Use of Shrink Wrapping for Interval Taylor Models in Algorithms of Computer-Assisted Proof of the Existence of Periodic Trajectories in Systems of Ordinary Differential Equations
Using interval Taylor models (TM), we construct algorithms for the computer-assisted proof of the existence of periodic trajectories in systems of ordinary differential equations (ODEs). Although TMs allow one to construct guaranteed estimates for families of solutions of systems of ODEs when integr...
Uloženo v:
| Vydáno v: | Differential equations Ročník 57; číslo 3; s. 391 - 407 |
|---|---|
| Hlavní autoři: | , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Moscow
Pleiades Publishing
01.03.2021
Springer |
| Témata: | |
| ISSN: | 0012-2661, 1608-3083 |
| On-line přístup: | Získat plný text |
| Tagy: |
Přidat tag
Žádné tagy, Buďte první, kdo vytvoří štítek k tomuto záznamu!
|
| Abstract | Using interval Taylor models (TM), we construct algorithms for the computer-assisted proof of the existence of periodic trajectories in systems of ordinary differential equations (ODEs). Although TMs allow one to construct guaranteed estimates for families of solutions of systems of ODEs when integrating ODEs over large time intervals, the interval residual included in the TMs begins to grow exponentially and becomes the dominant part of the estimate of the solution pencil, making it practically unusable. To eliminate this deficiency, the creators of the TM—K. Makino and M. Berz—proposed the idea of so-called “shrink wrapping.” We formalize the original algorithm within the framework of the TM definitions we have adopted and propose our own version of the “shrink wrapping,” more accurately adapted to the problem of the computer-aided proof of the existence of periodic trajectories. |
|---|---|
| AbstractList | Using interval Taylor models (TM), we construct algorithms for the computer-assistedproof of the existence of periodic trajectories in systems of ordinary differential equations (ODEs).Although TMs allow one to construct guaranteed estimates for families of solutions of systems ofODEs when integrating ODEs over large time intervals, the interval residual included in the TMsbegins to grow exponentially and becomes the dominant part of the estimate of the solutionpencil, making it practically unusable. To eliminate this deficiency, the creators of theTM-K. Makino and M. Berz-proposed the idea of so-called "shrink wrapping."We formalize the original algorithm within the framework of the TM definitions we have adoptedand propose our own version of the "shrink wrapping," more accurately adapted to the problem ofthe computer-aided proof of the existence of periodic trajectories. Using interval Taylor models (TM), we construct algorithms for the computer-assisted proof of the existence of periodic trajectories in systems of ordinary differential equations (ODEs). Although TMs allow one to construct guaranteed estimates for families of solutions of systems of ODEs when integrating ODEs over large time intervals, the interval residual included in the TMs begins to grow exponentially and becomes the dominant part of the estimate of the solution pencil, making it practically unusable. To eliminate this deficiency, the creators of the TM—K. Makino and M. Berz—proposed the idea of so-called “shrink wrapping.” We formalize the original algorithm within the framework of the TM definitions we have adopted and propose our own version of the “shrink wrapping,” more accurately adapted to the problem of the computer-aided proof of the existence of periodic trajectories. |
