p 8 = D p 9 = L p 10 = A Pinakac 14. Timèc twn paramètrwn p 5 p 10 thc PerÐptwshc 3 Timèc paramètrwn (PerÐptwsh 3) p 11 = ω 1 p 12 = ω 2 p 13 = c 1 p 14 = c 2 p 15 = L Pinakac 15. Timèc twn paramètrwn p 10 p 15 thc PerÐptwshc PerÐptwsh 4 Exetˆzoume thn perðptwsh ìpou diathroôme thn arqik tim tou parˆgonta apìsbeshc p 6 = ζ 1 kai allˆzoume tic timèc twn p 2 = k 1 kai p 4 = m 1. DiathroÔme dhlad thn arqik tim thc apìsbeshc p 11 = c 1 = 0.03 kai auxˆnoume thn tim thc suqnìthtac p 11 = ω 1 = Ston PÐnaka 26 blèpoume pwc den paðrnoume timèc twn idiotim n thc morf c pou jèloume. 'Ara h aôxhsh thc apìsbeshc eðnai aparaðthth prokeimènou na èqoume èna prìblhma thc morf c pou perigrˆfetai sthn parˆgrafo StoÔc pðnakec 16, 17, 18 parousiˆzontai oi timèc twn paramètrwn p 1 p 15 Timèc paramètrwn (PerÐptwsh 4) p 1 = k p 2 = k 1 p 3 = k 2 p 4 = m 1 p 5 = m Pinakac 16. Timèc twn paramètrwn p 1 p 5 thc PerÐptwshc 4 Timèc paramètrwn (PerÐptwsh 4) p 6 = ζ 1 p 7 = ζ 2 p 8 = D p 9 = L p 10 = A Pinakac 17. Timèc twn paramètrwn p 5 p 10 thc PerÐptwshc 4

37 KEFŸALAIO 4. DIEREŸUNHSH 37 Timèc paramètrwn (PerÐptwsh 4) p 11 = ω 1 p 12 = ω 2 p 13 = c 1 p 14 = c 2 p 15 = L Pinakac 18. Timèc twn paramètrwn p 10 p 15 thc PerÐptwshc Sumpèrasma Genikˆ ìpwc mporoôme na doôme ston PÐnaka 19 h aôxhsh tou parˆgonta apìsbeshc ζ 1 h opoða odhgeð se aôxhsh tou suntelest apìsbeshc c 1, h aôxhsh thc stajerˆc elathrðou k 1 kai h elˆttwsh thc mˆzac m 1 pou odhgoôn se aôxhsh thc suqnìthtac w 1, eðnai proc thn kateôjunsh twn susthmˆtwn pou anafèrjhkan ston prìlogo kai sumfwnoôn me thn anˆlush thc paragrˆfou sqetikˆ me th zhtoômenh morf twn idiotim n. Pio sugkekrimèna ìpwc mporoôme na doôme stic Peript seic 1-4 kai ston PÐnaka 19 h aôxhsh thc apìsbeshc c 1 mporeð apì mình thc na mac d sei sôsthma thc morf c pou jèloume, kˆti to opoðo den isqôei gia thn aôxhsh thc suqnìthtac w 1. Ston PÐnaka 19 parousiˆzontai oi metabolèc twn paramètrwn gia ìlec tic Peript seic. Peript seic Timèc paramètrwn p 6 = ζ 1 p 2 = k 1 p 4 = m 1 p 13 = c 1 p 11 = ω 1 c 1 /c 2 ω 1 /ω 2 Arqik perðptwsh PerÐptwsh PerÐptwsh PerÐptwsh PerÐptwsh Pinakac 19. Metabolèc twn paramètrwn gia ìlec tic Peript seic

38 Τιμές παραμέτρων ΙΔΙΟΤΙΜΕΣ ΙΑΚΩΒΙΑΝΗΣ λ 1r λ 1i λ 2r λ 2i λ 3r λ 3i λ 4r λ 4i Αρχικό πρόβλημα -7.07Ε Ε Ε Ε Ε Ε Ε Ε+00 p 1 = k(x10 3 ) -6.72Ε Ε Ε Ε Ε Ε Ε Ε+00 p 1 = k(x10 3 ) -7.07Ε Ε Ε Ε Ε Ε Ε Ε+00 p 2 = k 1 (x10 3 ) -2.24Ε Ε Ε Ε Ε Ε Ε Ε+00 p 2 = k 1 (x10 3 ) -1.16Ε Ε Ε Ε Ε Ε Ε Ε+00 p 3 = k 2 (x10 3 ) -4.08Ε Ε Ε Ε Ε Ε Ε Ε+00 p 3 = k 2 (x10 3 ) -7.07Ε Ε Ε Ε Ε Ε Ε Ε-01 p 4 = m 1 (x10 3 ) -1.29Ε Ε Ε Ε Ε Ε Ε Ε-01 p 4 = m 1 (x10 3 ) -2.24Ε Ε Ε Ε Ε Ε Ε Ε+00 p 5 = m 2 (x10 3 ) -7.07Ε Ε Ε Ε Ε Ε Ε Ε-02 p 5 = m 2 (x10 3 ) -4.08Ε Ε Ε Ε Ε Ε Ε Ε+00 p 6 = ζ 1 (x10 3 ) -1.41Ε Ε Ε Ε Ε Ε Ε Ε+00 p 6 = ζ 1 (x10 3 ) -1.29Ε Ε Ε Ε Ε Ε Ε Ε+00 p 7 = ζ 2 (x10 3 ) -2.57Ε Ε Ε Ε Ε Ε Ε Ε+00 p 7 = ζ 2 (x10 3 ) -7.07Ε Ε Ε Ε Ε Ε Ε Ε+00 p 8 = D(x10 3 ) -7.07Ε Ε Ε Ε Ε Ε Ε Ε+00 p 8 = D(x10 3 ) -7.07Ε Ε Ε Ε Ε Ε Ε Ε+00 p 9 = L(x10 3 ) -7.07Ε Ε Ε Ε Ε Ε Ε Ε+00 p 9 = L(x10 3 ) -7.07Ε Ε Ε Ε Ε Ε Ε Ε+00 p 10 = A(x10 3 ) -7.07Ε Ε Ε Ε Ε Ε Ε Ε+00 p 10 = A(x10 3 ) -7.07Ε Ε Ε Ε Ε Ε Ε Ε+00 Pinakac 20. Idiotimèc Iakwbian c gia metabol twn paramètrwn p 1 p 10 katˆ treðc (3) tˆxeic megèjouc. ( 10 ±3 ) Τιμές παραμέτρων ΙΔΙΟΤΙΜΕΣ ΙΑΚΩΒΙΑΝΗΣ (Περίπτωση 1) λ 1r λ 1i λ 2r λ 2i λ 3r λ 3i λ 4r λ 4i p 6 = ζ 1 = Ε Ε Ε Ε Ε Ε Ε Ε+00 PÐnakac 21. Idiotimèc Iakwbian c gia thn PerÐptwsh 1 (p 6 = ζ 1 = 3.00) KEFŸALAIO 4. DIEREŸUNHSH 38

39 Τιμές παραμέτρων ΙΔΙΟΤΙΜΕΣ ΙΑΚΩΒΙΑΝΗΣ λ 1r λ 1i λ 2r λ 2i λ 3r λ 3i λ 4r λ 4i p 6 = ζ 1 = Ε Ε Ε Ε Ε Ε Ε Ε+00 p 1 = k(x10 2 ) -1.99Ε Ε Ε Ε Ε Ε Ε Ε+00 p 1 = k(x10 2 ) -4.12Ε Ε Ε Ε Ε Ε Ε Ε+00 p 2 = k 1 (x10 2 ) -4.12Ε Ε Ε Ε Ε Ε Ε Ε+00 p 2 = k 1 (x10 2 ) -2.04Ε Ε Ε Ε Ε Ε Ε Ε+00 p 3 = k 2 (x10 2 ) -4.11Ε Ε Ε Ε Ε Ε Ε Ε+01 p 3 = k 2 (x10 2 ) -4.11Ε Ε Ε Ε Ε Ε Ε Ε-01 p 4 = m 1 (x10 2 ) -4.11Ε Ε Ε Ε Ε Ε Ε Ε+00 p 4 = m 1 (x10 2 ) -4.11Ε Ε Ε Ε Ε Ε Ε Ε+00 p 5 = m 2 (x10 2 ) -4.11Ε Ε Ε Ε Ε Ε Ε Ε-01 p 5 = m 2 (x10 2 ) -4.11Ε Ε Ε Ε Ε Ε Ε Ε+01 p 7 = ζ 2 (x10 2 ) -4.11Ε Ε Ε Ε Ε Ε Ε Ε+00 p 7 = ζ 2 (x10 2 ) -4.11Ε Ε Ε Ε Ε Ε Ε Ε+00 p 8 = D(x10 2 ) -4.11Ε Ε Ε Ε Ε Ε Ε Ε+00 p 8 = D(x10 2 ) -4.11Ε Ε Ε Ε Ε Ε Ε Ε+00 p 9 = L(x10 2 ) -4.12Ε Ε Ε Ε Ε Ε Ε Ε+00 p 9 = L(x10 2 ) -4.11Ε Ε Ε Ε Ε Ε Ε Ε+00 p 10 = A(x10 2 ) -4.11Ε Ε Ε Ε Ε Ε Ε Ε+00 p 10 = A(x10 2 ) -4.11Ε Ε Ε Ε Ε Ε Ε Ε+00 Pinakac 22. Idiotimèc Iakwbian c gia metabol twn paramètrwn ektìc thc p 6 katˆ dôo (2) tˆxeic megèjouc. ( 10 ±2 ) Τιμές παραμέτρων p 2 = k 1 = 100 p 6 = ζ 1 = 3 ΙΔΙΟΤΙΜΕΣ ΙΑΚΩΒΙΑΝΗΣ (Περίπτωση 2) λ 1r λ 1i λ 2r λ 2i λ 3r λ 3i λ 4r λ 4i -1.30Ε Ε Ε Ε Ε Ε Ε Ε+00 PÐnakac 23. Idiotimèc Iakwbian c gia thn PerÐptwsh 2 (p 6 = ζ 1 = 3.00, p 2 = k 1 = ) KEFŸALAIO 4. DIEREŸUNHSH 39

