{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,10]],"date-time":"2026-07-10T18:10:43Z","timestamp":1783707043278,"version":"3.55.0"},"reference-count":38,"publisher":"American Mathematical Society (AMS)","issue":"299","license":[{"start":{"date-parts":[[2016,9,28]],"date-time":"2016-09-28T00:00:00Z","timestamp":1475020800000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DSM 1318172"],"award-info":[{"award-number":["DSM 1318172"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    Judicious use of interval arithmetic, combined with careful pen and paper estimates, leads to effective strategies for computer assisted analysis of nonlinear operator equations. The method of radii polynomials is an efficient tool for bounding the smallest and largest neighborhoods on which a Newton-like operator associated with a nonlinear equation is a contraction mapping. The method has been used to study solutions of ordinary, partial, and delay differential equations such as equilibria, periodic orbits, solutions of initial value problems, heteroclinic and homoclinic connecting orbits in the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper C Superscript bold k\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>C<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"bold\">k<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">C^{\\mathbf {k}}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    category of functions. In the present work we adapt the method of radii polynomials to the analytic category. For ease of exposition we focus on studying periodic solutions in Cartesian products of infinite sequence spaces. We derive the radii polynomials for some specific application problems and give a number of computer assisted proofs in the analytic framework.\n                  <\/p>","DOI":"10.1090\/mcom\/3046","type":"journal-article","created":{"date-parts":[[2015,9,28]],"date-time":"2015-09-28T15:22:48Z","timestamp":1443453768000},"page":"1427-1459","source":"Crossref","is-referenced-by-count":65,"title":["Rigorous numerics for analytic solutions of differential equations: the radii polynomial approach"],"prefix":"10.1090","volume":"85","author":[{"given":"Allan","family":"Hungria","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jean-Philippe","family":"Lessard","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"J.","family":"Mireles James","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2015,9,28]]},"reference":[{"key":"1","isbn-type":"print","doi-asserted-by":"publisher","first-page":"331","DOI":"10.1007\/978-1-4020-6356-5_11","article-title":"Numerical computation of coherent structures","author":"Champneys, Alan R.","year":"2007","ISBN":"https:\/\/id.crossref.org\/isbn\/9781402063558"},{"issue":"3","key":"2","doi-asserted-by":"publisher","first-page":"427","DOI":"10.1090\/S0273-0979-1982-15008-X","article-title":"A computer-assisted proof of the Feigenbaum conjectures","volume":"6","author":"Lanford, Oscar E., III","year":"1982","journal-title":"Bull. Amer. Math. Soc. (N.S.)","ISSN":"https:\/\/id.crossref.org\/issn\/0273-0979","issn-type":"print"},{"issue":"4","key":"3","doi-asserted-by":"publisher","first-page":"565","DOI":"10.1137\/S0036144595284180","article-title":"Computer-assisted proofs in analysis and programming in logic: a case study","volume":"38","author":"Koch, Hans","year":"1996","journal-title":"SIAM Rev.","ISSN":"https:\/\/id.crossref.org\/issn\/1095-7200","issn-type":"print"},{"issue":"2","key":"4","doi-asserted-by":"publisher","first-page":"117","DOI":"10.1137\/030600210","article-title":"A rigorous numerical method for the global analysis of infinite-dimensional discrete dynamical systems","volume":"3","author":"Day, S.","year":"2004","journal-title":"SIAM J. Appl. Dyn. Syst."},{"key":"5","doi-asserted-by":"crossref","unstructured":"J. D. Mireles-James and K. Mischaikow, Computational proofs in dynamics, Encyclopedia of Applied Computational Mathematics, 2016. To appear.","DOI":"10.1007\/978-3-540-70529-1_322"},{"key":"6","doi-asserted-by":"publisher","first-page":"287","DOI":"10.1017\/S096249291000005X","article-title":"Verification methods: rigorous results using floating-point arithmetic","volume":"19","author":"Rump, Siegfried M.","year":"2010","journal-title":"Acta Numer.","ISSN":"https:\/\/id.crossref.org\/issn\/0962-4929","issn-type":"print"},{"issue":"3-4","key":"7","doi-asserted-by":"publisher","first-page":"321","DOI":"10.1081\/NFA-100105107","article-title":"Numerical verification methods for solutions of ordinary and partial differential equations","volume":"22","author":"Nakao, Mitsuhiro T.","year":"2001","journal-title":"Numer. Funct. Anal. Optim.","ISSN":"https:\/\/id.crossref.org\/issn\/0163-0563","issn-type":"print"},{"issue":"1","key":"8","doi-asserted-by":"publisher","first-page":"59","DOI":"10.1016\/j.jde.2004.03.014","article-title":"Covering relations for multidimensional dynamical systems. II","volume":"202","author":"Gidea, Marian","year":"2004","journal-title":"J. Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0022-0396","issn-type":"print"},{"issue":"4","key":"9","doi-asserted-by":"publisher","first-page":"1398","DOI":"10.1137\/050645968","article-title":"Validated continuation for equilibria of PDEs","volume":"45","author":"Day, Sarah","year":"2007","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"9","key":"10","doi-asserted-by":"publisher","first-page":"2237","DOI":"10.1016\/j.jde.2010.07.002","article-title":"Analytic estimates and rigorous continuation for equilibria of higher-dimensional PDEs","volume":"249","author":"Gameiro, Marcio","year":"2010","journal-title":"J. Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0022-0396","issn-type":"print"},{"issue":"3","key":"11","doi-asserted-by":"publisher","first-page":"988","DOI":"10.1137\/070709128","article-title":"Chaotic braided solutions via rigorous numerics: chaos in the Swift-Hohenberg equation","volume":"7","author":"van den Berg, Jan Bouwe","year":"2008","journal-title":"SIAM J. Appl. Dyn. Syst."},{"issue":"4","key":"12","doi-asserted-by":"publisher","first-page":"3093","DOI":"10.1016\/j.jde.2011.11.020","article-title":"Computational fixed-point theory for differential delay equations with multiple time lags","volume":"252","author":"Kiss, G\u00e1bor","year":"2012","journal-title":"J. Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0022-0396","issn-type":"print"},{"issue":"4","key":"13","doi-asserted-by":"publisher","first-page":"2063","DOI":"10.1137\/110836651","article-title":"Efficient rigorous numerics for higher-dimensional PDEs via one-dimensional estimates","volume":"51","author":"Gameiro, Marcio","year":"2013","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"12","key":"14","doi-asserted-by":"publisher","first-page":"4131","DOI":"10.1016\/j.na.2011.03.046","article-title":"Existence of secondary bifurcations or isolas for PDEs","volume":"74","author":"Gameiro, Marcio","year":"2011","journal-title":"Nonlinear Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0362-546X","issn-type":"print"},{"key":"15","doi-asserted-by":"publisher","first-page":"113","DOI":"10.1007\/s10440-013-9823-6","article-title":"Global bifurcation diagrams of steady states of systems of PDEs via rigorous numerics: a 3-component reaction-diffusion system","volume":"128","author":"Breden, Maxime","year":"2013","journal-title":"Acta Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0167-8019","issn-type":"print"},{"issue":"5","key":"16","doi-asserted-by":"publisher","first-page":"992","DOI":"10.1016\/j.jde.2009.11.008","article-title":"Recent advances about the uniqueness of the slowly oscillating periodic solutions of Wright\u2019s equation","volume":"248","author":"Lessard, Jean-Philippe","year":"2010","journal-title":"J. Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0022-0396","issn-type":"print"},{"issue":"4","key":"17","doi-asserted-by":"publisher","first-page":"1368","DOI":"10.1016\/j.matcom.2008.03.014","article-title":"Validated continuation over large parameter ranges for equilibria of PDEs","volume":"79","author":"Gameiro, Marcio","year":"2008","journal-title":"Math. Comput. Simulation","ISSN":"https:\/\/id.crossref.org\/issn\/0378-4754","issn-type":"print"},{"issue":"2","key":"18","doi-asserted-by":"publisher","first-page":"267","DOI":"10.1007\/s10884-014-9367-0","article-title":"Computer assisted proof of transverse saddle-to-saddle connecting orbits for first order vector fields","volume":"26","author":"Lessard, Jean-Philippe","year":"2014","journal-title":"J. Dynam. Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/1040-7294","issn-type":"print"},{"issue":"1","key":"19","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1137\/13090883X","article-title":"Rigorous numerics for nonlinear differential equations using Chebyshev series","volume":"52","author":"Lessard, Jean-Philippe","year":"2014","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"20","unstructured":"R. Castelli and H. Teismann, Rigorous numerics for NLS: bound states, spectra, and controllability. Preprint, 2013."},{"issue":"4","key":"21","doi-asserted-by":"publisher","first-page":"1557","DOI":"10.1137\/100812008","article-title":"Rigorous numerics for symmetric connecting orbits: even homoclinics of the Gray-Scott equation","volume":"43","author":"van den Berg, Jan Bouwe","year":"2011","journal-title":"SIAM J. Math. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1410","issn-type":"print"},{"issue":"1","key":"22","doi-asserted-by":"publisher","first-page":"423","DOI":"10.1137\/140987973","article-title":"Rigorous computation of a radially symmetric localized solution in a Ginzburg-Landau problem","volume":"14","author":"van den Berg, J. B.","year":"2015","journal-title":"SIAM J. Appl. Dyn. Syst."},{"issue":"1","key":"23","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1017\/S0956792514000308","article-title":"Coexistence of nontrivial solutions of the one-dimensional Ginzburg-Landau equation: a computer-assisted proof","volume":"26","author":"Correc, Ana\u00efs","year":"2015","journal-title":"European J. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0956-7925","issn-type":"print"},{"issue":"1","key":"24","doi-asserted-by":"publisher","first-page":"204","DOI":"10.1137\/120873960","article-title":"Rigorous numerics in Floquet theory: computing stable and unstable bundles of periodic orbits","volume":"12","author":"Castelli, Roberto","year":"2013","journal-title":"SIAM J. Appl. Dyn. Syst."},{"issue":"1","key":"25","doi-asserted-by":"publisher","first-page":"132","DOI":"10.1137\/140960207","article-title":"Parameterization of invariant manifolds for periodic orbits I: Efficient numerics via the Floquet normal form","volume":"14","author":"Castelli, Roberto","year":"2015","journal-title":"SIAM J. Appl. Dyn. Syst."},{"issue":"271","key":"26","doi-asserted-by":"publisher","first-page":"1565","DOI":"10.1090\/S0025-5718-10-02325-2","article-title":"Global smooth solution curves using rigorous branch following","volume":"79","author":"van den Berg, Jan Bouwe","year":"2010","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"27","doi-asserted-by":"crossref","unstructured":"M. Gameiro, J.-P. Lessard, and A. Pugliese, Computation of smooth manifolds of solutions of PDEs via rigorous multi-parameter continuation. To appear in Foundations of Computational Mathematics, 2015.","DOI":"10.1007\/s10208-015-9259-7"},{"issue":"4","key":"28","doi-asserted-by":"publisher","first-page":"753","DOI":"10.1007\/s00211-010-0350-3","article-title":"Rigorous computation of smooth branches of equilibria for the three dimensional Cahn-Hilliard equation","volume":"117","author":"Gameiro, Marcio","year":"2011","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"289","key":"29","doi-asserted-by":"publisher","first-page":"vi+122","DOI":"10.1090\/memo\/0289","article-title":"A computer-assisted proof of universality for area-preserving maps","volume":"47","author":"Eckmann, J.-P.","year":"1984","journal-title":"Mem. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-9266","issn-type":"print"},{"key":"30","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/b97227","volume-title":"A short course on spectral theory","volume":"209","author":"Arveson, William","year":"2002","ISBN":"https:\/\/id.crossref.org\/isbn\/0387953000"},{"issue":"2","key":"31","doi-asserted-by":"publisher","first-page":"942","DOI":"10.1137\/140984506","article-title":"Stationary coexistence of hexagons and rolls via rigorous computations","volume":"14","author":"van den Berg, Jan Bouwe","year":"2015","journal-title":"SIAM J. Appl. Dyn. Syst."},{"key":"32","unstructured":"M. Gameiro R. de la Llave, J.-L. Figueras and J.-P. Lessard, Theoretical results on the numerical computation and a-posteriori verification of invariant objects of evolution equations, in preparation, 2014."},{"key":"33","series-title":"Addison-Wesley Series in Computer Science and Information Processing","isbn-type":"print","volume-title":"The art of computer programming. Vol. 2","author":"Knuth, Donald E.","year":"1981","ISBN":"https:\/\/id.crossref.org\/isbn\/0201038226","edition":"2"},{"issue":"1","key":"34","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1080\/10586458.2005.10128904","article-title":"A software package for the numerical integration of ODEs by means of high-order Taylor methods","volume":"14","author":"Jorba, \u00c0ngel","year":"2005","journal-title":"Experiment. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1058-6458","issn-type":"print"},{"issue":"1","key":"35","doi-asserted-by":"publisher","first-page":"66","DOI":"10.1090\/S0273-0979-1995-00558-6","article-title":"Chaos in the Lorenz equations: a computer-assisted proof","volume":"32","author":"Mischaikow, Konstantin","year":"1995","journal-title":"Bull. Amer. Math. Soc. (N.S.)","ISSN":"https:\/\/id.crossref.org\/issn\/0273-0979","issn-type":"print"},{"issue":"2","key":"36","doi-asserted-by":"publisher","first-page":"280","DOI":"10.1137\/050626429","article-title":"Rigorous computations of homoclinic tangencies","volume":"5","author":"Arai, Zin","year":"2006","journal-title":"SIAM J. Appl. Dyn. Syst."},{"key":"37","doi-asserted-by":"crossref","unstructured":"J. B. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A, 15(1) (1977).","DOI":"10.1103\/PhysRevA.15.319"},{"key":"38","doi-asserted-by":"crossref","unstructured":"S. M. Rump, INTLAB - INTerval LABoratory, in Tibor Csendes, editor, Developments in Reliable Computing, Kluwer Academic Publishers, Dordrecht, 1999, pp. 77\u2013104. http:\/\/www.ti3.tu-harburg.de\/rump\/.","DOI":"10.1007\/978-94-017-1247-7_7"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2016-85-299\/S0025-5718-2015-03046-4\/S0025-5718-2015-03046-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2016-85-299\/S0025-5718-2015-03046-4\/S0025-5718-2015-03046-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:45:38Z","timestamp":1776797138000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2016-85-299\/S0025-5718-2015-03046-4\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,9,28]]},"references-count":38,"journal-issue":{"issue":"299","published-print":{"date-parts":[[2016,5]]}},"alternative-id":["S0025-5718-2015-03046-4"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3046","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2015,9,28]]}}}