| Audience | Academic |
| Author | Ryabkov, O. I. Shul’min, D. A. Evstigneev, N. M. |
| Author_xml | – sequence: 1 givenname: N. M. surname: Evstigneev fullname: Evstigneev, N. M. email: evstigneevnm@yandex.ru organization: Federal Research Center “Computer Science and Control,” Russian Academy of Sciences – sequence: 2 givenname: O. I. surname: Ryabkov fullname: Ryabkov, O. I. email: oleg.ryabkov.87@gmail.com, oi-techsup@yandex.ru organization: Federal Research Center “Computer Science and Control,” Russian Academy of Sciences – sequence: 3 givenname: D. A. surname: Shul’min fullname: Shul’min, D. A. email: dnssh@mail.ru organization: Lomonosov Moscow State University |
| BookMark | eNp9kd1qGzEQhUVJoE6aB8idXmAT_exqdy-N66aBlATskMtF0Y5suWvJleQQv1afsLN2r1oIEgidme9oOLogZz54IOSasxvOZXm7YIwLoRQXnEmG0icy4Yo1hWSNPCOTsVyM9c_kIqUNY6yteTUhv58T0GDpYh2d_0lfot7tnF9RGyK99xnimx7oUh8GvP8IPQyJOk-nwypEl9fbNLKzsN3tsbWYpuRShp4-xYA67rwGOn8fRW-O7zxBdKF3hi6j3oDJaANHy8UBm05-j7F3XscD_eqshQg-Oxxi_muvsws-fSHnVg8Jrv6el-T523w5-148PN7dz6YPhZE1ywU3tmIlA_6qdS20VKJvuCqNUkKBaZpW96DaSis7ZlaVui5tbbiCVyvaVgl5SW5Ovis9QOe8DTlqg6uHrTMYv3WoT2uupORV3SBQnwATQ0oRbGdcPs6MoBs6zrrxr7r__gpJ_g-5i26LEXzIiBOTsNevIHabsI8eE_kA-gMyX6lf |
| CitedBy_id | crossref_primary_10_3390_math11112536 crossref_primary_10_3390_math11204336 |
| Cites_doi | 10.1007/s002080010018 10.1134/S0012266118040092 10.1023/A:1024467732637 10.1007/11558958_8 10.1137/050638448 10.1070/RM1985v040n04ABEH003624 10.1023/A:1011423909873 10.1137/1.9781611970906 10.1016/j.apnum.2006.10.006 10.1134/S0012266119090088 10.1137/1.9780898717716 |
| ContentType | Journal Article |
| Copyright | Pleiades Publishing, Ltd. 2021 COPYRIGHT 2021 Springer |
| Copyright_xml | – notice: Pleiades Publishing, Ltd. 2021 – notice: COPYRIGHT 2021 Springer |
| DBID | AAYXX CITATION |
| DOI | 10.1134/S0012266121030113 |
| DatabaseName | CrossRef |
| DatabaseTitle | CrossRef |
| DatabaseTitleList | |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Sciences (General) Mathematics |
| EISSN | 1608-3083 |
| EndPage | 407 |
| ExternalDocumentID | A716331578 10_1134_S0012266121030113 |
| GroupedDBID | --Z -5D -5G -BR -EM -Y2 -~C -~X .86 .VR 04Q 04W 06D 0R~ 0VY 1N0 29G 2J2 2JN 2JY 2KG 2KM 2LR 2P1 2VQ 2~H 30V 3V. 4.4 408 409 40D 40E 5GY 5VS 67Z 6NX 6TJ 78A 7WY 8FE 8FG 8FL 8G5 8TC 8UJ 95- 95. 95~ 96X AAAVM AABHQ AACDK AAHNG AAIAL AAJBT AAJKR AANZL AARHV AARTL AASML AATNV AATVU AAUYE AAWCG AAYIU AAYQN AAYTO AAYZH ABAKF ABBBX ABBXA ABDBF ABDZT ABECU ABEFU ABFTV ABHQN ABJCF ABJNI ABJOX ABKCH ABKTR ABLLD ABMNI ABMQK ABNWP ABQBU ABQSL ABSXP ABTEG ABTHY ABTKH ABTMW ABULA ABUWG ABWNU ABXPI ACAOD ACBXY ACDTI ACGFS ACHSB ACHXU ACIWK ACKNC ACMDZ ACMLO ACNCT ACOKC ACOMO ACPIV ACUHS ACZOJ ADHHG ADHIR ADINQ ADKNI ADKPE ADRFC ADTPH ADURQ ADYFF ADZKW AEBTG AEFQL AEGAL AEGNC AEJHL AEJRE AEMSY AENEX AEOHA AEPYU AETLH AEVLU AEXYK AFBBN AFFNX AFGCZ AFKRA AFLOW AFQWF AFWTZ AFZKB AGAYW AGDGC AGJBK AGMZJ AGQMX AGRTI AGWIL AGWZB AGYKE AHAVH AHBYD AHKAY AHSBF AHYZX AI. AIAKS AIGIU AIIXL AILAN AITGF AJBLW AJRNO ALMA_UNASSIGNED_HOLDINGS ALWAN AMKLP AMXSW AMYLF AMYQR AOCGG ARAPS ARMRJ ASPBG AVWKF AXYYD AZFZN AZQEC B-. B0M BA0 BAPOH BDATZ BENPR BEZIV BGLVJ BGNMA BPHCQ BSONS CAG CCPQU COF CS3 CSCUP DDRTE DL5 DNIVK DPUIP DU5 DWQXO EAD EAP EBLON EBS EIOEI EJD EMK EPL ESBYG ESX FEDTE FERAY FFXSO FIGPU FINBP FNLPD FRNLG FRRFC FSGXE FWDCC GGCAI GGRSB GJIRD GNUQQ GNWQR GQ6 GQ7 GQ8 GROUPED_ABI_INFORM_COMPLETE GUQSH GXS H13 HCIFZ HF~ HG6 HMJXF HQYDN HRMNR HVGLF HZ~ IAO IHE IJ- IKXTQ ITM IWAJR IXC IZIGR IZQ I~X I~Z J-C JBSCW JCJTX JZLTJ K60 K6V K6~ K7- KDC KOV L6V LAK LLZTM M0C M0N M2O M4Y M7S MA- N2Q NB0 NPVJJ NQJWS NU0 O9- O93 O9J OAM OHT OVD P2P P62 P9R PADUT PF0 PKN PQBIZ PQBZA PQQKQ PROAC PT4 PTHSS Q2X QOS R89 R9I RNI RNS ROL RPX RSV RZC RZE S16 S1Z S27 S3B SAP SDH SHX SISQX SJYHP SMT SNE SNPRN SNX SOHCF SOJ SPISZ SRMVM SSLCW STPWE SZN T13 TEORI TN5 TSG TSK TSV TUC TUS TWZ U2A UG4 UOJIU UPT UTJUX UZXMN VC2 VFIZW VH1 W23 W48 WH7 WK8 XU3 YLTOR ~8M ~A9 AAPKM AAYXX ABDBE ABFSG ABRTQ ACSTC ADHKG AEZWR AFDZB AFFHD AFHIU AFOHR AGQPQ AHPBZ AHWEU AIXLP AMVHM ATHPR CITATION M2P PHGZM PHGZT PQGLB |
| ID | FETCH-LOGICAL-c370t-1cf5040e1baa72a362d8164c6626ec889ade695a6f121054a74f7c16ebf299623 |
| IEDL.DBID | RSV |
| ISICitedReferencesCount | 2 |
| ISICitedReferencesURI | http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000640213600011&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| ISSN | 0012-2661 |
| IngestDate | Sat Nov 29 09:58:22 EST 2025 Tue Nov 18 22:26:18 EST 2025 Sat Nov 29 01:43:29 EST 2025 Fri Feb 21 02:48:27 EST 2025 |
| IsDoiOpenAccess | false |
| IsOpenAccess | true |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | 3 |
| Language | English |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-c370t-1cf5040e1baa72a362d8164c6626ec889ade695a6f121054a74f7c16ebf299623 |
| OpenAccessLink | https://link.springer.com/content/pdf/10.1134/S0012266121030113.pdf |
| PageCount | 17 |
| ParticipantIDs | gale_infotracacademiconefile_A716331578 crossref_citationtrail_10_1134_S0012266121030113 crossref_primary_10_1134_S0012266121030113 springer_journals_10_1134_S0012266121030113 |
| PublicationCentury | 2000 |
| PublicationDate | 20210300 2021-03-00 20210301 |
| PublicationDateYYYYMMDD | 2021-03-01 |
| PublicationDate_xml | – month: 3 year: 2021 text: 20210300 |
| PublicationDecade | 2020 |
| PublicationPlace | Moscow |
| PublicationPlace_xml | – name: Moscow |
| PublicationTitle | Differential equations |
| PublicationTitleAbbrev | Diff Equat |
| PublicationYear | 2021 |
| Publisher | Pleiades Publishing Springer |
| Publisher_xml | – name: Pleiades Publishing – name: Springer |
| References | MooreR.E.Methods and Applications of Interval Analysis1979ProvidenceSIAM10.1137/1.9781611970906 Pilarczyk, P., Topological-numerical approach to the existence of periodic trajectories in ODE’s, Proc. Fourth Int. Conf. Dyn. Syst. Differ. Equat. (Wilmington, May 24–27, 2002), pp. 701–708. BerzM.MakinoK.Verified integration of ODEs and flows using differential algebraic methods on high-order Taylor modelsReliab. Comput.19984361369165214710.1023/A:1024467732637 BerzM.HoefkensJ.Verified