40 Τιμές παραμέτρων ΙΔΙΟΤΙΜΕΣ ΙΑΚΩΒΙΑΝΗΣ λ 1r λ 1i λ 2r λ 2i λ 3r λ 3i λ 4r λ 4i p 2 = k 1 = 100 p 6 = ζ 1 = Ε Ε Ε Ε Ε Ε Ε Ε+00 p 1 = k(x10 2 ) -1.26Ε Ε Ε Ε Ε Ε Ε Ε+00 p 1 = k(x10 2 ) -1.30Ε Ε Ε Ε Ε Ε Ε Ε+00 p 3 = k 2 (x10 2 ) -1.30Ε Ε Ε Ε Ε Ε Ε Ε+01 p 3 = k 2 (x10 2 ) -1.30Ε Ε Ε Ε Ε Ε Ε Ε-01 p 4 = m 1 (x10 2 ) -1.30Ε Ε Ε Ε Ε Ε Ε Ε+00 p 4 = m 1 (x10 2 ) -1.30Ε Ε Ε Ε Ε Ε Ε Ε+00 p 5 = m 2 (x10 2 ) -1.30Ε Ε Ε Ε Ε Ε Ε Ε-01 p 5 = m 2 (x10 2 ) -1.30Ε Ε Ε Ε Ε Ε Ε Ε+01 p 7 = ζ 2 (x10 2 ) -1.30Ε Ε Ε Ε Ε Ε Ε Ε-01 p 7 = ζ 2 (x10 2 ) -1.30Ε Ε Ε Ε Ε Ε Ε Ε+00 p 8 = D(x10 2 ) -1.30Ε Ε Ε Ε Ε Ε Ε Ε+00 p 8 = D(x10 2 ) -1.30Ε Ε Ε Ε Ε Ε Ε Ε+00 p 9 = L(x10 2 ) -1.30Ε Ε Ε Ε Ε Ε Ε Ε+00 p 9 = L(x10 2 ) -1.30Ε Ε Ε Ε Ε Ε Ε Ε+00 p 10 = A(x10 2 ) -1.30Ε Ε Ε Ε Ε Ε Ε Ε+00 p 10 = A(x10 2 ) -1.30Ε Ε Ε Ε Ε Ε Ε Ε+00 Pinakac 24. Idiotimèc Iakwbian c gia metabol twn paramètrwn ektìc twn p 6, p 2 katˆ dôo (2) tˆxeic megèjouc. ( 10 ±2 ) Τιμές παραμέτρων p 6 = ζ 1 = 3, p 2 = k 1 = 100 p 4 = m 1 = 0.02 ΙΔΙΟΤΙΜΕΣ ΙΑΚΩΒΙΑΝΗΣ (Περίπτωση 3) λ 1r λ 1i λ 2r λ 2i λ 3r λ 3i λ 4r λ 4i -4.12Ε Ε Ε Ε Ε Ε Ε Ε+00 PÐnakac 25. Idiotimèc Iakwbian c gia thn PerÐptwsh 3 (p 6 = ζ 1 = 3.00, p 2 = k 1 = , p 4 = m 1 = 0.02) KEFŸALAIO 4. DIEREŸUNHSH 40

41 Τιμές παραμέτρων p 2 = k 1 = 100 p 4 = m 1 = 0.02 ΙΔΙΟΤΙΜΕΣ ΙΑΚΩΒΙΑΝΗΣ (Περίπτωση 4) λ 1r λ 1i λ 2r λ 2i λ 3r λ 3i λ 4r λ 4i -7.07Ε Ε Ε Ε Ε Ε Ε Ε+00 PÐnakac 26. Idiotimèc Iakwbian c gia thn PerÐptwsh 4 (p 2 = k 1 = , p 4 = m 1 = 0.02) KEFŸALAIO 4. DIEREŸUNHSH 41

42 KEFŸALAIO 4. DIEREŸUNHSH SÔgkrish twn Peript sewn 1,2, Genikˆ Me th bo jeia tou k dika LSODE, gia tic arqikèc sunj kec pou dðdontai sthn parˆgrafo 3.4 kai gia tic Peript seic (1,2,3) pou orðsthkan sthn parˆgrafo 4.1 ja prosdiorðsoume tic metatopðseic u 1, u 2 twn maz n m 1, m 2, tic idiotimèc λ k thc Iakwbian c, tic qronoklðmakec τ k kai ta gr gora plˆth f kr pou antistoiqoôn se kˆje metablht. Katìpin gia kˆje perðptwsh ja d soume arqikˆ tic grafikèc parastˆseic twn metatopðsewn u 1, u 2 kai katìpin, sunart sei tou qrìnou, tic grafikèc parastˆseic twn pragmatik n mer n twn idiotim n λ kr ìpou faðnontai oi metaxô touc diaforèc. AkoloujoÔn, pˆnta sunart sei tou qrìnou, oi grafikèc parastˆseic twn qronoklimˆkwn τ k ìpou blèpoume tic timèc pou paðrnoun katˆ thn exèlixh tou fainìmenou kai orðzetai h qarakthristik qronoklðmaka τ char h opoða eðnai h pio gr gorh apì tic argèc qronoklðmakec. OrÐzetai epðshc to e to opoðo eðnai o lìgoc thc pio arg c apì tic gr gorec qronoklðmakec proc thn pio gr gorh apì tic argèc τ char kai apoteleð èna mètro tou kenoô metaxô gr gorwn kai arg n qronoklimˆkwn. 'Oso megalôtero eðnai autì to kenì tìso kalôterh eðnai h akrðbeia prosdiorismoô thc AAP kai tou AS. DÐdontai epðshc oi grafikèc parastˆseic twn plat n f kr ìpou blèpoume poia apì autˆ teðnoun sto mhdèn kai me poiˆ taqôthta. Apì ta parapˆnw mporoôme na sumperˆnoume an èqoume èna dôskampto prìblhma ìpou upˆrqoun argèc kai gr gorec aposbetikèc qronoklðmakec oi opoðec exantloôntai sôntoma. Tèloc oi mh amelhtèec timèc tou CSP pointer mac epishmaðnoun tic gr gorec metablhtèc tic opoðec ofeðloume na gnwrðzoume prokeimènou na qrhsimopoi soume thn tropopoihmènh mèjodo CSP. Oi qronoklðmakec orðzontai wc: τ k = 1, λ 2 kr + λ 2 ki ìpou λ kr kai λ ki eðnai antðstoiqa to pragmatikì kai to fantastikì mèroc thc idiotim c λ k = λ kr + iλ ki. Oi idiotimèc λ k = λ kr + iλ ki eðnai oi idiotimèc thc Iakwbian c J= q 1 q 1 q 1 q 1 x 1 x 2 x 3 x 4 q 2 q 2 q 2 q 2 x 1 x 2 x 3 x 4 q 3 q 3 q 3 q 3 x 1 x 2 x 3 x 4 q 4 q 4 q 4 q 4 x 1 x 2 x 3 x 4 Ta plˆth èqoun oristeð sthn parˆgrafo 2.2 wc:

43 KEFŸALAIO 4. DIEREŸUNHSH 43 f r = b r g = [ b r r, b r s f s = b s g = [ b s r, b s s ] [ g r g s ] [ g r g s ] = b r rg r + b r sg s ] = b s rg r + b s sg s Oi metablhtèc tou probl matoc orðsthkan sthn parˆgrafo 3.5 c akoloôjwc: Metatìpish tou talantwt 1: x 1 = u 1 TaqÔthta tou talantwt 1: x 2 = u 1 Metatìpish tou talantwt 2: x 3 = u 2 TaqÔthta tou talantwt 2: x 4 = u 2 Genikˆ sta graf mata ìpou qrhsimopoioôntai qr mata isqôoun ta akìlouja: 1 = MaÔro, 2 = Kìkkino, 3 = Prˆsino, 4 = Mple Arqikì prìblhma Oi timèc twn paramètrwn kai oi idiotimèc thc Iakwbian c gia tic arqikèc sunj kec pou dðdontai sthn parˆgrafo 3.4 eðnai autèc pou prosdiorðsthkan sthn parˆgrafo (Arqikì prìblhma) kai faðnontai stouc PÐnakec 27, 28, 29 kai 30, 31 antðstoiqa. Timèc paramètrwn (Arqikì prìblhma) p 1 = k p 2 = k 1 p 3 = k 2 p 4 = m 1 p 5 = m Pinakac 27. Timèc twn paramètrwn p 1 p 5 tou arqikoô probl matoc Timèc paramètrwn (Arqikì prìblhma) p 6 = ζ 1 p 7 = ζ 2 p 8 = D p 9 = L p 10 = A Pinakac 28. Timèc twn paramètrwn p 6 p 10 tou arqikoô probl matoc Timèc paramètrwn (Arqikì prìblhma) p 11 = ω 1 p 12 = ω 2 p 13 = c 1 p 14 = c 2 p 15 = L Pinakac 29. Timèc twn paramètrwn p 10 p 15 tou probl matoc IDIOTIMES IAKWBIANHS (Arqikì prìblhma) λ 1r λ 1i λ 2r λ 2i -7.07E E E E+00 Pinakac 30. Idiotimèc Iakwbian c λ 1, λ 2 arqikoô probl matoc IDIOTIMES IAKWBIANHS (Arqikì prìblhma) λ 3r λ 3i λ 4r λ 4i -1.29E E E E+00 PÐnakac 31. Idiotimèc Iakwbian c λ 3, λ 4 arqikoô probl matoc

