{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T15:23:00Z","timestamp":1772292180623,"version":"3.50.1"},"reference-count":50,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2020,2,15]],"date-time":"2020-02-15T00:00:00Z","timestamp":1581724800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,2,15]],"date-time":"2020-02-15T00:00:00Z","timestamp":1581724800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/100000086","name":"Directorate for Mathematical and Physical Sciences","doi-asserted-by":"publisher","award":["1818769"],"award-info":[{"award-number":["1818769"]}],"id":[{"id":"10.13039\/100000086","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2020,3]]},"DOI":"10.1007\/s10915-020-01160-w","type":"journal-article","created":{"date-parts":[[2020,2,15]],"date-time":"2020-02-15T17:02:51Z","timestamp":1581786171000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["An Adaptive Homotopy Method for Computing Bifurcations of Nonlinear Parametric Systems"],"prefix":"10.1007","volume":"82","author":[{"given":"Wenrui","family":"Hao","sequence":"first","affiliation":[]},{"given":"Chunyue","family":"Zheng","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,2,15]]},"reference":[{"key":"1160_CR1","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719154","volume-title":"Introduction to Numerical Continuation Methods","author":"EL Allgower","year":"2003","unstructured":"Allgower, E.L., Georg, K.: Introduction to Numerical Continuation Methods, vol. 45. SIAM, Philadelphia (2003)"},{"issue":"4","key":"1160_CR2","doi-asserted-by":"publisher","first-page":"451","DOI":"10.1007\/s11075-011-9463-8","volume":"58","author":"D Bates","year":"2011","unstructured":"Bates, D., Hauenstein, J., Sommese, A.: Efficient path tracking methods. Numer. Algorithms 58(4), 451\u2013459 (2011)","journal-title":"Numer. Algorithms"},{"key":"1160_CR3","unstructured":"Bates, D., Hauenstein, J., Sommese, A., Wampler, C.: Bertini: Software for numerical algebraic geometry (2006)"},{"issue":"2","key":"1160_CR4","doi-asserted-by":"publisher","first-page":"722","DOI":"10.1137\/060658862","volume":"46","author":"D Bates","year":"2008","unstructured":"Bates, D., Hauenstein, J., Sommese, A., Wampler, C.: Adaptive multiprecision path tracking. SIAM J. Numer. Anal. 46(2), 722\u2013746 (2008)","journal-title":"SIAM J. Numer. Anal."},{"key":"1160_CR5","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611972702","volume-title":"Numerically Solving Polynomial Systems with Bertini","author":"D Bates","year":"2013","unstructured":"Bates, D., Hauenstein, J., Sommese, A., Wampler, C.: Numerically Solving Polynomial Systems with Bertini, vol. 25. SIAM, Philadelphia (2013)"},{"key":"1160_CR6","doi-asserted-by":"publisher","first-page":"25","DOI":"10.1090\/conm\/556\/11006","volume":"556","author":"J Bates","year":"2011","unstructured":"Bates, J., Hauenstein, D., Sommese, A.: A parallel endgame. Contemp. Math. 556, 25\u201335 (2011)","journal-title":"Contemp. Math."},{"key":"1160_CR7","doi-asserted-by":"publisher","DOI":"10.1515\/9781400884339","volume-title":"Analytic Theory of Global Bifurcation: An Introduction","author":"B Buffoni","year":"2003","unstructured":"Buffoni, B., Toland, J., Toland, J.F.: Analytic Theory of Global Bifurcation: An Introduction. Princeton University Press, Princeton (2003)"},{"issue":"3","key":"1160_CR8","doi-asserted-by":"publisher","first-page":"361","DOI":"10.1007\/s00285-008-0166-2","volume":"57","author":"X Chen","year":"2008","unstructured":"Chen, X., Hambrock, R., Lou, Y.: Evolution of conditional dispersal: a reaction\u2013diffusion\u2013advection model. J. Math. Biol. 57(3), 361\u2013386 (2008)","journal-title":"J. Math. Biol."},{"issue":"2","key":"1160_CR9","doi-asserted-by":"publisher","first-page":"407","DOI":"10.1006\/jdeq.1994.1110","volume":"112","author":"C Chicone","year":"1994","unstructured":"Chicone, C.: Lyapunov\u2013Schmidt reduction and melnikov integrals for bifurcation of periodic solutions in coupled oscillators. J. Differ. Equ. 112(2), 407\u2013447 (1994)","journal-title":"J. Differ. Equ."