{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,7]],"date-time":"2025-10-07T11:49:32Z","timestamp":1759837772711},"reference-count":35,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2011,2,25]],"date-time":"2011-02-25T00:00:00Z","timestamp":1298592000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2011,4]]},"DOI":"10.1007\/s00211-011-0365-4","type":"journal-article","created":{"date-parts":[[2011,2,24]],"date-time":"2011-02-24T04:15:54Z","timestamp":1298520954000},"page":"779-811","source":"Crossref","is-referenced-by-count":52,"title":["Sliding motion on discontinuity surfaces of high co-dimension. A construction for selecting a Filippov vector field"],"prefix":"10.1007","volume":"117","author":[{"given":"Luca","family":"Dieci","sequence":"first","affiliation":[]},{"given":"Luciano","family":"Lopez","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2011,2,25]]},"reference":[{"key":"365_CR1","volume-title":"Numerical methods for nonsmooth dynamical systems. Applications in Mechanics and Electronics, Lecture Notes in Applied and Computational Mechanics","author":"V. Acary","year":"2008","unstructured":"Acary V., Brogliato B.: Numerical methods for nonsmooth dynamical systems. Applications in Mechanics and Electronics, Lecture Notes in Applied and Computational Mechanics. Springer, Berlin (2008)"},{"key":"365_CR2","first-page":"545","volume":"24","author":"J.C. Alexander","year":"1998","unstructured":"Alexander J.C., Seidman T.: Sliding modes in intersecting switching surfaces, I: Blending. Houston J. Math. 24, 545\u2013569 (1998)","journal-title":"Houston J. Math."},{"key":"365_CR3","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-69512-4","volume-title":"Differential inclusions","author":"J.-P. Aubin","year":"1984","unstructured":"Aubin J.-P., Cellina A.: Differential inclusions. Springer, Berlin (1984)"},{"key":"365_CR4","doi-asserted-by":"crossref","first-page":"984","DOI":"10.1109\/TCSI.2003.815189","volume":"50","author":"G. Bartolini","year":"2003","unstructured":"Bartolini G., Ferrara A.: A switching controller for systems with hard uncertainties. IEEE Trans. Circuits Syst. I 50, 984\u2013990 (2003)","journal-title":"IEEE Trans. Circuits Syst. I"},{"key":"365_CR5","doi-asserted-by":"crossref","first-page":"875","DOI":"10.1080\/0020717031000099010","volume":"76","author":"G. Bartolini","year":"2003","unstructured":"Bartolini G., Pisano A., Punta E., Usai E.: A survey of applications of second-order sliding mode control mechanical systems. Int. J. Control 76, 875\u2013892 (2003)","journal-title":"Int. J. Control"},{"key":"365_CR6","first-page":"134","volume-title":"HSCC, LNCS.","author":"G. Batt","year":"2005","unstructured":"Batt G., Rogers D., de Jong H., Geiselmann J.: Qualitative analysis and verification of hybrid models of genetics regulatory networks: Nutritional stress response in escherichia coli. In: Morari, M., Thiele, L. (eds) HSCC, LNCS., pp. 134\u2013150. Springer, Berlin (2005)"},{"key":"365_CR7","volume-title":"Discontinuous Control Systems","author":"I. Boiko","year":"2009","unstructured":"Boiko I.: Discontinuous Control Systems. Birkhauser, Boston (2009)"},{"key":"365_CR8","unstructured":"Brogliato, B.: Implicit euler numerical simulation of sliding mode systems. In: Conference SDS 2010. http:\/\/www.dm.uniba.it\/~delbuono\/sds10\/LectureBrogliatoSDS2010.pdf"},{"key":"365_CR9","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1007\/s00285-005-0338-2","volume":"52","author":"R. Casey","year":"2006","unstructured":"Casey R., de Jong H., Gouze J.-L.: Piecewise-linear models of genetics regulatory networks: Equilibria and their stability. J. Math. Biol. 52, 27\u201356 (2006)","journal-title":"J. Math. Biol."},{"key":"365_CR10","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1016\/j.automatica.2006.07.018","volume":"43","author":"M.L. Corradini","year":"2007","unstructured":"Corradini M.L., Orlando G.: Linear unstable plants with saturating actuators: robust stabilization by a time varying sliding surface. Automatica 43, 88\u201394 (2007)","journal-title":"Automatica"},{"key":"365_CR11","doi-asserted-by":"crossref","first-page":"1785","DOI":"10.1109\/TAC.2005.858636","volume":"50","author":"J. Davila","year":"2005","unstructured":"Davila J., Fridman L., Levant A.: Second order sliding mode observers for mechanical systems. IEEE Trans. Autom. Control 50, 1785\u20131789 (2005)","journal-title":"IEEE Trans. Autom. Control"},{"key":"365_CR12","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1016\/j.bulm.2003.08.010","volume":"66","author":"H. Jong de","year":"2004","unstructured":"de Jong H., Gouze J.