44 KEFŸALAIO 4. DIEREŸUNHSH 44 ParathroÔme pwc to pragmatikì mèroc twn dôo pr twn idiotim n eðnai arnhtikì allˆ kat' apìluth tim mikrìtero apì to fantastikì. Den èqoume dhlad èna dôskampto prìblhma sto opoðo upˆrqoun argèc kai gr gorec qronoklðmakec. Sto Sq. 5 parousiˆzontai oi metatopðseic u 1, u 2 twn maz n m 1, m 2. Blèpoume pwc kaj c h mˆza m 1 teðnei sth jèsh isorropðac h mˆza m 2 arqðzei thn talˆntws thc. Sq ma 5. Exèlixh sto qrìno twn metatopðsewn u 1 u 2 twn maz n m 1, m PerÐptwsh 1 Oi timèc twn paramètrwn kai oi idiotimèc thc Iakwbian c gia tic arqikèc sunj kec pou dðdontai sthn parˆgrafo 3.4 eðnai autèc pou prosdiorðsthkan sthn parˆgrafo (PerÐptwsh 1) kai faðnontai stouc PÐnakec 32, 33, 34 kai 35, 36 antðstoiqa. Timèc paramètrwn (PerÐptwsh 1) p 1 = k p 2 = k 1 p 3 = k 2 p 4 = m 1 p 5 = m Pinakac 32. Timèc twn paramètrwn p 1 p 5 gia thn PerÐptwsh 1 Timèc paramètrwn (PerÐptwsh 1) p 6 = ζ 1 p 7 = ζ 2 p 8 = D p 9 = L p 10 = A Pinakac 33. Timèc twn paramètrwn p 6 p 10 gia thn PerÐptwsh 1 Timèc paramètrwn (PerÐptwsh 1) p 11 = ω 1 p 12 = ω 2 p 13 = c 1 p 14 = c 2 p 15 = L Pinakac 34. Timèc twn paramètrwn p 10 p 15 gia thn PerÐptwsh 1 Se sqèsh me to arqikì prìblhma upˆrqei h ex c allag : p 6 = ζ 1 :

45 KEFŸALAIO 4. DIEREŸUNHSH 45 IDIOTIMES IAKWBIANHS (PerÐptwsh 1) λ 1r λ 1i λ 2r λ 2i -4.11E E E E+00 Pinakac 35. Idiotimèc Iakwbian c λ 1, λ 2 gia thn PerÐptwsh 1 IDIOTIMES IAKWBIANHS (PerÐptwsh 1) λ 3r λ 3i λ 4r λ 4i -1.85E E E E+00 Pinakac 36. Idiotimèc Iakwbian c λ 3, λ 4 gia thn PerÐptwsh 1 ParathroÔme pwc to pragmatikì mèroc twn dôo pr twn idiotim n eðnai arnhtikì kai kat' apìluth tim megalôtero apì to fantastikì, en oi epìmenec dôo idiotimèc sqhmatðzoun èna migadikì zeugˆri thc morf c λ 3r + iλ 3i, λ 4r iλ 4i ìpou to fantastikì mèroc eðnai polô megalôtero apì to pragmatikì. 'Eqoume dhlad èna dôskampto prìblhma sto opoðo upˆrqoun argèc kai gr gorec qronoklðmakec. Sto Sq. 6 parousiˆzontai oi metatopðseic u 1, u 2 twn maz n m 1, m 2. Xekinˆme me ton talantwt (1) na talant netai kai ton (2) na eðnai akðnhtoc (u 1 = 1, u 2 = 0) Sq. 6(a) kai katal goume na talant netai o (2) se plˆth polô megalôtera apì autˆ pou talant netai o (1) (u1 = ±0.002, u2 = ±0.03) Sq. 6(b). (a) (b) Sq ma 6. (a) Exèlixh sto qrìno twn metatopðsewn u 1 u 2 twn maz n m 1, m 2 kai (b) megèjunsh gôrw apì to 0. Sto Sq. 7 parousiˆzontai ta pragmatikˆ mèrh twn idiotim n thc Iakwbian c sunart sei tou qrìnou. ParathroÔme pwc upˆrqei kenì metaxô twn λ 1r (maôro) kai λ 2r (kìkkino) ta opoða paðrnoun arnhtikèc timèc polô mikrìterec apì autèc twn λ 3r (prˆsino) kai λ 4r (mple), ta opoða sumpðptoun kai brðskontai polô kontˆ sto 0. Pio sugkekrimèna to λ 1r paðrnei timèc polô mikrìterec apì to λ 2r, to opoðo brðsketai kontˆ sta λ 3r kai λ 4r.

46 KEFŸALAIO 4. DIEREŸUNHSH 46 (a) (b) Sq ma 7. (a) Exèlixh sto qrìno twn pragmatik n mer n λ kr twn tessˆrwn Idiotim n λ k thc Iakwbian c kai (b) megèjunsh gôrw apì to 0. Sto Sq. 8 parousiˆzontai to pragmatikì kai ta fantastikˆ mèrh twn dôo idiotim n λ 3 kai λ 4 sunart sei tou qrìnou, oi opoðec sqhmatðzoun èna migadikì zeugˆri thc morf c λ 3r + iλ 3i, λ 4r iλ 4i. Blèpoume pwc to fantastikì mèroc (λ 3i = λ 4i 1.4) eðnai polô megalôtero apì to pragmatikì (λ 3r = λ 4r 0) kai perðpou Ðso me th suqnìthta thc talˆntwshc (ω = ω 2 = 1.29). Apì ta parapˆnw sumperaðnoume pwc prèpei na upˆrqoun dôo gr gorec qronoklðmakec oi opoðec ja exantloôntai polô pio sôntoma apì tic argèc. Sq ma 8. Exèlixh sto qrìno tou pragmatikoô λ 3r = λ 4r kai twn fantastik n mer n λ 3i kai λ 4i tou migadikoô zeôgouc twn idiotim n. Sto Sq. 9 parousiˆzontai oi qronoklðmakec, pou antistoiqoôn stic metablhtèc tou probl - matoc, sunart sei tou qrìnou. ParathroÔme pwc upˆrqei kenì metaxô thc qronoklðmakac τ 1 (maôro) h opoða sqetðzetai me thn idiotim λ 1 kai twn τ 2 (kìkkino), τ 3 (prˆsino) kai τ 4 (mple) oi opoðec sqetðzontai me tic idiotimèc λ 2, λ 3 kai λ 4. H qronoklðmaka τ 1 paðrnei timèc polô mikrìterec apì autèc thc τ 2 h opoða paðrnei timèc lðgo megalôterec apì tic τ 3, τ 4 oi opoðec tautðzontai. Upˆrqei dhlad mða mìno gr gorh qronoklðmaka.

47 KEFŸALAIO 4. DIEREŸUNHSH 47 Sq ma 9. Exèlixh sto qrìno twn tessˆrwn Qronoklimˆkwn τ k Sta Sq. 10 kai Sq. 11 parousiˆzontai ta gr gora plˆth f kr, pou antistoiqoôn stic metablhtèc tou probl matoc, sunart sei tou qrìnou. ParathroÔme pwc apì ta tèssera plˆth, ta plˆth f 1r (maôro) kai f 2r (kìkkino) ta opoða sqetðzontai me tic qronoklðmakec τ 1 kai τ 2 teðnoun sto mhdèn. Pio sugkekrimèna blèpoume pwc to plˆtoc f 1r teðnei gr gora sto mhdèn en to f 2r akoloujeð, allˆ saf c pio argˆ. (a) (b) Sq ma 10. (a) Exèlixh sto qrìno twn tessˆrwn gr gorwn plat n f kr kai (b) megèjunsh gôrw apì to 0. Sq ma 11. Exèlixh sto qrìno twn dôo pio gr gorwn plat n f 1r kai f 2r O pointer èqei mh mhdenikèc timèc gia tic metablhtèc x 1, x 2.

48 KEFŸALAIO 4. DIEREŸUNHSH PerÐptwsh 2 Oi timèc twn paramètrwn kai oi idiotimèc thc Iakwbian c gia tic arqikèc sunj kec pou dðdontai sthn parˆgrafo 3.4 eðnai autèc pou prosdiorðsthkan sthn parˆgrafo (PerÐptwsh 2) kai faðnontai stouc PÐnakec 37, 38, 39 kai 40, 41 antðstoiqa. Timèc paramètrwn (PerÐptwsh 2) p 1 = k p 2 = k 1 p 3 = k 2 p 4 = m 1 p 5 = m Pinakac 37. Timèc twn paramètrwn p 1 p 5 gia thn PerÐptwsh 2 Timèc paramètrwn (PerÐptwsh 2) p 6 = ζ 1 p 7 = ζ 2 p 8 = D p 9 = L p 10 = A Pinakac 38. Timèc twn paramètrwn p 6 p 10 gia thn PerÐptwsh 2 Timèc paramètrwn (PerÐptwsh 2) p 11 = ω 1 p 12 = ω 2 p 13 = c 1 p 14 = c 2 p 15 = L Pinakac 39. Timèc twn paramètrwn p 10 p 15 gia thn PerÐptwsh 2 Se sqèsh me to arqikì prìblhma upˆrqoun oi ex c allagèc: p 6 = ζ 1 : ,p 2 = k 1 : IDIOTIMES IAKWBIANHS (PerÐptwsh 2) λ 1r λ 1i λ 2r λ 2i -1.30E E E E+00 Pinakac 40. Idiotimèc Iakwbian c λ 1, λ 2 gia thn perðptwsh 2 IDIOTIMES IAKWBIANHS (PerÐptwsh 2) λ 3r λ 3i λ 4r λ 4i -1.33E E E E+00 Pinakac 41. Idiotimèc Iakwbian c λ 3, λ 4 gia thn perðptwsh 2 ParathroÔme pwc to pragmatikì mèroc twn dôo pr twn idiotim n eðnai arnhtikì kai kat' a- pìluth tim polô megalôtero apì to fantastikì, en oi epìmenec dôo idiotimèc sqhmatðzoun èna migadikì zeugˆri thc morf c λ 3r + iλ 3i, λ 4r iλ 4i ìpou to fantastikì mèroc eðnai polô megalôtero apì to pragmatikì. 'Eqoume dhlad èna dôskampto prìblhma sto opoðo upˆrqoun argèc kai gr gorec qronoklðmakec. Sto Sq. 12 parousiˆzontai oi metatopðseic u 1, u 2 twn maz n m 1, m 2. Xekinˆme me ton talantwt (1) na talant netai kai ton (2) na eðnai akðnhtoc (u 1 = 1, u 2 = 0) Sq.12(a) kai katal goume na talant netai o (2) se plˆth polô megalôtera apì autˆ pou talant netai o (1) (u1 ± , u2 = ±0.02) Sq.12(b).