},{"issue":"6","key":"1160_CR10","doi-asserted-by":"publisher","first-page":"647","DOI":"10.1016\/0098-1354(95)00199-9","volume":"20","author":"S Choi","year":"1996","unstructured":"Choi, S., Harney, D., Book, N.: A robust path tracking algorithm for homotopy continuation. Comput. Chem. Eng. 20(6), 647\u2013655 (1996)","journal-title":"Comput. Chem. Eng."},{"key":"1160_CR11","doi-asserted-by":"crossref","unstructured":"Dayton, B., Zeng, Z.: Computing the multiplicity structure in solving polynomial systems. In: Proceedings of the 2005 International Symposium on Symbolic and Algebraic Computation. ACM, pp. 116\u2013123 (2005)","DOI":"10.1145\/1073884.1073902"},{"key":"1160_CR12","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-23899-4","volume-title":"Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms","author":"P Deuflhard","year":"2011","unstructured":"Deuflhard, P.: Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms, vol. 35. Springer, Berlin (2011)"},{"key":"1160_CR13","volume-title":"Plane Algebraic Curves","author":"G Fischer","year":"2001","unstructured":"Fischer, G.: Plane Algebraic Curves, vol. 15. American Mathematical Society, Providence (2001)"},{"issue":"5","key":"1160_CR14","doi-asserted-by":"publisher","first-page":"758","DOI":"10.1007\/s11538-014-0010-3","volume":"77","author":"A Friedman","year":"2015","unstructured":"Friedman, A., Hao, W.: A mathematical model of atherosclerosis with reverse cholesterol transport and associated risk factors. Bull. Math. Biol. 77(5), 758\u2013781 (2015)","journal-title":"Bull. Math. Biol."},{"issue":"2","key":"1160_CR15","doi-asserted-by":"publisher","first-page":"293","DOI":"10.1007\/s00205-005-0408-z","volume":"180","author":"A Friedman","year":"2006","unstructured":"Friedman, A., Hu, B.: Bifurcation from stability to instability for a free boundary problem arising in a tumor model. Arch. Ration. Mech. Anal. 180(2), 293\u2013330 (2006)","journal-title":"Arch. Ration. Mech. Anal."},{"issue":"1","key":"1160_CR16","doi-asserted-by":"publisher","first-page":"174","DOI":"10.1137\/060656292","volume":"39","author":"A Friedman","year":"2007","unstructured":"Friedman, A., Hu, B.: Bifurcation for a free boundary problem modeling tumor growth by stokes equation. SIAM J. Math. Anal. 39(1), 174\u2013194 (2007)","journal-title":"SIAM J. Math. Anal."},{"issue":"4","key":"1160_CR17","doi-asserted-by":"publisher","first-page":"651","DOI":"10.1016\/0005-1098(90)90044-I","volume":"26","author":"R Haber","year":"1990","unstructured":"Haber, R., Unbehauen, H.: Structure identification of nonlinear dynamic systems survey on input\/output approaches. Automatica 26(4), 651\u2013677 (1990)","journal-title":"Automatica"},{"issue":"8","key":"1160_CR18","doi-asserted-by":"publisher","first-page":"1793","DOI":"10.1007\/s11538-009-9425-7","volume":"71","author":"R Hambrock","year":"2009","unstructured":"Hambrock, R., Lou, Y.: The evolution of conditional dispersal strategies in spatially heterogeneous habitats. Bull. Math. Biol. 71(8), 1793\u20131817 (2009)","journal-title":"Bull. Math. Biol."},{"issue":"45","key":"1160_CR19","doi-asserted-by":"publisher","first-page":"16065","DOI":"10.1073\/pnas.1417789111","volume":"111","author":"W Hao","year":"2014","unstructured":"Hao, W., Crouser, E., Friedman, A.: Mathematical model of sarcoidosis. Proc. Natl. Acad. Sci. 111(45), 16065\u201316070 (2014)","journal-title":"Proc. Natl. Acad. Sci."