high-order inversion of functional dependencies and interval Newton methodsReliab. Comput.200175379398186292010.1023/A:1011423909873 Tucker, W., A Rigorous ODE solver and Smale’s 14th problem, Found. Comput. Math., 2002, no. 2, pp. 53–117. MooreR.E.KearfottR.B.CloudM.J.Introduction to Interval Analysis2009ProvidenceSIAM10.1137/1.9780898717716 EvstigneevN.M.RyabkovO.I.Algorithms for constructing isolating sets of phase flows and computer-assisted proofs with the use of interval Taylor modelsDiffer. Equations201955911981217402804210.1134/S0012266119090088 Rihm, R., Interval methods for initial value problems in ODEs, in Topics in Validated Computations, Herzberger, J., Ed., Amsterdam: Elsevier, 1994, pp. 173–207. EvstigneevN.M.RyabkovO.I.Applicability of the interval Taylor model to the computational proof of existence of periodic trajectories in systems of ordinary differential equationsDiffer. Equations2018544525538381377610.1134/S0012266118040092 SharyiS.P.Konechnomernyi interval’nyi analiz (Finite-Dimensional Interval Analysis)2010NovosibirskNovosib. Gos. Univ. BerzM.MakinoK.Suppression of the wrapping effect by Taylor model-based verified integrators: long-term stabilization by shrink wrappingInt. J. Differ. Equat. Appl.20051043854031133.65045 BerzM.MakinoK.Performance of Taylor model methods for validated integration of ODEsLect. Notes Comput. Sci.20063732657410.1007/11558958_8 Evstigneev, N.M. and Ryabkov, O.I., On the implementation of Taylor models on multiple graphics processing units for rigorous computations, in Parallel Computational Technologies. PCT 2020. Commun. Comput. Inf. Sci. Vol. 1263 , Sokolinsky, L. and Zymbler, M., Eds., Cham: Springer, 2020, pp. 85–99. BabenkoK.I.On computational proofs and mathematical experiments on computersRuss. Math. Surv.198540415315410.1070/RM1985v040n04ABEH003624 LinY.StadtherrM.A.Validated solutions of initial value problems for parametric ODEsAppl. Numer. Math.20075711451162234540510.1016/j.apnum.2006.10.006 ShokinYu.I.Interval’nyi analiz (Interval Analysis)1981NovosibirskNauka BerzM.MakinoK.Suppression of the wrapping effect by Taylor model-based verified integrators: long-term stabilization by preconditioningInt. J. Differ. Equat. Appl.200510435338423218181133.65045 Makino, K. and Berz, M., The method of shrink wrapping for the validated solution of ODEs, Michigan State Univ. Rep. MSU HEP 020510, 2002. DobronetsB.S.Interval’naya matematika (Interval Mathematics)2004KrasnoyarskSib. Gos. Tekh. Univ. NeherM.JacksonK.R.NedialkovN.S.On Taylor model based integration of ODEsSIAM J. Numer. Anal.2007451236262228585310.1137/050638448 2194_CR10 2194_CR11 M. Berz (2194_CR14) 2006; 3732 B.S. Dobronets (2194_CR4) 2004 K.I. Babenko (2194_CR6) 1985; 40 Y. Lin (2194_CR15) 2007; 57 2194_CR17 2194_CR18 M. Berz (2194_CR20) 2005; 10 N.M. Evstigneev (2194_CR8) 2018; 54 M. Berz (2194_CR13) 2005; 10 M. Berz (2194_CR16) 2001; 7 S.P. Sharyi (2194_CR5) 2010 R.E. Moore (2194_CR1) 1979 R.E. Moore (2194_CR2) 2009 N.M. Evstigneev (2194_CR9) 2019; 55 M. Berz (2194_CR12) 1998; 4 2194_CR7 M. Neher (2194_CR19) 2007; 45 Yu.I. Shokin (2194_CR3) 1981 |