49 KEFŸALAIO 4. DIEREŸUNHSH 49 (a) (b) Sq ma 12. (a) Exèlixh sto qrìno twn metatopðsewn u 1 u 2 twn maz n m 1, m 2 kai (b) megèjunsh gôrw apì to 0. Sto Sq. 13 parousiˆzontai ta pragmatikˆ mèrh twn idiotim n thc Iakwbian c sunart sei tou qrìnou. ParathroÔme pwc upˆrqei kenì metaxô twn λ 1r (maôro) kai λ 2r (kìkkino) ta opoða paðrnoun arnhtikèc timèc polô mikrìterec apì autèc twn λ 3r (prˆsino) kai λ 4r (mple), ta opoða sumpðptoun kai brðskontai polô kontˆ sto 0. EpÐshc parathroôme pwc to fainìmeno autì eðnai saf c pio èntono apì thn PerÐptwsh 1. Pio sugkekrimèna to kenì metaxô twn λ 1r kai λ 2r eðnai megalôtero, en kai ta dôo brðskontai pio makriˆ apì ta λ 3r kai λ 4r. (a) (b) Sq ma 13. (a) Exèlixh sto qrìno twn pragmatik n mer n λ kr twn tessˆrwn Idiotim n λ k thc Iakwbian c kai (b) megèjunsh gôrw apì to 0. Sto Sq. 14 parousiˆzontai to pragmatikì kai ta fantastikˆ mèrh twn dôo idiotim n λ 3 kai λ 4 sunart sei tou qrìnou, oi opoðec sqhmatðzoun èna migadikì zeugˆri thc morf c λ 3r + iλ 3i, λ 4r iλ 4i. Blèpoume pwc to fantastikì mèroc (λ 3i = λ 4i 1.4) eðnai polô megalôtero apì to pragmatikì kai perðpou Ðso me th suqnìthta thc talˆntwshc (ω = ω ). Apì ta parapˆnw sumperaðnoume pwc upˆrqoun dôo gr gorec qronoklðmakec oi opoðec ja exantloôntai polô pio sôntoma apì tic argèc.

50 KEFŸALAIO 4. DIEREŸUNHSH 50 Sq ma 14. Exèlixh sto qrìno tou pragmatikoô λ 3r = λ 4r kai twn fantastik n mer n λ 3i, λ 4i tou migadikoô zeôgouc twn idiotim n. Sto Sq. 15 parousiˆzontai oi qronoklðmakec τ k, pou antistoiqoôn stic metablhtèc tou probl matoc, sunart sei tou qrìnou. ParathroÔme pwc upˆrqei kenì metaxô twn qronoklimˆkwn τ 1 (maôro) kai τ 2 (kìkkino) oi opoðec sqetðzontai me tic idiotimèc λ 1 kai λ 2, en oi qronoklðmakec τ 3 (prˆsino) kai τ 4 (mple) oi opoðec sqetðzontai me tic idiotimèc λ 3 kai λ 4 sumpðptoun. H qronoklðmaka τ 1 paðrnei timèc mikrìterec apì autèc thc τ 2, allˆ kai oi dôo brðskontai makriˆ apì tic τ 3 kai τ 4. Genikˆ kai oi dôo qronoklðmakec τ 1 kai τ 2 paðrnoun timèc kontˆ sto mhdèn kai makriˆ apì thn qarakthristik qronoklðmaka h opoða orðzetai wc τ char = τ 3 = τ 4. Sth sugkekrimènh perðptwsh èqoume e = τ 2 /τ char = (a) (b) Sq ma 15. (a) Exèlixh sto qrìno twn tessˆrwn qronoklimˆkwn τ k kai (b) exèlixh sto qrìno twn dôo pio gr gorwn qronoklimˆkwn τ 1, τ 2. Sta Sq. 16 kai Sq. 17 parousiˆzontai ta gr gora plˆth f kr, pou antistoiqoôn stic metablhtèc tou probl matoc, sunart sei tou qrìnou. ParathroÔme pwc apì ta tèssera plˆth, ta plˆth f 1r (maôro) kai f 2r (kìkkino) ta opoða sqetðzontai me tic qronoklðmakec τ 1 kai τ 2 teðnoun sto mhdèn. Pio sugkekrimèna blèpoume pwc to plˆtoc f 1r teðnei gr gora sto mhdèn en to f 2r akoloujeð pio argˆ allˆ saf c kai ta dôo teðnoun sto mhdèn pio gr gora apì thn PerÐptwsh 1. O pointer èqei mh mhdenikèc timèc gia tic metablhtèc x 1, x 2 prˆgma to opoðo mac deðqnei pwc autèc eðnai oi gr gorec metablhtèc tou probl matoc.

51 KEFŸALAIO 4. DIEREŸUNHSH 51 (a) (b) Sq ma 16. (a) Exèlixh sto qrìno twn tessˆrwn gr gorwn plat n f kr kai (b) megèjush gôrw apì to 0. Sq ma 17. Exèlixh sto qrìno twn dôo pio gr gorwn plat n f 1r kai f 2r PerÐptwsh 3 Oi timèc twn paramètrwn kai oi idiotimèc thc Iakwbian c gia tic arqikèc sunj kec pou dðdontai sthn parˆgrafo 3.4 eðnai autèc pou prosdiorðsthkan sthn parˆgrafo (PerÐptwsh 3) kai faðnontai stouc PÐnakec 42, 43, 44 kai 45, 46 antðstoiqa. Timèc paramètrwn (PerÐptwsh 3) p 1 = k p 2 = k 1 p 3 = k 2 p 4 = m 1 p 5 = m Pinakac 42. Timèc twn paramètrwn p 1 p 5 gia thn PerÐptwsh 3 Timèc paramètrwn (PerÐptwsh 3) p 6 = ζ 1 p 7 = ζ 2 p 8 = D p 9 = L p 10 = A Pinakac 43. Timèc twn paramètrwn p 6 p 10 gia thn PerÐptwsh 3 Timèc paramètrwn (PerÐptwsh 3) p 11 = ω 1 p 12 = ω 2 p 13 = c 1 p 14 = c 2 p 15 = L Pinakac 44. Timèc twn paramètrwn p 10 p 15 gia thn PerÐptwsh 3

52 KEFŸALAIO 4. DIEREŸUNHSH 52 Se sqèsh me to arqikì prìblhma upˆrqoun oi ex c allagèc: p 6 = ζ 1 : ,p 2 = 1: , p 4 = m 1 : IDIOTIMES IAKWBIANHS (PerÐptwsh 3) λ 1r λ 1i λ 2r λ 2i -4.12E E E E+00 Pinakac 45. Idiotimèc Iakwbian c λ 1, λ 2 gia thn PerÐptwsh 3 IDIOTIMES IAKWBIANHS (PerÐptwsh 3) λ 3r λ 3i λ 4r λ 4i -1.30E E E E+00 Pinakac 46. Idiotimèc Iakwbian c λ 3, λ 4 gia thn PerÐptwsh 3 ParathroÔme pwc to pragmatikì mèroc twn dôo pr twn idiotim n eðnai arnhtikì kai kat' a- pìluth tim polô megalôtero apì to fantastikì, en oi epìmenec dôo idiotimèc sqhmatðzoun èna migadikì zeugˆri thc morf c λ 3r + iλ 3i, λ 4r iλ 4i ìpou to fantastikì mèroc eðnai polô megalôtero apì to pragmatikì. 'Eqoume dhlad èna dôskampto prìblhma sto opoðo upˆrqoun argèc kai gr gorec qronoklðmakec. Sto Sq. 18 parousiˆzontai oi metatopðseic u 1, u 2 twn maz n m 1, m 2. Xekinˆme me ton talantwt (1) na talant netai kai ton (2) na eðnai akðnhtoc (u 1 = 1, u 2 = 0) Sq.18(a) kai katal goume na talant netai o (2) se plˆth polô megalôtera apì autˆ pou talant netai o (1) (u1 = ±4E 05, u2 = ±0.006) Sq.18(b). (a) (b) Sq ma 18. (a) Exèlixh sto qrìno twn metatopðsewn u 1 u 2 twn maz n m 1, m 2 kai (b) megèjunsh gôrw apì to 0. Sto Sq. 19 parousiˆzontai ta pragmatikˆ mèrh twn idiotim n λ k thc Iakwbian c sunart sei tou qrìnou. ParathroÔme pwc upˆrqei kenì metaxô twn λ 1r (maôro) kai λ 2r (kìkkino) ta opoða paðrnoun arnhtikèc timèc polô mikrìterec apì autèc twn λ 3r (prˆsino) kai λ 4r (mple), ta opoða sumpðptoun kai paðrnoun timèc polô kontˆ sto mhdèn. EpÐshc parathroôme pwc to fainìmeno autì eðnai saf c pio èntono apì tic dôo prohgoômenec peript seic. Pio sugkekrimèna to kenì metaxô twn λ 1r kai λ 2r eðnai megalôtero, en kai ta dôo brðskontai pio makriˆ apì ta λ 3r kai λ 4r.