},{"issue":"3","key":"1160_CR20","doi-asserted-by":"publisher","first-page":"e90497","DOI":"10.1371\/journal.pone.0090497","volume":"9","author":"W Hao","year":"2014","unstructured":"Hao, W., Friedman, A.: The ldl\u2013hdl profile determines the risk of atherosclerosis: a mathematical model. PLoS ONE 9(3), e90497 (2014)","journal-title":"PLoS ONE"},{"issue":"2","key":"1160_CR21","doi-asserted-by":"publisher","first-page":"694","DOI":"10.1016\/j.nonrwa.2011.08.010","volume":"13","author":"W Hao","year":"2012","unstructured":"Hao, W., Hauenstein, J., Hu, B., Liu, Y., Sommese, A., Zhang, Y.-T.: Bifurcation for a free boundary problem modeling the growth of a tumor with a necrotic core. Nonlinear Anal. Real World Appl. 13(2), 694\u2013709 (2012)","journal-title":"Nonlinear Anal. Real World Appl."},{"issue":"6","key":"1160_CR22","doi-asserted-by":"crossref","first-page":"2661","DOI":"10.1016\/j.amc.2011.08.006","volume":"218","author":"W Hao","year":"2011","unstructured":"Hao, W., Hauenstein, J., Hu, B., Sommese, A.: A three-dimensional steady-state tumor system. Appl. Math. Comput. 218(6), 2661\u20132669 (2011)","journal-title":"Appl. Math. Comput."},{"issue":"5","key":"1160_CR23","doi-asserted-by":"publisher","first-page":"332","DOI":"10.1016\/j.jcp.2013.05.008","volume":"250","author":"W Hao","year":"2013","unstructured":"Hao, W., Hauenstein, J., Shu, C.-W., Sommese, A., Xu, Z., Zhang, Y.-T.: A homotopy method based on weno schemes for solving steady state problems of hyperbolic conservation laws. J. Comput. Phys. 250(5), 332\u2013346 (2013)","journal-title":"J. Comput. Phys."},{"key":"1160_CR24","unstructured":"Hao, W., Lam, K.Y., Lou, Y.: Concentration phenomena in an integro-pde model for evolution of conditional dispersal. Indiana Univ. Math. J. (2017, to appear)"},{"issue":"5","key":"1160_CR25","doi-asserted-by":"publisher","first-page":"052113","DOI":"10.1103\/PhysRevE.88.052113","volume":"88","author":"W Hao","year":"2013","unstructured":"Hao, W., Nepomechie, R., Sommese, A.: Completeness of solutions of Bethe\u2019s equations. Phys. Rev. E 88(5), 052113 (2013)","journal-title":"Phys. Rev. E"},{"issue":"3","key":"1160_CR26","doi-asserted-by":"publisher","first-page":"P03024","DOI":"10.1088\/1742-5468\/2014\/03\/P03024","volume":"2014","author":"W Hao","year":"2014","unstructured":"Hao, W., Nepomechie, R., Sommese, A.: Singular solutions, repeated roots and completeness for higher-spin chains. J. Stat. Mech. Theory Exp. 2014(3), P03024 (2014)","journal-title":"J. Stat. Mech. Theory Exp."},{"issue":"3","key":"1160_CR27","doi-asserted-by":"publisher","first-page":"371","DOI":"10.1007\/s10208-013-9147-y","volume":"13","author":"J Hauenstein","year":"2013","unstructured":"Hauenstein, J., Wampler, C.: Isosingular sets and deflation. Found. Comput. Math. 13(3), 371\u2013403 (2013)","journal-title":"Found. Comput. Math."},{"issue":"2","key":"1160_CR28","doi-asserted-by":"publisher","first-page":"302","DOI":"10.1006\/jcph.2000.6626","volume":"169","author":"T Hou","year":"2001","unstructured":"Hou, T., Lowengrub, J., Shelley, M.: Boundary integral methods for multicomponent fluids and multiphase materials. J. Comput. Phys. 169(2), 302\u2013362 (2001)","journal-title":"J. Comput. Phys."},{"issue":"1","key":"1160_CR29","doi-asserted-by":"publisher","first-page":"91","DOI":"10.1023\/A:1019163811284","volume":"18","author":"B Huber","year":"1998","unstructured":"Huber, B., Verschelde, J.: Polyhedral end games for polynomial continuation. Numer. Algorithms 18(1), 91\u2013108 (1998)","journal-title":"Numer. Algorithms"},{"key":"1160_CR30","volume-title":"Nonlinear Systems","author":"H Khalil","year":"2002","unstructured":"Khalil, H.: Nonlinear Systems. Prentice Hall, Upper Saddle River (2002)"},{"issue":"4","key":"1160_CR31","doi-asserted-by":"publisher","first-page":"851","DOI":"10.1007\/s00285-013-0650-1","volume":"68","author":"KY Lam","year":"2014","unstructured":"Lam, K.Y., Lou, Y.: Evolution of conditional dispersal: evolutionarily stable strategies in spatial models. J. Math. Biol. 68(4), 851\u2013877 (2014)","journal-title":"J. Math. Biol."