{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T15:13:43Z","timestamp":1772291623416,"version":"3.50.1"},"reference-count":37,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2017,10,30]],"date-time":"2017-10-30T00:00:00Z","timestamp":1509321600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Found Comput Math"],"published-print":{"date-parts":[[2018,12]]},"DOI":"10.1007\/s10208-017-9369-5","type":"journal-article","created":{"date-parts":[[2017,10,30]],"date-time":"2017-10-30T14:44:09Z","timestamp":1509374649000},"page":"1299-1332","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":["Algorithm for Rigorous Integration of Delay Differential Equations and the Computer-Assisted Proof of Periodic Orbits in the Mackey\u2013Glass Equation"],"prefix":"10.1007","volume":"18","author":[{"given":"Robert","family":"Szczelina","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Piotr","family":"Zgliczy\u0144ski","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2017,10,30]]},"reference":[{"key":"9369_CR1","unstructured":"CAPD DynSys library. http:\/\/capd.ii.uj.edu.pl , 2014. Accessed: 2017-07-11."},{"key":"9369_CR2","doi-asserted-by":"crossref","first-page":"1119","DOI":"10.1137\/10078298X","volume":"9","author":"G Arioli","year":"2010","unstructured":"G.\u00a0Arioli and H.\u00a0Koch. Integration of dissipative partial differential equations: A case study. SIAM J. Appl. Dyn. Syst., 9:1119\u20131133, 2010.","journal-title":"SIAM J. Appl. Dyn. Syst."},{"key":"9369_CR3","doi-asserted-by":"crossref","first-page":"537","DOI":"10.1137\/120904226","volume":"13","author":"B B\u00e1nhelyi","year":"2014","unstructured":"B.\u00a0B\u00e1nhelyi, T.\u00a0Csendes, A.\u00a0Neumaier, and T.\u00a0Krisztin. Global attractivity of the zero solution for Wright\u2019s equation. J. Appl. Dyn. Syst., 13:537\u2013563, 2014.","journal-title":"J. Appl. Dyn. Syst."},{"key":"9369_CR4","doi-asserted-by":"crossref","first-page":"2043","DOI":"10.1080\/10236198.2013.804916","volume":"19","author":"FA Bartha","year":"2013","unstructured":"F.A. Bartha, A.\u00a0Garab, and T.\u00a0Krisztin. Local stability implies global stability for the 2-dimensional Ricker map. J. Difference Equ. Appl., 19:2043\u20132078, 2013.","journal-title":"J. Difference Equ. Appl."},{"key":"9369_CR5","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4684-9467-9","volume-title":"Ordinary and Delay Differential Equations","author":"RD Driver","year":"1977","unstructured":"R.D. Driver. Ordinary and Delay Differential Equations. Springer-Verlag, New York, 1977."},{"issue":"1","key":"9369_CR6","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1016\/j.jde.2004.03.014","volume":"202","author":"M Gidea","year":"2004","unstructured":"M.\u00a0Gidea and P.\u00a0Zgliczy\u0144ski. Covering relations for multidimensional dynamical systems. J. Differential Equations, 202(1):32\u201358, 2004.","journal-title":"J. Differential Equations"},{"issue":"6","key":"9369_CR7","doi-asserted-by":"crossref","first-page":"1899","DOI":"10.1088\/0951-7715\/16\/6\/302","volume":"16","author":"T Kapela","year":"2003","unstructured":"T.\u00a0Kapela and P.\u00a0Zgliczy\u0144ski. The existence of simple choreographies for the n -body problem-a computer-assisted proof. Nonlinearity, 16(6):1899, 2003.","journal-title":"Nonlinearity"},{"issue":"4","key":"9369_CR8","doi-asserted-by":"crossref","first-page":"3093","DOI":"10.1016\/j.jde.2011.11.020","volume":"252","author":"G Kiss","year":"2012","unstructured":"G. Kiss and J.P. Lessard. Computational fixed-point theory for differential delay equations with multiple time lags. J. of Differential Equations, 252(4):3093\u20133115, 2012.","journal-title":"J. of Differential Equations"},{"issue":"1","key":"9369_CR9","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1007\/s10998-008-5083-x","volume":"56","author":"T Krisztin","year":"2008","unstructured":"T.