{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,24]],"date-time":"2026-04-24T02:31:14Z","timestamp":1776997874286,"version":"3.51.4"},"reference-count":23,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2023,3,13]],"date-time":"2023-03-13T00:00:00Z","timestamp":1678665600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,3,13]],"date-time":"2023-03-13T00:00:00Z","timestamp":1678665600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100006389","name":"University of Geneva","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100006389","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2023,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We consider piecewise-smooth dynamical systems, i.e., systems of ordinary differential equations switching between different sets of equations on distinct domains, separated by hyper-surfaces. As is well-known, when the solution approaches a discontinuity manifold, a classical solution may cease to exist. For this reason, starting with the pioneering work of Filippov, a concept of weak solution (also known as sliding mode) has been introduced and studied. Nowadays, the solution of piecewise-smooth dynamical systems in and close to discontinuity manifolds is well understood, if the manifold consists locally of a single discontinuity hyper-surface or of the intersection of two discontinuity hyper-surfaces. The present work presents partial results on the solution in and close to discontinuity manifolds of codimension 3 and higher.<\/jats:p>","DOI":"10.1007\/s11075-023-01499-9","type":"journal-article","created":{"date-parts":[[2023,3,13]],"date-time":"2023-03-13T12:06:10Z","timestamp":1678709170000},"page":"257-273","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Sliding modes of high codimension in piecewise-smooth dynamical systems"],"prefix":"10.1007","volume":"94","author":[{"given":"Nicola","family":"Guglielmi","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ernst","family":"Hairer","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,3,13]]},"reference":[{"key":"1499_CR1","doi-asserted-by":"crossref","unstructured":"Acary, V., Brogliato, B.: Numerical methods for nonsmooth dynamical systems: applications in mechanics and electronics. Lecture Notes in Applied and Computational Mechanics, vol. 35 Springer-Verlag (2008)","DOI":"10.1007\/978-3-540-75392-6"},{"key":"1499_CR2","unstructured":"di Bernardo, M., Budd, C.J., Champneys, A.R., Kowalczyk, P.: Piecewise-smooth dynamical systems, vol. 163 of Applied Mathematical Sciences. Springer-Verlag London Ltd., London. Theory and applications (2008)"},{"key":"1499_CR3","doi-asserted-by":"publisher","first-page":"161","DOI":"10.1016\/j.cam.2013.10.055","volume":"262","author":"L Dieci","year":"2014","unstructured":"Dieci, L., Difonzo, F.: A comparison of Filippov sliding vector fields in codimension 2. J. Comput. Appl Math. 262, 161\u2013179 (2014)","journal-title":"J. Comput. Appl Math."},{"key":"1499_CR4","doi-asserted-by":"publisher","first-page":"732","DOI":"10.1016\/j.cam.2015.02.026","volume":"292","author":"L Dieci","year":"2016","unstructured":"Dieci, L., Difonzo, F.: Minimum variation solutions for sliding vector fields on the intersection of two surfaces in $\\mathbb {R},^{3}$. J. Comput. Appl Math. 292, 732\u2013745 (2016)","journal-title":"J. Comput. Appl Math."},{"issue":"93","key":"1499_CR5","first-page":"99","volume":"51","author":"AF Filippov","year":"1960","unstructured":"Filippov, A.F.: Differential equations with discontinuous right-hand side. Mat. Sb. (N.S.) 51(93), 99\u2013128 (1960). Amer. Math. Soc. Transl. Ser. 2, Vol. 42, pp. 199\u2013231","journal-title":"Mat. Sb. (N.S.)"},{"key":"1499_CR6","unstructured":"Filippov, A.F.: Differential equations with discontinuous righthand sides, vol. 18 of Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht. Translated from the Russian (1988)"},{"issue":"4","key":"1499_CR7","doi-asserted-by":"publisher","first-page":"2428","DOI":"10.1137\/100801238","volume":"44","author":"N Guglielmi","year":"2012","unstructured":"Guglielmi, N., Hairer, E.: Asymptotic expansions for regularized state-dependent neutral delay equations. SIAM J. Math Anal. 44(4), 2428\u20132458 (2012)","journal-title":"SIAM J. Math Anal."},{"issue":"3","key":"1499_CR8","doi-asserted-by":"publisher","first-page":"1454","DOI":"10.1137\/15100326X","volume":"14","author":"N Guglielmi","year":"2015","unstructured":"Guglielmi, N., Hairer, E.: Classification of hidden dynamics in discontinuous dynamical systems. SIAM J. Appl. Dyn Syst. 14(3), 1454\u20131477 (2015)","journal-title":"SIAM J. Appl. Dyn Syst."},{"issue":"2","key":"1499_CR9","doi-asserted-by":"publisher","first-page":"1427","DOI":"10.1007\/s11071-016-3320-1","volume":"88","author":"N Guglielmi","year":"2017","unstructured":"Guglielmi, N., Hairer, E.: Solutions leaving a codimension-2 sliding. Nonlinear Dyn. 88(2), 1427\u20131439 (2017)","journal-title":"Nonlinear Dyn."