| References_xml | – reference: LinY.StadtherrM.A.Validated solutions of initial value problems for parametric ODEsAppl. Numer. Math.20075711451162234540510.1016/j.apnum.2006.10.006 – reference: DobronetsB.S.Interval’naya matematika (Interval Mathematics)2004KrasnoyarskSib. Gos. Tekh. Univ. – reference: EvstigneevN.M.RyabkovO.I.Applicability of the interval Taylor model to the computational proof of existence of periodic trajectories in systems of ordinary differential equationsDiffer. Equations2018544525538381377610.1134/S0012266118040092 – reference: Pilarczyk, P., Topological-numerical approach to the existence of periodic trajectories in ODE’s, Proc. Fourth Int. Conf. Dyn. Syst. Differ. Equat. (Wilmington, May 24–27, 2002), pp. 701–708. – reference: BerzM.MakinoK.Verified integration of ODEs and flows using differential algebraic methods on high-order Taylor modelsReliab. Comput.19984361369165214710.1023/A:1024467732637 – reference: BerzM.MakinoK.Suppression of the wrapping effect by Taylor model-based verified integrators: long-term stabilization by shrink wrappingInt. J. Differ. Equat. Appl.20051043854031133.65045 – reference: BerzM.HoefkensJ.Verified high-order inversion of functional dependencies and interval Newton methodsReliab. Comput.200175379398186292010.1023/A:1011423909873 – reference: ShokinYu.I.Interval’nyi analiz (Interval Analysis)1981NovosibirskNauka – reference: NeherM.JacksonK.R.NedialkovN.S.On Taylor model based integration of ODEsSIAM J. Numer. Anal.2007451236262228585310.1137/050638448 – reference: Tucker, W., A Rigorous ODE solver and Smale’s 14th problem, Found. Comput. Math., 2002, no. 2, pp. 53–117. – reference: EvstigneevN.M.RyabkovO.I.Algorithms for constructing isolating sets of phase flows and computer-assisted proofs with the use of interval Taylor modelsDiffer. Equations201955911981217402804210.1134/S0012266119090088 – reference: Makino, K. and Berz, M., The method of shrink wrapping for the validated solution of ODEs, Michigan State Univ. Rep. MSU HEP 020510, 2002. – reference: BerzM.MakinoK.Suppression of the wrapping effect by Taylor model-based verified integrators: long-term stabilization by preconditioningInt. J. Differ. Equat. Appl.200510435338423218181133.65045 – reference: SharyiS.P.Konechnomernyi interval’nyi analiz (Finite-Dimensional Interval Analysis)2010NovosibirskNovosib. Gos. Univ. – reference: Rihm, R., Interval methods for initial value problems in ODEs, in Topics in Validated Computations, Herzberger, J., Ed., Amsterdam: Elsevier, 1994, pp. 173–207. – reference: MooreR.E.KearfottR.B.CloudM.J.Introduction to Interval Analysis2009ProvidenceSIAM10.1137/1.9780898717716 – reference: BerzM.MakinoK.Performance of Taylor model methods for validated integration of ODEsLect. Notes Comput. Sci.20063732657410.1007/11558958_8 – reference: Evstigneev, N.M. and Ryabkov, O.I., On the implementation of Taylor models on multiple graphics processing units for rigorous computations, in Parallel Computational Technologies. PCT 2020. Commun. Comput. Inf. Sci. Vol. 1263 , Sokolinsky, L. and Zymbler, M., Eds., Cham: Springer, 2020, pp. 85–99. – reference: MooreR.E.Methods and Applications of Interval Analysis1979ProvidenceSIAM10.1137/1.9781611970906 – reference: BabenkoK.I.On computational proofs and mathematical experiments on