\u00a0Krisztin. Global dynamics of delay differential equations. Period. Math. Hungar., 56(1):83\u201395, 2008.","journal-title":"Period. Math. Hungar."},{"issue":"4","key":"9369_CR10","doi-asserted-by":"crossref","first-page":"727","DOI":"10.1007\/s10884-011-9225-2","volume":"23","author":"T Krisztin","year":"2011","unstructured":"T.\u00a0Krisztin and G.\u00a0Vas.Large-Amplitude Periodic Solutions for Differential Equations with Delayed Monotone Positive Feedback. J. Dyn. Diff. Eq., 23(4):727\u2013790, 2011.","journal-title":"J. Dyn. Diff. Eq."},{"key":"9369_CR11","volume-title":"Shape, smoothness and invariant stratification of an attracting set for delayed monotone positive feedback","author":"T Krisztin","year":"1999","unstructured":"T.\u00a0Krisztin, H.O. Walther, and J.\u00a0Wu. Shape, smoothness and invariant stratification of an attracting set for delayed monotone positive feedback, volume\u00a011. American Mathematical Society, Providence, RI, 1999."},{"key":"9369_CR12","doi-asserted-by":"crossref","unstructured":"B.\u00a0Lani-Wayda and R.\u00a0Srzednicki. A generalized lefschetz fixed point theorem and symbolic dynamics in delay equations. Ergodic Theory Dynam. Systems, 22:1215\u20131232, 8 2002.","DOI":"10.1017\/S0143385702000639"},{"issue":"1","key":"9369_CR13","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1002\/mana.3211800109","volume":"180","author":"B Lani-Wayda","year":"1996","unstructured":"B.\u00a0Lani-Wayda and H-O. Walther. Chaotic motion generated by delayed negative feedback part ii: Construction of nonlinearities. Math. Nachr., 180(1):141\u2013211, 1996.","journal-title":"Math. Nachr."},{"issue":"5","key":"9369_CR14","doi-asserted-by":"crossref","first-page":"992","DOI":"10.1016\/j.jde.2009.11.008","volume":"248","author":"JP Lessard","year":"2010","unstructured":"J.P. Lessard. Recent advances about the uniqueness of the slowly oscillating periodic solutions of Wright\u2019s equation. J. of Differential Equations, 248(5):992\u20131016, 2010.","journal-title":"J. of Differential Equations"},{"key":"9369_CR15","unstructured":"R.J. Lohner, J.R. Cach\u00a0(ed), and I.\u00a0Gladwel\u00a0(ed). Computation of Guaranteed Enclosures for the Solutions of Ordinary Initial and Boundary Value Problems: in Computational Ordinary Differential Equations. Clarendon Press, Oxford, New York, 1992."},{"key":"9369_CR16","unstructured":"M.\u00a0C. Mackey and L.\u00a0Glass. Mackey\u2013Glass equation, article on scholarpedia. http:\/\/www.scholarpedia.org\/article\/Mackey-Glass_equation ."},{"issue":"4300","key":"9369_CR17","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1126\/science.267326","volume":"197","author":"MC Mackey","year":"1977","unstructured":"M.\u00a0C. Mackey and L.\u00a0Glass. Oscillation and chaos in physiological control systems. Science, 197(4300):287\u2013289, 1977.","journal-title":"Science"},{"issue":"4","key":"9369_CR18","doi-asserted-by":"crossref","first-page":"2468","DOI":"10.1137\/13091943X","volume":"46","author":"J Mallet-Paret","year":"2014","unstructured":"J.\u00a0Mallet-Paret and R.\u00a0D. Nussbaum. Analyticity and nonanalyticity of solutions of delay-differential equations. SIAM J. Math. Anal., 46(4):2468\u20132500, 2014.","journal-title":"SIAM J. Math. Anal."