53 KEFŸALAIO 4. DIEREŸUNHSH 53 (a) (b) Sq ma 19. (a) Exèlixh sto qrìno twn pragmatik n mer n λ kr twn tessˆrwn Idiotim n λ k thc Iakwbian c kai (b) megèjunsh gôrw apì to 0. Sto Sq. 20 parousiˆzontai to pragmatikì kai ta fantastikˆ mèrh twn dôo epìmenwn i- diotim n λ k sunart sei tou qrìnou, oi opoðec sqhmatðzoun èna migadikì zeugˆri thc morf c λ 3r +iλ 3i, λ 4r iλ 4i. Blèpoume pwc to fantastikì mèroc (λ 3i = λ 4i 1.4) eðnai polô megalôtero apì to pragmatikì kai perðpou Ðso me th suqnìthta thc talˆntwshc (ω =ω 2 = 1.29). Apì ta parapˆnw sumperaðnoume pwc upˆrqoun dôo gr gorec qronoklðmakec oi opoðec ja exantloôntai polô pio gr gora apì tic argèc. Sq ma 20. Exèlixh sto qrìno tou pragmatikoô λ 3r = λ 4r kai twn fantastik n mer n λ 3i, λ 4i tou migadikoô zeôgouc twn idiotim n. Sto Sq. 21 parousiˆzontai oi qronoklðmakec τ k, pou antistoiqoôn stic metablhtèc tou probl matoc, sunart sei tou qrìnou. ParathroÔme pwc upˆrqei kenì metaxô twn qronoklimˆkwn τ 1 (maôro) kai τ 2 (kìkkino) oi opoðec sqetðzontai me tic idiotimèc λ 1 kai λ 2, en oi qronoklðmakec τ 3 (prˆsino) kai τ 4 (mple) oi opoðec sqetðzontai me tic idiotimèc λ 3 kai λ 4 sumpðptoun. H qronoklðmaka τ 1 paðrnei timèc mikrìterec apì autèc thc τ 2, en kai oi dôo brðskontai makriˆ apì tic τ 3 kai τ 4 allˆ me saf c pio èntono trìpo apì thn PerÐptwsh 2. Genikˆ kai oi dôo qronoklðmakec τ 1 kai τ 2 paðrnoun timèc kontˆ sto mhdèn kai makriˆ apì thn qarakthristik qronoklðmaka h opoða orðzetai wc τ char = τ 3 = τ 4. Sthn sugkekrimènh perðptwsh èqoume e = τ 2 /τ char = 0.11 to opoðo eðnai mikrìtero apì to e = 0.36 thc PerÐptwshc 2 prˆgma to opoðo shmaðnei pwc to kenì metaxô thc pio arg c τ 2 apì tic gr gorec qronoklðmakec kai thc qarakthristik c τ char eðnai megalôtero kai sunep c h akrðbeia prosdiorismoô thc AAP kai tou AS eðnai kalôterh apì aut thc PerÐptwshc 2.

54 KEFŸALAIO 4. DIEREŸUNHSH 54 (a) (b) Sq ma 21. (a) Exèlixh sto qrìno twn qronoklimˆkwn τ k kai (b) exèlixh sto qrìno twn dôo pio gr gorwn qronoklimˆkwn τ 1, τ 2. Sta Sq. 22 kai Sq. 23 parousiˆzontai ta gr gora plˆth f kr, pou antistoiqoôn stic metablhtèc tou probl matoc, sunart sei tou qrìnou. ParathroÔme pwc apì ta tèssera plˆth, ta plˆth f 1r (maôro) kai f 2r (kìkkino) ta opoða sqetðzontai me tic qronoklðmakec τ 1 kai τ 2 teðnoun sto mhdèn. Pio sugkekrimèna blèpoume pwc to plˆtoc f 1r teðnei gr gora sto mhdèn en to f 2r akoloujeð pio argˆ allˆ kai ta dôo teðnoun sto mhdèn saf c pio gr gora apì tic dôo prohgoômenec peript seic. O pointer èqei mh mhdenikèc timèc gia tic metablhtèc x 1, x 2 prˆgma to opoðo mac deðqnei pwc autèc eðnai oi gr gorec metablhtèc tou probl matoc. (a) (b) Sq ma 22. (a) Exèlixh sto qrìno twn tessˆrwn gr gorwn plat n f kr kai (b) megèjunsh gôrw apì to 0.

55 KEFŸALAIO 4. DIEREŸUNHSH 55 Sq ma 23. Exèlixh sto qrìno twn dôo gr gorwn plat n f 1r, f 2r Eklog PerÐptwshc StoÔc PÐnakec pou akoloujoôn parousiˆzontai ta apotelèsmata thc diereônhshc bˆsei twn o- poðwn ja eklegeð h PerÐptwsh gia thn opoða ja upologðsoume thn AAP. Pio sugkekrimèna stouc PÐnakec 47, 48, 49 parousiˆzontai oi timèc twn paramètrwn tou arqikoô probl matoc, en ston PÐnaka 50 parousiˆzontai oi allagèc twn paramètrwn gia kˆje perðptwsh. Sth sunèqeia stouc PÐnakec 51, 52 parousiˆzontai oi idiotimèc thc Iakwbian c (pragmatikˆ kai fantastikˆ mèrh antðstoiqa) en stouc PÐnakec 54, 55 parousiˆzontai oi qronoklðmakec, to e kai ta plˆth gia ìlec tic peript seic. Timèc paramètrwn p 1 = k p 2 = k 1 p 3 = k 2 p 4 = m 1 p 5 = m Pinakac 47. Timèc twn paramètrwn p 1 p 5 tou probl matoc Timèc paramètrwn p 6 = ζ 1 p 7 = ζ 2 p 8 = D p 9 = L p 10 = A Pinakac 48. Timèc twn paramètrwn p 5 p 10 tou probl matoc Timèc paramètrwn p 11 = ω 1 p 12 = ω 2 p 13 = c 1 p 14 = c 2 p 15 = L Pinakac 49. Timèc twn paramètrwn p 10 p 15 tou probl matoc Peript seic Timèc paramètrwn p 6 = ζ 1 p 2 = k 1 p 4 = m 1 p 13 = c 1 p 11 = ω 1 c 1 /c 2 ω 1 /ω 2 Arqik perðptwsh PerÐptwsh PerÐptwsh PerÐptwsh PerÐptwsh Pinakac 50. Metabolèc twn paramètrwn gia ìlec tic Peript seic

56 KEFŸALAIO 4. DIEREŸUNHSH 56 Peript seic MetatopÐseic 1 2 PerÐptwsh E E-02 PerÐptwsh E E-02 PerÐptwsh E E-03 PÐnakac 51. MetatopÐseic ìlwn twn Peript sewn Peript seic Idiotimèc (pragmatikì mèroc) Arqik PerÐptwsh E E E E-02 PerÐptwsh E E E E-02 PerÐptwsh E E E E-02 PerÐptwsh E E E E-02 PerÐptwsh E E E E-02 PÐnakac 52. Pragmatikˆ mèrh Idiotim n ìlwn twn Peript sewn Peript seic Idiotimèc (fantastikì mèroc) Arqik PerÐptwsh 7.42 E E E E+00 PerÐptwsh E E+00 PerÐptwsh E E+00 PerÐptwsh E E+00 PerÐptwsh E E E E+00 PÐnakac 53. Fantastikˆ mèrh Idiotim n ìlwn twn Peript sewn Peript seic QronoklÐmakec e PerÐptwsh E E E E-01 PerÐptwsh E E E E PerÐptwsh E E E E PÐnakac 54. QronoklÐmakec ìlwn twn Peript sewn Peript seic Plˆth (mèsec timèc gia t = 10sec) PerÐptwsh E E E E-02 PerÐptwsh E E E-02 PerÐptwsh E E E-02 PÐnakac 55. Plˆth ìlwn twn Peript sewn Sto prìblhma to opoðo mac apasqoleð se kˆje qronik stigm kai se kˆje perðptwsh (1,2,3) upˆrqoun dôo idiotimèc me pragmatikì mèroc arnhtikì kai kat' apìluth tim polô megalôtero apì to fantastikì, me to fainìmeno autì na parousiˆzetai pio èntono sthn perðptwsh 3 ìpwc mporeð na faneð sta sqetikˆ sq mata kai stouc PÐnakec 52, 53. Oi dôo epìmenec idiotimèc sqhmatðzoun èna migadikì zeugˆri thc morf c λ 3r + iλ 3i, λ 4r iλ 4i ìpou se kˆje perðptwsh (1,2,3) to fantastikì mèroc (λ 3i = λ 4i 1.4) eðnai polô megalôtero apì to pragmatikì kai perðpou Ðso me th suqnìthta thc talˆntwshc (ω = ω ). Dedomènou ìti h qronoklðmaka pou