},{"issue":"3","key":"1499_CR10","doi-asserted-by":"publisher","first-page":"1311","DOI":"10.1007\/s11075-021-01154-1","volume":"89","author":"N Guglielmi","year":"2022","unstructured":"Guglielmi, N., Hairer, E.: An efficient algorithm for solving piecewise-smooth dynamical systems. Numer Algorithms 89(3), 1311\u20131334 (2022)","journal-title":"Numer Algorithms"},{"issue":"1","key":"1499_CR11","doi-asserted-by":"publisher","first-page":"94","DOI":"10.1016\/0022-0396(89)90097-1","volume":"80","author":"MW Hirsch","year":"1989","unstructured":"Hirsch, M.W., Systems of differential equations: that are competitive or cooperative. V. Convergence in 3-dimensional systems. J. Differential Equations 80(1), 94\u2013106 (1989)","journal-title":"J. Differential Equations"},{"key":"1499_CR12","doi-asserted-by":"publisher","first-page":"34","DOI":"10.1016\/j.physd.2014.02.003","volume":"274\u2013274","author":"MR Jeffrey","year":"2014","unstructured":"Jeffrey, MR: Hidden dynamics in models of discontinuity and switching. Phys. D 274\u2013274, 34\u201345 (2014)","journal-title":"Phys. D"},{"key":"1499_CR13","doi-asserted-by":"crossref","unstructured":"Jeffrey, M.R.: Hidden dynamics. Springer, Cham. The mathematics of switches, decisions and other discontinuous behaviour (2018)","DOI":"10.1007\/978-3-030-02107-8"},{"issue":"1","key":"1499_CR14","doi-asserted-by":"publisher","first-page":"508","DOI":"10.1137\/080722886","volume":"8","author":"J Llibre","year":"2009","unstructured":"Llibre, J., da Silva, P.R., Teixeira, M.A.: Study of singularities in nonsmooth dynamical systems via singular perturbation. SIAM J. Appl. Dyn Syst. 8(1), 508\u2013526 (2009)","journal-title":"SIAM J. Appl. Dyn Syst."},{"issue":"2","key":"1499_CR15","doi-asserted-by":"publisher","first-page":"493","DOI":"10.1088\/0951-7715\/28\/2\/493","volume":"28","author":"J Llibre","year":"2015","unstructured":"Llibre, J., da Silva, P.R., Teixeira, M.A.: Sliding vector fields for non-smooth dynamical systems having intersecting switching manifolds. Nonlinearity 28(2), 493\u2013507 (2015)","journal-title":"Nonlinearity"},{"issue":"3","key":"1499_CR16","doi-asserted-by":"publisher","first-page":"882","DOI":"10.1016\/j.na.2010.09.039","volume":"74","author":"A Machina","year":"2011","unstructured":"Machina, A., Ponosov, A.: Filippov solutions in the analysis of piecewise linear models describing gene regulatory networks. Nonlinear Anal. 74 (3), 882\u2013900 (2011)","journal-title":"Nonlinear Anal."},{"issue":"2","key":"1499_CR17","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":"5","key":"1499_CR18","doi-asserted-by":"publisher","first-page":"954","DOI":"10.1016\/j.matcom.2010.10.003","volume":"81","author":"A Pisano","year":"2011","unstructured":"Pisano, A., Usai, E.: Sliding mode control: a survey with applications in math. Math. Comput. Simulation 81(5), 954\u2013979 (2011)","journal-title":"Math. Comput. Simulation"},{"issue":"1-2","key":"1499_CR19","doi-asserted-by":"publisher","first-page":"150","DOI":"10.1016\/j.physd.2004.11.014","volume":"201","author":"E Plahte","year":"2005","unstructured":"Plahte, E., Kj\u00f8glum, S.: Analysis and generic properties of gene regulatory networks with graded response functions. Phys. D 201(1-2), 150\u2013176 (2005)","journal-title":"Phys. D"},{"issue":"4","key":"1499_CR20","doi-asserted-by":"publisher","first-page":"321","DOI":"10.1007\/s002850050103","volume":"36","author":"E Plahte","year":"1998","unstructured":"Plahte, E., Mestl, T., Omholt, S.W.: A methodological basis for description and analysis of systems with complex switch-like interactions. J. Math. Biol. 36(4), 321\u2013348 (1998)","journal-title":"J. Math. Biol."},{"key":"1499_CR21","doi-asserted-by":"crossref","unstructured":"Shafarevich, I.R.: Basic algebraic geometry. Springer Study Edition. Springer-Verlag, Berlin-New York, 1977. Translated from the Russian by K. A. Hirsch, Revised printing of Grundlehren der mathematischen Wissenschaften, vol. 213 (1974)","DOI":"10.1007\/978-3-642-96200-4"},{"key":"1499_CR22","unstructured":"Sotomayor, J., Teixeira, M.A.: Regularization of discontinuous vector fields. In: International conference on differential equations (Lisboa, 1995), pp. 207\u2013223. World Sci. Publ., River Edge, NJ (1998)"},{"key":"1499_CR23","doi-asserted-by":"crossref","unstructured":"Utkin, V.I.: Sliding modes in control and optimization. Communications and Control Engineering Series. Springer-Verlag, Berlin, 1992. Translated and revised from the 1981 Russian original (1981)","DOI":"10.1007\/978-3-642-84379-2_17"}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-023-01499-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-023-01499-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-023-01499-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,12,8]],"date-time":"2023-12-08T20:52:45Z","timestamp":1702068765000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-023-01499-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,13]]},"references-count":23,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2023,9]]}},"alternative-id":["1499"],"URL":"https:\/\/doi.org\/10.1007\/s11075-023-01499-9","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,3,13]]},"assertion":[{"value":"5 December 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 January 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 March 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"<!--Emphasis Type='Bold' removed-->Competing interests"}}]}}