},{"issue":"2","key":"9369_CR19","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1006\/jdeq.1996.0037","volume":"125","author":"J Mallet-Paret","year":"1996","unstructured":"J. Mallet-Paret and G. R. Sell. The Poincar\u00e9-Bendixson Theorem for Monotone Cyclic Feedback Systems with Delay. J. of Differential Equations, 125(2):441\u2013489, 1996.","journal-title":"J. of Differential Equations"},{"issue":"1","key":"9369_CR20","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1006\/jdeq.2000.3894","volume":"169","author":"K Mischaikow","year":"2001","unstructured":"K. Mischaikow, M. Mrozek, and A. Szymczak. Chaos in the lorenz equations: A computer assisted proof part iii: Classical parameter values. J. of Differential Equations, 169(1):17\u201356, 2001.","journal-title":"J. of Differential Equations"},{"key":"9369_CR21","unstructured":"R.E. Moore. Interval Analysis. Prentice Hall, 1966."},{"key":"9369_CR22","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1307\/mmj\/1029001104","volume":"20","author":"RD Nussbaum","year":"1973","unstructured":"R.\u00a0D. Nussbaum. Periodic solutions of analytic functional differential equations are analytic. Michigan Math. J., 20:249\u2013255, 1973.","journal-title":"Michigan Math. J."},{"key":"9369_CR23","unstructured":"Szczelina R. Source codes for the computer assisted proofs. http:\/\/scirsc.org\/p\/mackeyglass . Accessed: 2017-07-11."},{"key":"9369_CR24","unstructured":"Szczelina R. Rigorous Integration of Delay Differential Equations. PhD Thesis, Jagiellonian University, Krak\u00f3w, Poland, 2015, materials published online: http:\/\/scirsc.org\/p\/phd , 2015. Accessed: 2017-07-11."},{"key":"9369_CR25","doi-asserted-by":"crossref","first-page":"1","DOI":"10.14232\/ejqtde.2016.1.83","volume":"83","author":"R Szczelina","year":"2016","unstructured":"Szczelina R. A computer assisted proof of multiple periodic orbits in some first order non-linear delay differential equation. Electron. J. Qual. Theory Differ. Equ., 83:1\u201319, 2016.","journal-title":"Electron. J. Qual. Theory Differ. Equ."},{"key":"9369_CR26","doi-asserted-by":"crossref","unstructured":"L.B. Rall. Automatic Differentiation: Techniques and Applications. In: Lecture Notes in Computer Science, vol 120. Springer Verlag, 1981.","DOI":"10.1007\/3-540-10861-0"},{"key":"9369_CR27","doi-asserted-by":"crossref","first-page":"171","DOI":"10.4064\/sm-2-1-171-180","volume":"2","author":"J Schauder","year":"1930","unstructured":"J.\u00a0Schauder. Der Fixpunktsatz in Funktionalr\u00e4umen. Studia Math., 2:171\u2013189, 1930.","journal-title":"Studia Math."},{"key":"9369_CR28","doi-asserted-by":"publisher","unstructured":"IEEE\u00a0Computer Society. IEEE Standard for Floating-Point Arithmetic. published online: http:\/\/ieeexplore.ieee.org\/servlet\/opac?punumber=4610933 , https:\/\/doi.org\/10.1109\/IEEESTD.2008.4610935 , ISBN 978-0-7381-5753-5, 2008. Accessed: 2017-07-11.","DOI":"10.1109\/IEEESTD.2008.4610935"},{"issue":"1","key":"9369_CR29","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1007\/s002080010018","volume":"2","author":"W Tucker","year":"2002","unstructured":"W.\u00a0Tucker. A rigorous ODE solver and Smale\u2019s 14th problem. Found. Comput. Math., 2(1):53\u2013117, 2002.","journal-title":"Found. Comput. Math."