57 KEFŸALAIO 4. DIEREŸUNHSH 57 antistoiqeð se aut thn idiotim mporeð na jewrhjeð wc h qarakthristik τ char tou sust matoc katal goume sto sumpèrasma ìti o mègistoc arijmìc twn exisorrop sewn kai kat' epèktash oi gr gorec qronoklðmakec oi opoðec sundèontai me autèc eðnai dôo. ParathroÔme epðshc (PÐnakac 54) pwc sthn pr th perðptwsh èqoume mìno mia gr gorh qronoklðmaka en sth deôterh perðptwsh èqoume dôo gr gorec qronoklðmakec me thn apìstash metaxô pr thc kai deôterhc qronoklðmakac eðnai saf c megalôterh apì thn apìstash metaxô thc deôterhc kai thc qarakthristik c qronoklðmakac. Sthn trðth perðptwsh h diaforˆ metaxô pr thc kai deôterhc eðnai mikrìterh en h diaforˆ metaxô deôterhc kai qarakthristik c qronoklðmakac eðnai saf c pio èntonh. Kai stic treðc Peript seic o pointer èqei mh mhdenikèc timèc gia tic metablhtèc x 1, x 2 prˆgma to opoðo mac deðqnei pwc autèc eðnai oi gr gorec metablhtèc tou sust matoc. Apì touc parapˆnw pðnakec kai thn anˆlush pou prohg jhke blèpoume pwc sthn PerÐptwsh 3 parousiˆzetai pio èntono to fainìmeno twn arg n kai gr gorwn aposbetik n qronoklimˆkwn oi opoðec exantloôntai sôntoma, prˆgma to opoðo mac bohjˆ na prosdiorðsoume thn AAP. EpÐshc to megalôtero kenì metaxô thc pio arg c apì tic gr gorec qronoklðmakec kai thc qarakthristik c, deðqnei ìti h akrðbeia prosdiorismoô thc AAP kai tou AS eðnai kalôterh se aut thn perðptwsh. 'Etsi ja qrhsimopoi soume thn PerÐptwsh 3, gia thn opoða isqôoun oi timèc twn paramètrwn pou faðnontai stouc PÐnakec 42-44, prokeimènou na upologðsoume thn AAP me th bo jeia thc tropopoihmènhc mejìdou CSP eklègontac wc gr gorec metablhtèc tic x 1, x 2.

58

59 Kefˆlaio 5 Upologismìc thc AAP 5.1 Genikˆ Sto kefˆlaio autì ja upologðsoume thn AAP gia thn PerÐptwsh 3, gia touc lìgouc pou a- nafèrjhkan sthn parˆgrafo 4.2.6, qrhsimopoi ntac thn tropopoihmènh mèjodo CSP ìpwc aut perigrˆfetai sto Kefˆlaio 2 Apì thn parˆgrafo 3.5 èqoume gia to diˆnusma twn exarthmènwn metablht n y: x 1 y= x 2 x 3 (5.1) x 4 An g eðna mia algebrik exðswsh tou y tìte èqoume gia to sôsthma twn tessˆrwn diaforik n exis sewn pou perigrˆfei thn exèlixh tou fainìmenou: ìpou: dy dt = g(y) = p 4 q 1 q 2 q 3 q 4 (5.2) q 1 = ẋ 1 = u 1 = x 2 (5.3) ( ) ( ) p1 p2 ( ) p q 2 = ẋ 2 = ü 1 = z 2 z 1 x 1 13 p 4 x 2 (5.4) p 5 p 4 q 3 = ẋ 3 = u 2 = x 4 (5.5) ( ) ( ) ( ) p1 p3 p14 q 4 = ẋ 4 = ü 2 = z 2 z 1 x 3 x 4 (5.6) me p i na eðnai oi parˆmetroi tou probl matoc ìpwc orðzontai sthn parˆgrafo 3.4 kai H Iakwbian tou sust matoc eðnai: p 5 z 0 = (p 8 + x 3 x 1 ) 2 + p 2 9 (5.7) z 1 = 1 p 15 z0 (5.8) z 2 = p 8 + x 3 x 1 (5.9) 59 p 5

60 KEFŸALAIO 5. UPOLOGISMŸOS THS AAP 60 J= q 1 q 1 q 1 q 1 x 1 x 2 x 3 x 4 q 2 q 2 q 2 q 2 x 1 x 2 x 3 x 4 q 3 q 3 q 3 q 3 x 1 x 2 x 3 x 4 q 4 q 4 q 4 q 4 x 1 x 2 x 3 x 4 Oi dôo qronoklðmakec pou exantloôntai eðnai autèc pou antistoiqoôn stic metablhtèc x 1, x 2 ìpwc prokôptei apì thn anˆlush sto kefˆlaio 4 en ta plˆth pou sundèontai me autèc tic qronoklðmakec perigrˆfontai sômfwna me thn parˆgrafo apì tic exis seic: ìpou: F r = B r g (5.11) B r = [I r r G r s], (5.12) kai [ ] G r s (n + 1) = (J r r G r s (n) J s r) 1 G r s (n) J s s J r s + dgr s (n) dt (5.13) H exðswsh F r = 0 perigrˆfei th morf thc AAP. Sto sugkekrimèno prìblhma ja pragmatopoi soume èna refinement. Opìte oi exis seic pou ja perigrˆfoun thn AAP prin kai metˆ to refinement ja eðnai antistoðqwc: kai F r1 = B r1 g = 0 (5.14) F r2 = B r2 g = 0 (5.15) AkoloujeÐ parousðash stouc q rouc twn fˆsewn me suntetagmènec x 3, x 4, x 1 kai x 3, x 4, x 2 thc AAP pou perigrˆfetai apì thn exðswsh F r1 = 0 F r2 = 0 stic parakˆtw perioqèc: 1) x 3, x 4 ( 0.01, 0.01) 2) x 3, x 4 ( 0.1, 0.1) 3) x 3, x 4 ( 1, 1) 4) x 3, x 4 ( 10, 10)

61 KEFŸALAIO 5. UPOLOGISMŸOS THS AAP Q roc fˆsewn:(x 3, x 4, x 1 ) AAP F r1 = 0 kai F r2 = 0 (refinement) Sta Sq. 24, 25 parousiˆzontai ston q ro twn fˆsewn x 3, x 4, x 1 h AAP pou perigrˆfetai apì thn exðswsh F r1 = 0 kai h AAP pou prokôptei metˆ to pr to refinement kai perigrˆfetai apì thn e- xðswsh F r2 = 0 se mia perioq gôrw apì to (0,0) me x 3, x 4 ( 0.01, 0.01) kai x 3, x 4 ( 0.1, 0.1) (a) (b) Sq ma 24. Q roc fˆsewn x 3, x 4, x 1 me x 3, x 4 ( 0.01, 0.01) Grafik anaparˆstash twn AAP (a) F r1 = 0 kai (b) F r2 = 0(refinement) (a) (b) Sq ma 25. Q roc fˆsewn x 3, x 4, x 1 me x 3, x 4 ( 0.1, 0.1) Grafik anaparˆstash twn AAP (a) F r1 = 0 kai (b) F r2 = 0(refinement)

62 KEFŸALAIO 5. UPOLOGISMŸOS THS AAP 62 Sta Sq. 26, 27 parousiˆzontai ston q ro twn fˆsewn x 3, x 4, x 1 h AAP pou perigrˆfetai apì thn exðswsh F r1 = 0 kai h AAP pou prokôptei metˆ to pr to refinement kai perigrˆfetai apì thn exðswsh F r2 = 0 se mia perioq gôrw apì to (0,0) me x 3, x 4 ( 1, 1) kai x 3, x 4 ( 10, 10). (a) (b) Sq ma 26. Q roc fˆsewn x 3, x 4, x 1 me x 3, x 4 ( 1, 1) Grafik anaparˆstash twn AAP (a) F r1 = 0 kai (b) F r2 = 0(refinement) (a) (b) Sq ma 27. Q roc fˆsewn x 3, x 4, x 1 me x 3, x 4 ( 10, 10) Grafik anaparˆstash twn AAP (a) F r1 = 0 kai (b) F r2 = 0(refinement)

63 KEFŸALAIO 5. UPOLOGISMŸOS THS AAP 63 Sto Sq. 28 parousiˆzontai anˆ zeôgh ston q ro twn fˆsewn x 3, x 4, x 1 h AAP pou perigrˆfetai apì thn exðswsh F r1 = 0 (prˆsino qr ma) kai h AAP pou prokôptei metˆ to pr to refinement kai perigrˆfetai apì thn exðswsh F r2 = 0 (mple qr ma) se mia perioq gôrw apì to (0,0) me x 3, x 4 ( 0.01, 0.01), x 3, x 4 ( 0.1, 0.1), x 3, x 4 ( 1, 1) kai x 3, x 4 ( 10, 10). (a) (b) (g) (d) Sq ma 28. Q roc fˆsewn x 3, x 4, x 1 Grafik anaparˆstash anˆ zeôgh twn AAP F r1 = 0 kai F r2 = 0(refinement) stic perioqèc: (a) x 3, x 4 ( 0.01, 0.01), (b) x 3, x 4 ( 0.1, 0.1), (g) x 3, x 4 ( 1, 1) kai (d) x 3, x 4 ( 10, 10).

64 KEFŸALAIO 5. UPOLOGISMŸOS THS AAP 64 Sto Sq. 29 parousiˆzontai anˆ zeôgh ston q ro twn fˆsewn x 3, x 4, x 1 h AAP pou perigrˆfetai apì thn exðswsh F r1 = 0 (prˆsino qr ma) kai h AAP pou prokôptei metˆ to pr to refinement kai perigrˆfetai apì thn exðswsh F r2 = 0 (mple qr ma) se perioqèc makriˆ a- pì to (0,0) me x 3, x 4 ( 0.009, 0.007), x 3, x 4 (0.007, 0.009), x 3, x 4 ( 0.09, 0.07) kai x 3, x 4 (0.07, 0.09). (a) (b) (g) (d) Sq ma 29. Q roc fˆsewn x 3, x 4, x 1 Grafik anaparˆstash anˆ zeôgh twn AAP F r1 = 0 kai F r2 = 0(refinement) stic perioqèc: (a) x 3, x 4 ( 0.009, 0.007), (b) x 3, x 4 (0.007, 0.009), (g) x 3, x 4 ( 0.09, 0.07) kai (d) x 3, x 4 (0.07, 0.09).