-L., Hernandez C., Page M., Sari T., Geiselmann J.: Qualitative simulation of genetic regulatory networks using piecewise-linear models. Bull. Math. Biol. 66, 301\u2013340 (2004)","journal-title":"Bull. Math. Biol."},{"key":"365_CR13","first-page":"725","volume-title":"Proceedings of the 15th European Conference on Artificial Intelligence.","author":"H. Jong de","year":"2002","unstructured":"de Jong H., Gouze J.-L., Hernandez C., Sari T., Geiselmann J.: Dealing with discontinuities in the qualitative simulation of genetic regulatory network. In: Harmelen, F. (ed.) Proceedings of the 15th European Conference on Artificial Intelligence., pp. 725\u2013730. IOS Press, Amsterdam (2002)"},{"key":"365_CR14","volume-title":"Piecewise-smooth Dynamical Systems. Theory and Applications, Applied Mathematical Sciences 163","author":"M. di Bernardo","year":"2008","unstructured":"di Bernardo M., Budd C.J., Champneys A.R., Kowalczyk P.: Piecewise-smooth Dynamical Systems. Theory and Applications, Applied Mathematical Sciences 163. Springer-Verlag, Berlin (2008)"},{"key":"365_CR15","unstructured":"Dieci, L., Guglelmi, N.: Regularization methods for discontinuous differential equations"},{"key":"365_CR16","doi-asserted-by":"crossref","first-page":"2023","DOI":"10.1137\/080724599","volume":"47","author":"L. Dieci","year":"2009","unstructured":"Dieci L., Lopez L.: Sliding motion in Filippov differential systems: Theoretical results and a computational approach. SIAM J. Numer. Anal. 47, 2023\u20132051 (2009)","journal-title":"SIAM J. Numer. Anal."},{"key":"365_CR17","doi-asserted-by":"crossref","DOI":"10.1201\/9781498701822","volume-title":"Sliding Mode Control: Theory and Applications","author":"C. Edwards","year":"1998","unstructured":"Edwards C., Spurgeron S.K.: Sliding Mode Control: Theory and Applications. Taylor and Francis, New York (1998)"},{"key":"365_CR18","doi-asserted-by":"crossref","DOI":"10.1007\/978-94-015-7793-9","volume-title":"Differential Equations with Discontinuous Right-Hand Sides, Mathematics and Its Applications","author":"A.F. Filippov","year":"1988","unstructured":"Filippov A.F.: Differential Equations with Discontinuous Right-Hand Sides, Mathematics and Its Applications. Kluwer Academic, Dordrecht (1988)"},{"key":"365_CR19","first-page":"263","volume-title":"Sliding Mode Control in Engineering.","author":"L. Fridman","year":"2002","unstructured":"Fridman L., Levant A.: Higher order sliding modes. In: Barbot, J.P., Perruguetti, W. (eds) Sliding Mode Control in Engineering., pp. 263\u2013293. Marcel Dekker, New York (2002)"},{"key":"365_CR20","doi-asserted-by":"crossref","first-page":"3230","DOI":"10.1016\/j.jde.2010.12.013","volume":"250","author":"G. Fusco","year":"2011","unstructured":"Fusco G., Guglielmi N.: A regularization for discontinuous differential equations with application to state-dependent delay differential equations of neutral. J. Differ. Equ. 250, 3230\u20133279 (2011)","journal-title":"J. Differ. Equ."},{"key":"365_CR21","doi-asserted-by":"crossref","first-page":"373","DOI":"10.1016\/S0045-7825(98)00025-5","volume":"163","author":"U. Galvanetto","year":"1998","unstructured":"Galvanetto U.: Computational techniques for nonlinear dynamics in multiple frictions oscillators. Comput. Methods Appl. Mech. Eng. 163, 373\u2013382 (1998)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"365_CR22","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1016\/S0045-7825(99)00021-3","volume":"178","author":"U. Galvanetto","year":"1999","unstructured":"Galvanetto U.: Non-linear dynamics of multiple frictions oscillators. Comput. Methods Appl. Mech. Eng. 178, 291\u2013306 (1999)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"365_CR23","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1080\/1468936021000041681","volume":"17","author":"J.-L. Gouze","year":"2002","unstructured":"Gouze J.-L., Sari T.: A class of piecewise linear differential equations arsing in biological models. Dyn. Syst. 17, 299\u2013319 (2002)","journal-title":"Dyn. Syst."