},{"issue":"2","key":"1160_CR32","doi-asserted-by":"publisher","first-page":"109","DOI":"10.1007\/s00607-008-0015-6","volume":"83","author":"T-L Lee","year":"2008","unstructured":"Lee, T.-L., Li, T.-Y., Tsai, C.-H.: Hom4ps-2.0: a software package for solving polynomial systems by the polyhedral homotopy continuation method. Computing 83(2), 109\u2013133 (2008)","journal-title":"Computing"},{"issue":"11","key":"1160_CR33","doi-asserted-by":"publisher","first-page":"1937","DOI":"10.1142\/S0218202507002522","volume":"17","author":"Y Lee","year":"2007","unstructured":"Lee, Y., Wu, J., Xu, J., Zikatanov, L.: Robust subspace correction methods for nearly singular systems. Math. Models Methods Appl. Sci. 17(11), 1937\u20131963 (2007)","journal-title":"Math. Models Methods Appl. Sci."},{"issue":"1\u20133","key":"1160_CR34","doi-asserted-by":"publisher","first-page":"111","DOI":"10.1016\/j.tcs.2006.02.018","volume":"359","author":"A Leykin","year":"2006","unstructured":"Leykin, A., Verschelde, J., Zhao, A.: Newton\u2019s method with deflation for isolated singularities of polynomial systems. Theor. Comput. Sci. 359(1\u20133), 111\u2013122 (2006)","journal-title":"Theor. Comput. Sci."},{"issue":"5","key":"1160_CR35","doi-asserted-by":"publisher","first-page":"1241","DOI":"10.1137\/0726069","volume":"26","author":"T-Y Li","year":"1989","unstructured":"Li, T.-Y., Sauer, T., Yorke, J.: The cheater\u2019s homotopy: an efficient procedure for solving systems of polynomial equations. SIAM J. Numer. Anal. 26(5), 1241\u20131251 (1989)","journal-title":"SIAM J. Numer. Anal."},{"issue":"200","key":"1160_CR36","doi-asserted-by":"publisher","first-page":"483","DOI":"10.1090\/S0025-5718-1992-1151113-4","volume":"59","author":"T-Y Li","year":"1992","unstructured":"Li, T.-Y., Zeng, Z.: Homotopy-determinant algorithm for solving nonsymmetric eigenvalue problems. Math. Comput. 59(200), 483\u2013502 (1992)","journal-title":"Math. Comput."},{"issue":"2","key":"1160_CR37","first-page":"115","volume":"24","author":"A Morgan","year":"1987","unstructured":"Morgan, A., Sommese, A.: Computing all solutions to polynomial systems using homotopy continuation. Appl. Math. Comput. 24(2), 115\u2013138 (1987)","journal-title":"Appl. Math. Comput."},{"issue":"2","key":"1160_CR38","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1016\/0096-3003(87)90063-4","volume":"24","author":"A Morgan","year":"1987","unstructured":"Morgan, A., Sommese, A.: A homotopy for solving general polynomial systems that respects m-homogeneous structures. Appl. Math. Comput. 24(2), 101\u2013113 (1987)","journal-title":"Appl. Math. Comput."},{"issue":"1","key":"1160_CR39","doi-asserted-by":"publisher","first-page":"391","DOI":"10.1007\/BF01385867","volume":"63","author":"A Morgan","year":"1992","unstructured":"Morgan, A., Sommese, A., Wampler, C.: A power series method for computing singular solutions to nonlinear analytic systems. Numer. Math. 63(1), 391\u2013409 (1992)","journal-title":"Numer. Math."},{"issue":"1","key":"1160_CR40","doi-asserted-by":"publisher","first-page":"681","DOI":"10.1007\/BF01386437","volume":"57","author":"P Rabier","year":"1990","unstructured":"Rabier, P., Rheinboldt, W.: On a computational method for the second fundamental tensor and its application to bifurcation problems. Numer. Math. 57(1), 681\u2013694 (1990)","journal-title":"Numer. Math."},{"issue":"1","key":"1160_CR41","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1137\/0715001","volume":"15","author":"W Rheinboldt","year":"1978","unstructured":"Rheinboldt, W.: Numerical methods for a class of finite dimensional bifurcation problems. SIAM J. Numer. Anal. 15(1), 1\u201311 (1978)","journal-title":"SIAM J. Numer. Anal."