{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T07:55:29Z","timestamp":1776844529049,"version":"3.51.2"},"reference-count":18,"publisher":"American Mathematical Society (AMS)","issue":"271","license":[{"start":{"date-parts":[[2011,3,11]],"date-time":"2011-03-11T00:00:00Z","timestamp":1299801600000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    In this paper, we present a new method for rigorously computing smooth branches of zeros of nonlinear operators\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"f colon double-struck upper R Superscript l 1 Baseline times upper B 1 right-arrow double-struck upper R Superscript l 2 Baseline times upper B 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo>:<\/mml:mo>\n                            <mml:msup>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"double-struck\">R<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:msub>\n                                  <mml:mi>l<\/mml:mi>\n                                  <mml:mn>1<\/mml:mn>\n                                <\/mml:msub>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo>\n                              \u00d7\n                              \n                            <\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>B<\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">\n                              \u2192\n                              \n                            <\/mml:mo>\n                            <mml:msup>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"double-struck\">R<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:msub>\n                                  <mml:mi>l<\/mml:mi>\n                                  <mml:mn>2<\/mml:mn>\n                                <\/mml:msub>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo>\n                              \u00d7\n                              \n                            <\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>B<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">f:\\mathbb {R}^{l_1} \\times B_1 \\rightarrow \\mathbb {R}^{l_2} \\times B_2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper B 1\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>B<\/mml:mi>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">B_1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper B 2\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>B<\/mml:mi>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">B_2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    are Banach spaces. The method is first introduced for parameter continuation and then generalized to pseudo-arclength continuation. Examples in the context of ordinary, partial and delay differential equations are given.\n                  <\/p>","DOI":"10.1090\/s0025-5718-10-02325-2","type":"journal-article","created":{"date-parts":[[2010,4,15]],"date-time":"2010-04-15T09:01:02Z","timestamp":1271322062000},"page":"1565-1584","source":"Crossref","is-referenced-by-count":52,"title":["Global smooth solution curves using rigorous branch following"],"prefix":"10.1090","volume":"79","author":[{"given":"Jan","family":"van den Berg","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jean-Philippe","family":"Lessard","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Konstantin","family":"Mischaikow","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2010,3,11]]},"reference":[{"issue":"3","key":"1","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. 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