},{"key":"365_CR24","doi-asserted-by":"crossref","first-page":"539","DOI":"10.1016\/S0005-1098(98)00160-5","volume":"35","author":"K.H. Johansson","year":"1999","unstructured":"Johansson K.H., Rantzer A., Astrom K.J.: Fast swiyches in relay feedback systems. Automatica 35, 539\u2013552 (1999)","journal-title":"Automatica"},{"key":"365_CR25","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1016\/j.physd.2006.08.021","volume":"223","author":"R.I. Leine","year":"2006","unstructured":"Leine R.I.: Bifurcations of equilibria in mechanical systems. Physica D 223, 121\u2013137 (2006)","journal-title":"Physica D"},{"key":"365_CR26","doi-asserted-by":"crossref","first-page":"138","DOI":"10.1007\/s00419-001-0190-9","volume":"72","author":"R.I. Leine","year":"2002","unstructured":"Leine R.I., van Campen D.H.: Discontinuous fold bifurcations in non-smooth dynamical systems. Arch. Appl. Mech. 72, 138\u2013146 (2002)","journal-title":"Arch. Appl. Mech."},{"key":"365_CR27","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1023\/A:1008384928636","volume":"23","author":"R.I. Leine","year":"2000","unstructured":"Leine R.I., van Campen D.H., van de Vrande B.L.: Bifurcations in nonlinear discontinuous systems. Nonlinear Dyn. 23, 105\u2013164 (2000)","journal-title":"Nonlinear Dyn."},{"key":"365_CR28","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1007\/s10884-006-9057-7","volume":"19","author":"J. Llibre","year":"2009","unstructured":"Llibre J., da Silva P.R., Teixeira M.A.: Regularization of discontinuous vector fields via singular perturbation. J. Dyn. Differ. Equ. 19, 309\u2013331 (2009)","journal-title":"J. Dyn. Differ. Equ."},{"key":"365_CR29","doi-asserted-by":"crossref","first-page":"508","DOI":"10.1137\/080722886","volume":"8","author":"J. Llibre","year":"2009","unstructured":"Llibre J., da Silva P.R., Teixeira M.A.: Studies of singularities of nonsmooth dynamical systems via singular perturbation. SIAM J. Appl. Dyn. Syst. 8, 508\u2013526 (2009)","journal-title":"SIAM J. Appl. Dyn. Syst."},{"key":"365_CR30","unstructured":"Sotomayor, J., Teixeira, M.A.: Regularization of discontinuous vector field. In: International Conference on Differential Equations, pp. 207\u2013223, (1996)"},{"key":"365_CR31","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1007\/BF01385627","volume":"58","author":"D.E. Stewart","year":"1990","unstructured":"Stewart D.E.: A high accuracy method for solving ODEs with discontinuity right-hand side. Numer. Math. 58, 299\u2013328 (1990)","journal-title":"Numer. Math."},{"key":"365_CR32","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1137\/S0036144599360110","volume":"42","author":"D.E. Stewart","year":"2000","unstructured":"Stewart D.E.: Rigid-body dynamics with friction and impact. SIAM Rev. 42, 3\u201339 (2000)","journal-title":"SIAM Rev."},{"key":"365_CR33","volume-title":"Sliding Modes and Their Application in Variable Structure Systems","author":"V.I. Utkin","year":"1978","unstructured":"Utkin V.I.: Sliding Modes and Their Application in Variable Structure Systems. MIR Publisher, Moskow (1978)"},{"key":"365_CR34","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-84379-2","volume-title":"Sliding Mode in Control and Optimization","author":"V.I. Utkin","year":"1992","unstructured":"Utkin V.I.: Sliding Mode in Control and Optimization. Springer, Berlin (1992)"},{"key":"365_CR35","doi-asserted-by":"crossref","first-page":"451","DOI":"10.1080\/10241230306724","volume":"8","author":"V.I. Utkin","year":"2002","unstructured":"Utkin V.I., Chang H.-C.: Sliding mode control on electro-mechanical systems. Math. Probl. Eng. 8, 451\u2013473 (2002)","journal-title":"Math. Probl. Eng."}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-011-0365-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00211-011-0365-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-011-0365-4","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,8]],"date-time":"2019-06-08T16:11:40Z","timestamp":1560010300000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00211-011-0365-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,2,25]]},"references-count":35,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2011,4]]}},"alternative-id":["365"],"URL":"https:\/\/doi.org\/10.1007\/s00211-011-0365-4","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"value":"0029-599X","type":"print"},{"value":"0945-3245","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,2,25]]}}}