},{"key":"1160_CR42","doi-asserted-by":"publisher","DOI":"10.1142\/5763","volume-title":"The Numerical Solution of Systems of Polynomials Arising in Engineering and Science","author":"A Sommese","year":"2005","unstructured":"Sommese, A., Wampler, C.: The Numerical Solution of Systems of Polynomials Arising in Engineering and Science, vol. 99. World Scientific, Singapore (2005)"},{"key":"1160_CR43","volume-title":"Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering","author":"S Strogatz","year":"2014","unstructured":"Strogatz, S.: Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Westview Press, New York (2014)"},{"issue":"2","key":"1160_CR44","doi-asserted-by":"publisher","first-page":"251","DOI":"10.1145\/317275.317286","volume":"25","author":"J Verschelde","year":"1999","unstructured":"Verschelde, J.: Algorithm 795: Phcpack: a general-purpose solver for polynomial systems by homotopy continuation. ACM Trans. Math. Softw. 25(2), 251\u2013276 (1999)","journal-title":"ACM Trans. Math. Softw."},{"issue":"3","key":"1160_CR45","doi-asserted-by":"publisher","first-page":"525","DOI":"10.1007\/s00285-015-0958-0","volume":"73","author":"X Wang","year":"2016","unstructured":"Wang, X., Golubitsky, M.: Singularity theory of fitness functions under dimorphism equivalence. J. Math. Biol. 73(3), 525\u2013573 (2016)","journal-title":"J. Math. Biol."},{"issue":"3","key":"1160_CR46","doi-asserted-by":"publisher","first-page":"281","DOI":"10.1145\/29380.214343","volume":"13","author":"L Watson","year":"1987","unstructured":"Watson, L., Billups, S., Morgan, A.: Algorithm 652: Hompack: a suite of codes for globally convergent homotopy algorithms. ACM Trans. Math. Softw. 13(3), 281\u2013310 (1987)","journal-title":"ACM Trans. Math. Softw."},{"issue":"4","key":"1160_CR47","doi-asserted-by":"publisher","first-page":"581","DOI":"10.1137\/1034116","volume":"34","author":"J Xu","year":"1992","unstructured":"Xu, J.: Iterative methods by space decomposition and subspace correction. SIAM Rev. 34(4), 581\u2013613 (1992)","journal-title":"SIAM Rev."},{"issue":"1","key":"1160_CR48","doi-asserted-by":"publisher","first-page":"599","DOI":"10.1007\/978-3-642-03413-8_14","volume":"1","author":"J Xu","year":"2009","unstructured":"Xu, J., Chen, L., Nochetto, R.: Optimal multilevel methods for h (grad), h (curl), and h (div) systems on graded and unstructured grids. Multiscale Nonlinear Adapt. Approx. 1(1), 599\u2013659 (2009)","journal-title":"Multiscale Nonlinear Adapt. Approx."},{"issue":"2","key":"1160_CR49","doi-asserted-by":"publisher","first-page":"218","DOI":"10.1145\/992200.992209","volume":"30","author":"Z Zeng","year":"2004","unstructured":"Zeng, Z.: Algorithm 835: Multroot\u2014a matlab package for computing polynomial roots and multiplicities. ACM Trans. Math. Softw. 30(2), 218\u2013236 (2004)","journal-title":"ACM Trans. Math. Softw."},{"issue":"250","key":"1160_CR50","doi-asserted-by":"publisher","first-page":"869","DOI":"10.1090\/S0025-5718-04-01692-8","volume":"74","author":"Z Zeng","year":"2005","unstructured":"Zeng, Z.: Computing multiple roots of inexact polynomials. Math. Comput. 74(250), 869\u2013903 (2005)","journal-title":"Math. Comput."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-020-01160-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10915-020-01160-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-020-01160-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,2,14]],"date-time":"2021-02-14T09:53:26Z","timestamp":1613296406000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10915-020-01160-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,2,15]]},"references-count":50,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2020,3]]}},"alternative-id":["1160"],"URL":"https:\/\/doi.org\/10.1007\/s10915-020-01160-w","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,2,15]]},"assertion":[{"value":"12 July 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 February 2020","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 February 2020","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 February 2020","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"53"}}