{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T13:33:43Z","timestamp":1762522423403,"version":"3.37.3"},"reference-count":32,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2020,3,10]],"date-time":"2020-03-10T00:00:00Z","timestamp":1583798400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,3,10]],"date-time":"2020-03-10T00:00:00Z","timestamp":1583798400000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Found Comput Math"],"published-print":{"date-parts":[[2020,12]]},"DOI":"10.1007\/s10208-020-09454-z","type":"journal-article","created":{"date-parts":[[2020,3,10]],"date-time":"2020-03-10T20:02:43Z","timestamp":1583870563000},"page":"1363-1400","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Preservation of Bifurcations of Hamiltonian Boundary Value Problems Under Discretisation"],"prefix":"10.1007","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0392-4957","authenticated-orcid":false,"given":"Robert I","family":"McLachlan","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5940-8057","authenticated-orcid":false,"given":"Christian","family":"Offen","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,3,10]]},"reference":[{"key":"9454_CR1","doi-asserted-by":"publisher","unstructured":"Arnold, V.I., Goryunov, V.V., Lyashko, O.V., Vasil\u2019ev, V.A.: Critical Points of Functions, pp. 10\u201350. Springer Berlin Heidelberg, Berlin, Heidelberg (1998). https:\/\/doi.org\/10.1007\/978-3-642-58009-3_1.","DOI":"10.1007\/978-3-642-58009-3_1"},{"key":"9454_CR2","unstructured":"Baek, J., Deopurkar, A., Redfield, K.: Finding geodesics on surfaces. [online] (2007). http:\/\/cs.stanford.edu\/people\/jbaek\/18.821.paper2.pdf"},{"key":"9454_CR3","doi-asserted-by":"publisher","unstructured":"Beyn, W.J., Th\u00fcmmler, V.: Phase Conditions, Symmetries and PDE Continuation, pp. 301\u2013330. Springer Netherlands, Dordrecht (2007). https:\/\/doi.org\/10.1007\/978-1-4020-6356-5_10.","DOI":"10.1007\/978-1-4020-6356-5_10"},{"issue":"4","key":"9454_CR4","doi-asserted-by":"publisher","first-page":"971","DOI":"10.1137\/0912052","volume":"12","author":"JR Cash","year":"1991","unstructured":"Cash, J.R., Wright, M.H.: A deferred correction method for nonlinear two-point boundary value problems: Implementation and numerical evaluation. SIAM Journal on Scientific and Statistical Computing 12(4), 971\u2013989 (1991). https:\/\/doi.org\/10.1137\/0912052.","journal-title":"SIAM Journal on Scientific and Statistical Computing"},{"key":"9454_CR5","unstructured":"Doedel, E.J., Champneys, A.R., Dercole, F., Fairgrieve, T.F., Yu, A., Oldeman, B., Paffenroth, R.C., Sandstede, B., Wang, X.J., Zhang, C.H.: AUTO-07P: Continuation and bifurcation software for ordinary differential equations (2007)"},{"key":"9454_CR6","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198535881.001.0001","volume-title":"The geometry of topological stability","author":"A Du Plessis","year":"1995","unstructured":"Du\u00a0Plessis, A., Wall, C.: The geometry of topological stability. London Mathematical Society Monographs. Clarendon Press (1995)"},{"issue":"2","key":"9454_CR7","doi-asserted-by":"publisher","first-page":"479","DOI":"10.1137\/S1064827593251496","volume":"17","author":"WH Enright","year":"1996","unstructured":"Enright, W.H., Muir, P.H.: Runge-Kutta software with defect control for boundary value odes. SIAM Journal on Scientific Computing 17(2), 479\u2013497 (1996). https:\/\/doi.org\/10.1137\/S1064827593251496.","journal-title":"SIAM Journal on Scientific Computing"},{"key":"9454_CR8","doi-asserted-by":"publisher","unstructured":"Flaherty, F., do\u00a0Carmo, M.: Riemannian Geometry. Mathematics: Theory & Applications. Birkh\u00e4user Boston (1992). https:\/\/doi.org\/10.1007\/978-1-4757-2201-7.","DOI":"10.1007\/978-1-4757-2201-7"},{"issue":"13","key":"9454_CR9","doi-asserted-by":"publisher","first-page":"2705","DOI":"10.1140\/epjst\/e2014-02287-6","volume":"223","author":"J Galan-Vioque","year":"2014","unstructured":"Galan-Vioque, J., Almaraz, F.J.M., Mac\u00edas, E.F.: Continuation of periodic orbits in symmetric Hamiltonian and conservative systems. The European Physical Journal Special Topics 223(13), 2705\u20132722 (2014). https:\/\/doi.org\/10.1140\/epjst\/e2014-02287-6.","journal-title":"The European Physical Journal Special Topics"},{"key":"9454_CR10","unstructured":"Gilmore, R.: Catastrophe Theory for Scientists and Engineers. Dover books on advanced mathematics. Dover Publications (1993). https:\/\/books.google.co.nz\/books?id=HbuecPcWxJUC"},{"key":"9454_CR11","doi-asserted-by":"publisher","unstructured":"Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. Springer Series in Computational Mathematics. Springer Berlin Heidelberg (2013). https:\/\/doi.org\/10.1007\/3-540-30666-8.","DOI":"10.1007\/3-540-30666-8"},{"issue":"03","key":"9454_CR12","doi-asserted-by":"publisher","first-page":"451","DOI":"10.1142\/S0218127402004498","volume":"12","author":"ME Henderson","year":"2002","unstructured":"Henderson, M.E.: Multiple parameter continuation: computing implicitly defined k-manifolds. International Journal of Bifurcation and Chaos 12(03), 451\u2013476 (2002). https:\/\/doi.org\/10.1142\/S0218127402004498.","journal-title":"International Journal of Bifurcation and Chaos"},{"key":"9454_CR13","unstructured":"H\u00e9non, M.: Numerical exploration of Hamiltonian systems. In: G.\u00a0Iooss, R.\u00a0Helleman, R.\u00a0Stora (eds.) Chaotic Behaviour of Deterministic Systems, Les Houches, XXXVI, pp. 53\u2013170. Elsevier (1983)"},{"key":"9454_CR14","doi-asserted-by":"publisher","unstructured":"Itoh, J.i., Kiyohara, K.: The cut loci and the conjugate loci on ellipsoids. manuscripta mathematica 114(2), 247\u2013264 (2004). https:\/\/doi.org\/10.1007\/s00229-004-0455-z.","DOI":"10.1007\/s00229-004-0455-z"},{"issue":"3","key":"9454_CR15","doi-asserted-by":"publisher","first-page":"299","DOI":"10.1145\/502800.502801","volume":"27","author":"J Kierzenka","year":"2001","unstructured":"Kierzenka, J., Shampine, L.F.: A bvp solver based on residual control and the Maltab PSE. ACM Trans. Math. Softw. 27(3), 299\u2013316 (2001). https:\/\/doi.org\/10.1145\/502800.502801.","journal-title":"ACM Trans. Math. Softw."},{"key":"9454_CR16","unstructured":"Kreusser, L.M., McLachlan, R., Offen, C.: Detection of high codimensional bifurcations in variational PDEs (2019). https:\/\/arxiv.org\/abs\/1903.02659"},{"key":"9454_CR17","doi-asserted-by":"publisher","unstructured":"Leimkuhler, B.J., Skeel, R.D.: Symplectic numerical integrators in constrained Hamiltonian systems. Journal of Computational Physics 112(1), 117\u2013125 (1994). https:\/\/doi.org\/10.1006\/jcph.1994.1085. http:\/\/www.sciencedirect.com\/science\/article\/pii\/S0021999184710850","DOI":"10.1006\/jcph.1994.1085"},{"key":"9454_CR18","doi-asserted-by":"publisher","unstructured":"Lu, Y.: Singularity theory and an introduction to catastrophe theory. Universitext (1979). Springer-Verlag (1976). https:\/\/doi.org\/10.1007\/978-1-4612-9909-7.","DOI":"10.1007\/978-1-4612-9909-7"},{"issue":"6","key":"9454_CR19","doi-asserted-by":"publisher","first-page":"2895","DOI":"10.1088\/1361-6544\/aab630","volume":"31","author":"RI McLachlan","year":"2018","unstructured":"McLachlan, R.I., Offen, C.: Bifurcation of solutions to Hamiltonian boundary value problems. Nonlinearity 31(6), 2895 (2018). https:\/\/doi.org\/10.1088\/1361-6544\/aab630","journal-title":"Nonlinearity"},{"key":"9454_CR20","doi-asserted-by":"crossref","first-page":"83","DOI":"10.53733\/34","volume":"48","author":"RI McLachlan","year":"2018","unstructured":"McLachlan, R.I., Offen, C.: Hamiltonian boundary value problems, conformal symplectic symmetries, and conjugate loci. New Zealand Journal of Mathematics (NZJM) 48, 83\u201399 (2018)","journal-title":"New Zealand Journal of Mathematics (NZJM)"},{"key":"9454_CR21","doi-asserted-by":"publisher","unstructured":"Mohsen, A.: A simple solution of the Bratu problem. Computers & Mathematics with Applications 67(1), 26\u201333 (2014). https:\/\/doi.org\/10.1016\/j.camwa.2013.10.003. http:\/\/www.sciencedirect.com\/science\/article\/pii\/S089812211300610X","DOI":"10.1016\/j.camwa.2013.10.003"},{"key":"9454_CR22","unstructured":"Offen, C.: Local intersections of Lagrangian manifolds correspond to catastrophe theory. Preprint (2018). https:\/\/arxiv.org\/abs\/1811.10165"},{"key":"9454_CR23","unstructured":"Offen, C.: Singularities illustrations cusp and swallowtail. https:\/\/www.youtube.com\/playlist?list=PLIp-UrijLTJ5m-3ZASHPurIkehiBuW_sO (2019). Accessed 13\/01\/20"},{"key":"9454_CR24","unstructured":"Offen, C.: Singularities illustrations elliptic umbilic. https:\/\/www.youtube.com\/playlist?list=PLIp-UrijLTJ7NUg6Vb3J_dV-mmdXFNPfN (2019). Accessed 13\/01\/20"},{"key":"9454_CR25","unstructured":"Offen, C.: Singularities illustrations hyperbolic umbilic. https:\/\/www.youtube.com\/watch?v=RU5DBPZTEG8&list=PLIp-UrijLTJ5C1dMN5vBaN63fWUiPg98e (2019). Accessed 13\/01\/20"},{"issue":"2","key":"9454_CR26","doi-asserted-by":"publisher","first-page":"121","DOI":"10.1023\/B:AGAG.0000018558.65790.db","volume":"25","author":"P Piccione","year":"2004","unstructured":"Piccione, P., Portaluri, A., Tausk, D.V.: Spectral flow, Maslov index and bifurcation of semi-Riemannian geodesics. Annals of Global Analysis and Geometry 25(2), 121\u2013149 (2004). https:\/\/doi.org\/10.1023\/B:AGAG.0000018558.65790.db.","journal-title":"Annals of Global Analysis and Geometry"},{"key":"9454_CR27","volume-title":"Chaos and integrability in nonlinear dynamics: an introduction","author":"M Tabor","year":"1989","unstructured":"Tabor, M.: Chaos and integrability in nonlinear dynamics: an introduction. Wiley-Interscience publication. Wiley (1989)"},{"issue":"2","key":"9454_CR28","doi-asserted-by":"publisher","first-page":"187","DOI":"10.1007\/s00371-014-1041-3","volume":"31","author":"H Thielhelm","year":"2015","unstructured":"Thielhelm, H., Vais, A., Wolter, F.E.: Geodesic bifurcation on smooth surfaces. The Visual Computer 31(2), 187\u2013204 (2015). https:\/\/doi.org\/10.1007\/s00371-014-1041-3.","journal-title":"The Visual Computer"},{"key":"9454_CR29","unstructured":"Uecker, H., Wetzel, D., Rademacher, J.: pde2path - a Matlab package for continuation and bifurcation in 2d elliptic systems (2012). https:\/\/arxiv.org\/abs\/1208.3112"},{"key":"9454_CR30","doi-asserted-by":"publisher","unstructured":"Wall, C.: Lectures on $${C}^\\infty $$ stability and classification. In: C.\u00a0Wall (ed.) Proceedings of Liverpool Singularities \u2014 Symposium I, pp. 178\u2013206. Springer Berlin Heidelberg, Berlin, Heidelberg (1971). https:\/\/doi.org\/10.1007\/BFb0066823.","DOI":"10.1007\/BFb0066823"},{"key":"9454_CR31","doi-asserted-by":"publisher","unstructured":"Waters, T.: Bifurcations of the conjugate locus. Journal of Geometry and Physics 119, 1\u20138 (2017). https:\/\/doi.org\/10.1016\/j.geomphys.2017.04.003. http:\/\/www.sciencedirect.com\/science\/article\/pii\/S039304401730102X","DOI":"10.1016\/j.geomphys.2017.04.003"},{"key":"9454_CR32","doi-asserted-by":"publisher","unstructured":"Weinstein, A.: Symplectic manifolds and their Lagrangian submanifolds. Advances in Mathematics 6(3), 329\u2013346 (1971). https:\/\/doi.org\/10.1016\/0001-8708(71)90020-X. http:\/\/www.sciencedirect.com\/science\/article\/pii\/000187087190020X","DOI":"10.1016\/0001-8708(71)90020-X"}],"container-title":["Foundations of Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10208-020-09454-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10208-020-09454-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10208-020-09454-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,1]],"date-time":"2024-08-01T23:03:47Z","timestamp":1722553427000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10208-020-09454-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,3,10]]},"references-count":32,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2020,12]]}},"alternative-id":["9454"],"URL":"https:\/\/doi.org\/10.1007\/s10208-020-09454-z","relation":{},"ISSN":["1615-3375","1615-3383"],"issn-type":[{"type":"print","value":"1615-3375"},{"type":"electronic","value":"1615-3383"}],"subject":[],"published":{"date-parts":[[2020,3,10]]},"assertion":[{"value":"23 April 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 January 2019","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 January 2020","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 March 2020","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}