65 KEFŸALAIO 5. UPOLOGISMŸOS THS AAP 65 Sto Sq. 30 parousiˆzontai anˆ zeôgh ston q ro twn fˆsewn x 3, x 4, x 1 h AAP pou perigrˆfetai apì thn exðswsh F r1 = 0 (prˆsino qr ma) kai h AAP pou prokôptei metˆ to pr to refinement kai perigrˆfetai apì thn exðswsh F r2 = 0 (mple qr ma) se perioqèc makriˆ apì to (0,0) me x 3, x 4 ( 0.9, 0.7), x 3, x 4 (0.7, 0.9), x 3, x 4 ( 9, 7) kai x 3, x 4 (7, 9). (a) (b) (g) (d) Sq ma 30. Q roc fˆsewn x 3, x 4, x 1 Grafik anaparˆstash anˆ zeôgh twn AAP F r1 = 0 kai F r2 = 0(refinement) stic perioqèc: (a) x 3, x 4 ( 0.9, 0.7), (b) x 3, x 4 (0.7, 0.9), (g) x 3, x 4 ( 9, 7) kai (d) x 3, x 4 (7, 9).

66 KEFŸALAIO 5. UPOLOGISMŸOS THS AAP Q roc fˆsewn:(x 3, x 4, x 2 ) AAP F r1 = 0 kai F r2 = 0 (refinement) Sta Sq. 31, 32 parousiˆzontai ston q ro twn fˆsewn x 3, x 4, x 2 h AAP pou perigrˆfetai apì thn exðswsh F r1 = 0 kai h AAP pou prokôptei metˆ to pr to refinement kai perigrˆfetai apì thn exðswsh F r2 = 0 se mia perioq gôrw apì to (0,0) me x 3, x 4 ( 0.01, 0.01) kai x 3, x 4 ( 0.1, 0.1). (a) (b) Sq ma 31. Q roc fˆsewn x 3, x 4, x 2 me x 3, x 4 ( 0.01, 0.01) Grafik anaparˆstash twn AAP (a) F r1 = 0 kai (b) F r2 = 0(refinement) (a) (b) Sq ma 32. Q roc fˆsewn x 3, x 4, x 2 me x 3, x 4 ( 0.1, 0.1) Grafik anaparˆstash twn AAP (a) F r1 = 0 kai (b) F r2 = 0(refinement)

67 KEFŸALAIO 5. UPOLOGISMŸOS THS AAP 67 Sta Sq. 33, 34 parousiˆzontai ston q ro twn fˆsewn x 3, x 4, x 2 h AAP pou perigrˆfetai apì thn exðswsh F r1 = 0 kai h AAP pou prokôptei metˆ to pr to refinement kai perigrˆfetai apì thn exðswsh F r2 = 0 se mia perioq gôrw apì to (0,0) me x 3, x 4 ( 1, 1) kai x 3, x 4 ( 10, 10). (a) (b) Sq ma 33. Q roc fˆsewn x 3, x 4, x 2 me x 3, x 4 ( 1, 1) Grafik anaparˆstash twn AAP (a) F r1 = 0 kai (b) F r2 = 0(refinement) (a) (b) Sq ma 34. Q roc fˆsewn x 3, x 4, x 2 me x 3, x 4 ( 10, 10) Grafik anaparˆstash twn AAP (a) F r1 = 0 kai (b) F r2 = 0(refinement)

68 KEFŸALAIO 5. UPOLOGISMŸOS THS AAP 68 Sto Sq. 35 parousiˆzontai anˆ zeôgh ston q ro twn fˆsewn x 3, x 4, x 2 h AAP pou perigrˆfetai apì thn exðswsh F r1 = 0 (prˆsino qr ma) kai h AAP pou prokôptei metˆ to pr to refinement kai perigrˆfetai apì thn exðswsh F r2 = 0 (mple qr ma) se mia perioq gôrw apì to (0,0) me x 3, x 4 ( 0.01, 0.01), x 3, x 4 ( 0.1, 0.1), x 3, x 4 ( 1, 1) kai x 3, x 4 ( 10, 10) (a) (b) (g) (d) Sq ma 35. Q roc fˆsewn x 3, x 4, x 2 Grafik anaparˆstash anˆ zeôgh twn AAP F r1 = 0 kai F r2 = 0(refinement) stic perioqèc: (a) x 3, x 4 ( 0.01, 0.01), (b) x 3, x 4 ( 0.1, 0.1), (g) x 3, x 4 ( 1, 1) kai (d) x 3, x 4 ( 10, 10)

69 KEFŸALAIO 5. UPOLOGISMŸOS THS AAP 69 Sto Sq. 36 parousiˆzontai anˆ zeôgh ston q ro twn fˆsewn x 3, x 4, x 2 h AAP pou perigrˆfetai apì thn exðswsh F r1 = 0 (prˆsino qr ma) kai h AAP pou prokôptei metˆ to pr to refinement kai perigrˆfetai apì thn exðswsh F r2 = 0 (mple qr ma) se mia perioq makriˆ a- pì to (0,0) me x 3, x 4 ( 0.009, 0.007), x 3, x 4 (0.007, 0.009), x 3, x 4 ( 0.09, 0.07) kai x 3, x 4 (0.07, 0.09) (a) (b) (g) (d) Sq ma 36. Q roc fˆsewn x 3, x 4, x 2 Grafik anaparˆstash anˆ zeôgh twn AAP F r1 = 0 kai F r2 = 0(refinement) stic perioqèc: (a) x 3, x 4 ( 0.009, 0.007), (b) x 3, x 4 (0.007, 0.009), (g) x 3, x 4 ( 0.09, 0.07) kai (d) x 3, x 4 (0.07, 0.09)

70 KEFŸALAIO 5. UPOLOGISMŸOS THS AAP 70 Sto Sq. 37 parousiˆzontai anˆ zeôgh ston q ro twn fˆsewn x 3, x 4, x 2 h AAP pou perigrˆfetai apì thn exðswsh F r1 = 0 (prˆsino qr ma) kai h AAP pou prokôptei metˆ to pr to refinement kai perigrˆfetai apì thn exðswsh F r2 = 0 (mple qr ma) se mia perioq makriˆ apì to (0,0) me x 3, x 4 ( 0.9, 0.7), x 3, x 4 (0.7, 0.9), x 3, x 4 ( 9, 7) kai x 3, x 4 (7, 9) (a) (b) (g) (d) Sq ma 37. Q roc fˆsewn x 3, x 4, x 2 Grafik anaparˆstash anˆ zeôgh twn AAP F r1 = 0 kai F r2 = 0(refinement) stic perioqèc: (a) x 3, x 4 ( 0.9, 0.7), (b) x 3, x 4 (0.7, 0.9), (g) x 3, x 4 ( 9, 7) kai (d) x 3, x 4 (7, 9)

71 KEFŸALAIO 5. UPOLOGISMŸOS THS AAP Sumpèrasma Blèpoume pwc h epðdrash tou refinement sthn perigraf thc AAP eðnai mikrìterh sthn perðptwsh thc anaparˆstashc thc AAP sto q ro twn fˆsewn x 3, x 4, x 1 se sôgkrish me ton q ro twn fˆsewn x 3, x 4, x 2. H arqik ektðmhs thc AAP sto q ro twn fˆsewn x 3, x 4, x 1 eðnai kalôterh se sqèsh me thn arqik thc ektðmhsh sto q ro twn fˆsewn x 3, x 4, x 2 afoô to plˆtoc f 1r thc metablhthc x 1 paðrnei pio gr gora timèc pio kontˆ sto mhdèn se sqèsh me to plˆtoc f 2r thc metablhthc x 2. Genikˆ blèpoume pwc kontˆ sto x 3, x 4 = (0,0) oi dôo AAP (prin kai metˆ to refinement) tautðzontai en kaj c apomakrunìmaste apì to x 3, x 4 = (0,0) oi dôo AAP arqðzoun na apoklðnoun. Autì parousiˆzetai ligìtera èntono sto q ro twn fˆsewn x 3, x 4, x 1 giatð ekeð h arqik ektðmhsh thc AAP eðnai kalôterh.