computersRuss. Math. Surv.198540415315410.1070/RM1985v040n04ABEH003624 – ident: 2194_CR7 doi: 10.1007/s002080010018 – volume: 10 start-page: 385 issue: 4 year: 2005 ident: 2194_CR13 publication-title: Int. J. Differ. Equat. Appl. – volume-title: Interval’nyi analiz (Interval Analysis) year: 1981 ident: 2194_CR3 – volume: 54 start-page: 525 issue: 4 year: 2018 ident: 2194_CR8 publication-title: Differ. Equations doi: 10.1134/S0012266118040092 – ident: 2194_CR10 – ident: 2194_CR11 – volume-title: Interval’naya matematika (Interval Mathematics) year: 2004 ident: 2194_CR4 – volume: 4 start-page: 361 year: 1998 ident: 2194_CR12 publication-title: Reliab. Comput. doi: 10.1023/A:1024467732637 – volume: 3732 start-page: 65 year: 2006 ident: 2194_CR14 publication-title: Lect. Notes Comput. Sci. doi: 10.1007/11558958_8 – volume: 45 start-page: 236 issue: 1 year: 2007 ident: 2194_CR19 publication-title: SIAM J. Numer. Anal. doi: 10.1137/050638448 – volume: 40 start-page: 153 issue: 4 year: 1985 ident: 2194_CR6 publication-title: Russ. Math. Surv. doi: 10.1070/RM1985v040n04ABEH003624 – volume: 7 start-page: 379 issue: 5 year: 2001 ident: 2194_CR16 publication-title: Reliab. Comput. doi: 10.1023/A:1011423909873 – ident: 2194_CR17 – ident: 2194_CR18 – volume-title: Methods and Applications of Interval Analysis year: 1979 ident: 2194_CR1 doi: 10.1137/1.9781611970906 – volume: 57 start-page: 1145 year: 2007 ident: 2194_CR15 publication-title: Appl. Numer. Math. doi: 10.1016/j.apnum.2006.10.006 – volume: 55 start-page: 1198 issue: 9 year: 2019 ident: 2194_CR9 publication-title: Differ. Equations doi: 10.1134/S0012266119090088 – volume: 10 start-page: 353 issue: 4 year: 2005 ident: 2194_CR20 publication-title: Int. J. Differ. Equat. Appl. – volume-title: Introduction to Interval Analysis year: 2009 ident: 2194_CR2 doi: 10.1137/1.9780898717716 – volume-title: Konechnomernyi interval’nyi analiz (Finite-Dimensional Interval Analysis) year: 2010 ident: 2194_CR5 |
| SSID | ssj0009715 |
| Score | 2.2000687 |
| Snippet | Using interval Taylor models (TM), we construct algorithms for the computer-assisted proof of the existence of periodic trajectories in systems of ordinary... Using interval Taylor models (TM), we construct algorithms for the computer-assistedproof of the existence of periodic trajectories in systems of ordinary... |
| SourceID | gale crossref springer |
| SourceType | Aggregation Database Enrichment Source Index Database Publisher |
| StartPage | 391 |
| SubjectTerms | Algorithms Analysis Difference and Functional Equations Differential equations Mathematics Mathematics and Statistics Numerical Methods Ordinary Differential Equations Partial Differential Equations |
| Title | Use of Shrink Wrapping for Interval Taylor Models in Algorithms of Computer-Assisted Proof of the Existence of Periodic Trajectories in Systems of Ordinary Differential Equations |
| URI | https://link.springer.com/article/10.1134/S0012266121030113 |
| Volume | 57 |
| WOSCitedRecordID | wos000640213600011&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| journalDatabaseRights | – providerCode: PRVAVX databaseName: Springer Standard Collection customDbUrl: eissn: 1608-3083 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0009715 issn: 0012-2661 databaseCode: RSV dateStart: 20000101 isFulltext: true titleUrlDefault: https://link.springer.com/search?facet-content-type=%22Journal%22 providerName: Springer Nature |