},{"issue":"3","key":"9369_CR30","doi-asserted-by":"crossref","first-page":"1850","DOI":"10.1016\/j.jde.2016.10.031","volume":"262","author":"Gabriella Vas","year":"2017","unstructured":"Gabriella Vas. Configurations of periodic orbits for equations with delayed positive feedback. J. of Differential Equations, 262(3):1850\u20131896, 2017.","journal-title":"J. of Differential Equations"},{"issue":"1","key":"9369_CR31","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1016\/j.jde.2003.07.001","volume":"195","author":"H-O Walther","year":"2003","unstructured":"H.-O. Walther. The solution manifold and $$C^1$$ C 1 -smoothness for differential equations with state-dependent delay. J. of Differential Equations, 195(1):46\u201365, 2003.","journal-title":"J. of Differential Equations"},{"issue":"4","key":"9369_CR32","doi-asserted-by":"crossref","first-page":"1632","DOI":"10.1137\/090759975","volume":"8","author":"D Wilczak","year":"2009","unstructured":"D.\u00a0Wilczak and P.\u00a0Zgliczy\u0144ski. Computer Assisted Proof of the Existence of Homoclinic Tangency for the H\u00e9non Map and for the Forced Damped Pendulum. SIAM J. Appl. Dyn. Syst., 8(4):1632\u20131663, 2009.","journal-title":"SIAM J. Appl. Dyn. Syst."},{"issue":"2","key":"9369_CR33","doi-asserted-by":"crossref","first-page":"373","DOI":"10.12775\/TMNA.2009.025","volume":"33","author":"M Zalewski","year":"2009","unstructured":"M.\u00a0Zalewski. Computer-assisted proof of a periodic solution in a nonlinear feedback DDE. Topol. Methods Nonlinear Anal., 33(2):373\u2013393, 2009.","journal-title":"Topol. Methods Nonlinear Anal."},{"key":"9369_CR34","volume-title":"Applied Functional Analysis: Applications to Mathematical Physics","author":"E Zeidler","year":"1995","unstructured":"E.\u00a0Zeidler. Applied Functional Analysis: Applications to Mathematical Physics. Springer New York, New York, NY, 1995."},{"key":"9369_CR35","doi-asserted-by":"crossref","unstructured":"P.\u00a0Zgliczy\u0144ski. $$C^1$$ C 1 -Lohner algorithm. Found. Comput. Math., 2, 2002.","DOI":"10.1007\/s102080010025"},{"issue":"2","key":"9369_CR36","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1007\/s10208-002-0080-8","volume":"4","author":"P Zgliczy\u0144ski","year":"2004","unstructured":"P.\u00a0Zgliczy\u0144ski. Rigorous Numerics for Dissipative Partial Differential Equations II. Periodic Orbit for the Kuramoto-Sivashinsky PDE; A Computer-Assisted Proof. Found. Comput. Math., 4(2):157\u2013185, April 2004.","journal-title":"Found. Comput. Math."},{"key":"9369_CR37","first-page":"197","volume":"36","author":"P Zgliczy\u0144ski","year":"2010","unstructured":"P.\u00a0Zgliczy\u0144ski. Rigorous Numerics for Dissipative Partial Differential Equations III. An effective algorithm for rigorous integration of dissipative PDEs. Topol. Methods Nonlinear Anal., 36:197\u2013262, 2010.","journal-title":"Topol. Methods Nonlinear Anal."}],"container-title":["Foundations of Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10208-017-9369-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10208-017-9369-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10208-017-9369-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,10,5]],"date-time":"2019-10-05T06:10:06Z","timestamp":1570255806000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10208-017-9369-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,10,30]]},"references-count":37,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2018,12]]}},"alternative-id":["9369"],"URL":"https:\/\/doi.org\/10.1007\/s10208-017-9369-5","relation":{},"ISSN":["1615-3375","1615-3383"],"issn-type":[{"value":"1615-3375","type":"print"},{"value":"1615-3383","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,10,30]]}}}