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73 Kefˆlaio 6 AAP kai akrib c lôsh 6.1 Genikˆ Sto Kefˆlaio autì ja parousiastoôn h AAP pou perigrˆfetai apì thn exðswsh F r1 = 0 F r2 = 0 kai h akrib c lôsh, ìpwc aut upologðzetai apì to upologistikì pakèto LSODE gia tic arqikèc sunj kec pou anafèrontai sthn parˆgrafo 3.4, bˆsei twn opoðwn ègine o upologismìc thc AAP. Ja dojoôn grafikèc parastˆseic thc AAP mazð me thn akrib lôsh kai ja upologisteð h apìklish d metaxô touc h opoða orðzetai wc ex c: ìpou: d = x ia x icsp x ia x ia : akrib c lôsh apì to pakèto LSODE, ìpou x 1A eðnai h akrib c lôsh sto q ro x 3, x 4, x 1 en x 2A eðnai h akrib c lôsh sto q ro x 3, x 4, x 2 x icsp : lôsh me th mèjodo CSP, ìpou x 1CSP eðnai h lôsh sto q ro x 3, x 4, x 1 en x 2CSP eðnai h lôsh sto q ro x 3, x 4, x 2 AkoloujeÐ parousðash stouc q rouc twn fˆsewn x 3, x 4, x 1 kai x 3, x 4, x 2 thc AAP pou perigrˆfetai apì thn exðswsh F r1 = 0 F r2 = 0 kai thc akriboôc lôshc stic parakˆtw perioqèc: 1) x 3, x 4 ( 0.01, 0.01) 2) x 3, x 4 ( 0.1, 0.1) 3) x 3, x 4 ( 1, 1) 4) x 3, x 4 ( 10, 10) Gia kˆje ènan apì touc q rouc twn fˆsewn me suntetagmènec x 3, x 4, x 1 kai x 3, x 4, x 2 parousiˆzetai h apìklish d metaxô AAP (gia F r1 = 0 kai F r2 = 0(refinement)) kai akriboôc lôshc 73

74 KEFŸALAIO 6. AAP KAI AKRIBŸHS LŸUSH Q roc fˆsewn:(x 3, x 4, x 1 ) AAP F r1 = 0, F r2 = 0(refinement) kai akrib c lôsh Sta Sq. 38 kai 39 parousiˆzontai ston q ro twn fˆsewn x 3, x 4, x 1, h akrib c lôsh gia 0 < t < 120 sec ìpwc upologðsthke apì to upologistikì pakèto LSODE kai oi AAP pou perigrˆfontai apì tic exis seic F r1 = 0 kai F r2 = 0(refinement) se mia perioq gôrw apì to (0,0) me x 3, x 4 ( 0.01, 0.01), x 3, x 4 ( 0.1, 0.1). ParathroÔme ìti h lôsh èlketai apì thn AAP. (a) (b) Sq ma 38 Q roc fˆsewn x 3, x 4, x 1 me x 3, x 4 ( 0.01, 0.01) Grafik anaparˆstash gia 0 < t < 120 (a) thc AAP F r1 = 0 kai thc akriboôc lôshc (b) thc AAP F r2 = 0(refinement) kai thc akriboôc lôshc (a) (b) Sq ma 39 Q roc fˆsewn x 3, x 4, x 1 me x 3, x 4 ( 0.1, 0.1) Grafik anaparˆstash gia 0 < t < 120 (a) thc AAP F r1 = 0 kai thc akriboôc lôshc (b) thc AAP F r2 = 0(refinement) kai thc akriboôc lôshc.

75 KEFŸALAIO 6. AAP KAI AKRIBŸHS LŸUSH 75 Sta Sq. 40 kai Sq. 41 parousiˆzontai ston q ro twn fˆsewn x 3, x 4, x 1, h akrib c lôsh gia 0 < t < 120 sec ìpwc upologðsthke apì to upologistikì pakèto LSODE kai oi AAP pou perigrˆfontai apì tic exis seic F r1 = 0 kai F r2 = 0(refinement) se mia perioq gôrw apì to (0,0) me x 3, x 4 ( 1, 1), x 3, x 4 ( 10, 10). ParathroÔme ìti h lôsh èlketai apì thn AAP. (a) (b) Sq ma 40 Q roc fˆsewn x 3, x 4, x 1 me x 3, x 4 ( 1, 1) Grafik anaparˆstash gia 0 < t < 120 (a) thc AAP F r1 = 0 kai thc akriboôc lôshc (b) thc AAP F r2 = 0(refinement) kai thc akriboôc lôshc (a) (b) Sq ma 41 Q roc fˆsewn x 3, x 4, x 1 me x 3, x 4 ( 10, 10) Grafik anaparˆstash gia 0 < t < 120 (a) thc AAP F r1 = 0 kai thc akriboôc lôshc (b) thc AAP F r2 = 0(refinement) kai thc akriboôc lôshc

76 KEFŸALAIO 6. AAP KAI AKRIBŸHS LŸUSH SÔgkrish AAP F r1 = 0 kai akriboôc lôshc Exetˆzoume eˆn h akrib c lôsh ìpwc aut prokôptei apì to pakèto LSODE pèftei epˆnw sthn AAP F r1 = 0 ìpwc aut parousiˆzetai sto q ro twn fˆsewn x 3, x 4, x 1. Sto Sq. 42 èqoume ston orizìntio ˆxona to qrìno kai ston katakìrufo ˆxona thn apìklish d, ìpwc aut orðsthke sthn parˆgrafo 6.1, metaxô AAP kai akriboôc lôshc. H sôgkrish gðnetai arqikˆ gia olìklhro to qronikì diˆsthma exèlixhc tou fainìmenou (0-120 sec). Katìpin blèpoume pwc exelðsetai arqikˆ h sôgklish (0-5 sec) opìte h akrib c lôsh arqðzei na pèftei pˆnw sthn AAP kai katìpin sto tèloc ( sec) opìte plèon h akrib c lôsh kineðtai stajerˆ pˆnw sthn AAP. (a) (b) (g) Sq ma 42. AAP F r1 = 0 kai akrib c lôsh sto q ro twn fˆsewn x 3, x 4, x 1. Grafik anaparˆstash gia (a) sec (b) 0-5 sec kai (g) sec Blèpoume pwc stì q ro twn fˆsewn x 3, x 4, x 1 h akrib c lôsh ìpwc aut prokôptei apì to pakèto LSODE pèftei epˆnw sthn AAP F r1 = 0 me th diaforˆ touc na brðsketai sto

77 KEFŸALAIO 6. AAP KAI AKRIBŸHS LŸUSH SÔgkrish AAP F r2 = 0(refinement) kai akriboôc lôshc Exetˆzoume eˆn h akrib c lôsh ìpwc aut prokôptei apì to pakèto LSODE pèftei epˆnw sthn AAP F r2 = 0(refinement) ìpwc aut parousiˆzetai sto q ro twn fˆsewn x 3, x 4, x 1. Sto Sq. 43 èqoume ston orizìntio ˆxona to qrìno kai ston katakìrufo ˆxona thn apìklish d, ìpwc aut orðsthke sthn parˆgrafo 6.1, metaxô AAP kai akriboôc lôshc. H sôgkrish gðnetai arqikˆ gia olìklhro to qronikì diˆsthma exèlixhc tou fainìmenou (0-120 sec). Katìpin blèpoume pwc exelðsetai arqikˆ h sôgklish (0-5 sec) opìte h akrib c lôsh arqðzei na pèftei pˆnw sthn AAP kai katìpin sto tèloc ( sec) opìte plèon h akrib c lôsh kineðtai stajerˆ pˆnw sthn AAP. (a) (b) (g) Sq ma 43. AAP F r2 = 0(refinement) kai akrib c lôsh sto q ro twn fˆsewn x 3, x 4, x 1. Grafik anaparˆstash gia (a) sec (b) 0-5 sec kai (g) sec Blèpoume pwc stì q ro twn fˆsewn x 3, x 4, x 1 h akrib c lôsh ìpwc aut prokôptei apì to pakèto LSODE pèftei epˆnw sthn AAP F r2 = 0(refinement) me th diaforˆ touc na brðsketai sto ParathroÔme pwc se sôgkrish me thn AAP pou prosdiorðsthke qwrðc refinement, h akrðbeia me thn opoða h AAP proseggðzei th lôsh eðnai megalôterh perðpou katˆ mia tˆxh megèjouc.

78 KEFŸALAIO 6. AAP KAI AKRIBŸHS LŸUSH Q roc fˆsewn:(x 3, x 4, x 2 ) AAP F r1 = 0, F r2 = 0(refinement) kai akrib c lôsh Sta Sq. 44 kai Sq. 45 parousiˆzontai ston q ro twn fˆsewn x 3, x 4, x 2, h akrib c lôsh gia 0 < t < 120 sec ìpwc upologðsthke apì to upologistikì pakèto LSODE kai oi AAP pou perigrˆfontai apì tic exis seic F r1 = 0 kai F r2 = 0(refinement) se mia perioq gôrw apì to (0,0) me x 3, x 4 ( 0.01, 0.01), x 3, x 4 ( 0.1, 0.1). ParathroÔme ìti h lôsh èlketai apì thn AAP. (a) (b) Sq ma 44 Q roc fˆsewn x 3, x 4, x 2 me x 3, x 4 ( 0.01, 0.01) Grafik anaparˆstash gia 0 < t < 120 (a) thc AAP F r1 = 0 kai thc akriboôc lôshc (b) thc AAP F r2 = 0(refinement) kai thc akriboôc lôshc (a) (b) Sq ma 45 Q roc fˆsewn x 3, x 4, x 2 me x 3, x 4 ( 0.1, 0.1) Grafik anaparˆstash gia 0 < t < 120 (a) thc AAP F r1 = 0 kai thc akriboôc lôshc (b) thc AAP F r2 = 0(refinement) kai thc akriboôc lôshc

79 KEFŸALAIO 6. AAP KAI AKRIBŸHS LŸUSH 79 Sta Sq. 46 kai Sq. 47 parousiˆzontai ston q ro twn fˆsewn x 3, x 4, x 2, h akrib c lôsh gia 0 < t < 120 sec ìpwc upologðsthke apì to upologistikì pakèto LSODE kai oi AAP pou perigrˆfontai apì tic exis seic F r1 = 0 kai F r2 = 0(refinement) se mia perioq gôrw apì to (0,0) me x 3, x 4 ( 1, 1), x 3, x 4 ( 10, 10). ParathroÔme ìti h lôsh èlketai apì thn AAP. (a) (b) Sq ma 46 Q roc fˆsewn x 3, x 4, x 2 me x 3, x 4 ( 1, 1) Grafik anaparˆstash gia 0 < t < 120 (a) thc AAP F r1 = 0 kai thc akriboôc lôshc (b) thc AAP F r2 = 0(refinement) kai thc akriboôc lôshc (a) (b) Sq ma 47 Q roc fˆsewn x 3, x 4, x 2 me x 3, x 4 ( 10, 10) Grafik anaparˆstash gia 0 < t < 120 (a) thc AAP F r1 = 0 kai thc akriboôc lôshc (b) thc AAP F r2 = 0(refinement) kai thc akriboôc lôshc