| link | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV1Lb9QwEB5B4QCHQgsVy0tzQOKlSOskjrPHFWzFAcqKUugtcvxoF5ZdWG-r_i5-ITOOU1TxkODq2GMrHs_D4_kG4JGsdFvWVmde8W2VE3WmlTfZSErrjJOtbWOi8Gu1t1cfHo6mKY879K_d-5BklNRd3ZGSc3pFzuokF9GMLy7DFdJ2NddreLf_4SfSrurLFuQZd0-hzN-SuKCMepF8MSAa9czujf9a4U3YTGYljjs-2IJLbrEN19-cY7KGbdhKxzjgk4Q1_fQWfD8IDpce949pps_4caUZr-EIyZTFeFlIjIidV49cNm0ecLbA8fxouZqtj78EHtsXhshoq5lpLE7JHPf8iabHyRk30sTcMCWGX9qZQdKRn2LAgDx1JpmQ07nPW-JZThLGl6l4CwmhOU6-daDk4TYc7E7ev3iVpTIOmSnUcJ0J4yWJCidarVWuSWPampw0U5Ev5Uxdj7R11UjqyvOPk6VWpVdGVK71pCvJPNuBjcVy4e4A-rw1WppSD1VbStdqKZwdSlsoL9g2HMCw38_GJIxzLrUxb6KvU5TNL3s0gGfnQ752AB9_6_yYmaThw090jU45DLQ6htFqxuR9FoUgKTiA5z2PNEkqhD_TvftPve_BtZxf1sSXcPdhY706cQ_gqjldz8LqYTwNPwDdngNQ |
| linkProvider | Springer Nature |
| linkToHtml | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV1LbxMxEB6VggQ9AC0gAhR8QOJRrRTvrtebY0RTFZGGiLbQ28rrRxtIkzYOiN_FL2TG621V8ZDg6rXH1no8D4_nG4DnolB1XhqVOEm3VZaXiZJOJz0hjNVW1KYOicJDORqVR0e9cczj9u1r9zYkGSR1U3ckp5xenpI6SXkw47NrcD1HhUWA-R_2P14i7cq2bEGaUPcYyvwtiSvKqBXJVwOiQc_s3PmvFd6F29GsZP2GD9Zhxc42YG3vApPVb8B6PMaevYxY06_uwY9Db9ncsf0TnOkL-7RQhNdwzNCUZeGyEBmRNV49o7JpU88mM9afHs8Xk-XJqaexbWGIBLeamMawMZrjjj7h9GzwnRpxYmoYI8PPzUQz1JGfQ8AAPXUiGZHTqc975FlKEmbbsXgLCqEpG5w3oOT-PhzuDA7e7CaxjEOiM9ldJlw7gaLC8lopmSrUmKZEJ00X6EtZXZY9ZWzRE6pw9ONErmTupOaFrR3qSjTPHsDqbD6zD4G5tNZK6Fx1ZZ0LWyvBrekKk0nHyTbsQLfdz0pHjHMqtTGtgq-T5dUve9SB1xdDzhqAj791fkFMUtHhR7paxRwGXB3BaFV99D6zjKMU7MBWyyNVlAr-z3Qf_VPvZ3Bz92BvWA3fjt49hlspvbIJr-KewOpy8dVuwg39bTnxi6fhZPwECyIGNA |
| linkToPdf | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV1Lb9QwEB6VglA5QFtALFDqAxIvRayTOM4eV-2uQJRlpVLoLXL8aLcs2bJOK34Xv5CZxGlV8ZAQV2c8tuLxPGzPNwBPRabKNDcqcpJOqyzPIyWdjgZCGKutKE3ZJArvyckkPzwcTEOdU9-9du-uJNucBkJpqurXp8aFGiQp5ffymExLzBuXPrkG11N6R0_h-v6nS9Rd2ZUwiCMiD9eav2VxxTB16vnq5Whjc8Z3_nu263A7uJts2MrHBqzYahNuvb_AavWbsBG2t2fPAwb1i7vw48BbtnBs_xhH_cI-LxXhOBwxdHFZc4iIAsraaJ9RObW5Z7OKDedHi-WsPv7qqW9XMCJCESBhMmyKbrqjTzg8G32nRhyYGqa4ERZmphnazpPmIgEjeGIZENWJ5gPKMiUPs91Q1AWV05yNvrVg5f4eHIxHH3feRKG8Q6QT2a8jrp1AFWJ5qZSMFVpSk2PwpjOMsazO84EyNhsIlTn6cSJVMnVS88yWDm0oum33YbVaVPYBMBeXWgmdqr4sU2FLJbg1fWES6Tj5jD3od2tb6IB9TiU45kUTAyVp8csa9eDlRZfTFvjjb8TPSGAKUgrIV6uQ24CzI3itYohRaZJw1I49eNXJSxG0hf8z34f_RL0NN6e742Lv7eTdI1iL6fFN81juMazWyzO7BTf0eT3zyyfNJvkJgckPGA |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Use+of+Shrink+Wrapping+for+Interval+Taylor+Models+in+Algorithms+of+Computer-Assisted+Proof+of+the+Existence+of+Periodic+Trajectories+in+Systems+of+Ordinary+Differential+Equations&rft.jtitle=Differential+equations&rft.au=Evstigneev%2C+N.+M.&rft.au=Ryabkov%2C+O.+I.&rft.au=Shul%E2%80%99min%2C+D.+A.&rft.date=2021-03-01&rft.pub=Pleiades+Publishing&rft.issn=0012-2661&rft.eissn=1608-3083&rft.volume=57&rft.issue=3&rft.spage=391&rft.epage=407&rft_id=info:doi/10.1134%2FS0012266121030113&rft.externalDocID=10_1134_S0012266121030113 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0012-2661&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0012-2661&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0012-2661&client=summon |