{"id":"https://openalex.org/W2259214864","doi":"https://doi.org/10.1137/140959031","title":"Necessary and Sufficient Conditions for the Noninvertibility of Fundamental Solution Matrices of a Discontinuous System","display_name":"Necessary and Sufficient Conditions for the Noninvertibility of Fundamental Solution Matrices of a Discontinuous System","publication_year":2016,"publication_date":"2016-01-01","ids":{"openalex":"https://openalex.org/W2259214864","doi":"https://doi.org/10.1137/140959031","mag":"2259214864"},"language":"en","primary_location":{"id":"doi:10.1137/140959031","is_oa":false,"landing_page_url":"https://doi.org/10.1137/140959031","pdf_url":null,"source":{"id":"https://openalex.org/S161160597","display_name":"SIAM Journal on Applied Dynamical Systems","issn_l":"1536-0040","issn":["1536-0040"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Applied Dynamical Systems","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"http://hdl.handle.net/11311/995134","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5086093589","display_name":"Federico Bizzarri","orcid":"https://orcid.org/0000-0002-6568-6491"},"institutions":[{"id":"https://openalex.org/I93860229","display_name":"Politecnico di Milano","ror":"https://ror.org/01nffqt88","country_code":"IT","type":"education","lineage":["https://openalex.org/I93860229"]}],"countries":["IT"],"is_corresponding":true,"raw_author_name":"Federico Bizzarri","raw_affiliation_strings":["Department of Electronics, Information, and Bioengineering, Politecnico di Milano, 20133 Milano, Italy,"],"affiliations":[{"raw_affiliation_string":"Department of Electronics, Information, and Bioengineering, Politecnico di Milano, 20133 Milano, Italy,","institution_ids":["https://openalex.org/I93860229"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5071970892","display_name":"Alessandro Colombo","orcid":"https://orcid.org/0000-0002-1937-3988"},"institutions":[{"id":"https://openalex.org/I93860229","display_name":"Politecnico di Milano","ror":"https://ror.org/01nffqt88","country_code":"IT","type":"education","lineage":["https://openalex.org/I93860229"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Alessandro Colombo","raw_affiliation_strings":["Department of Electronics, Information, and Bioengineering, Politecnico di Milano, 20133 Milano, Italy"],"affiliations":[{"raw_affiliation_string":"Department of Electronics, Information, and Bioengineering, Politecnico di Milano, 20133 Milano, Italy","institution_ids":["https://openalex.org/I93860229"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5069148682","display_name":"Fabio Dercole","orcid":"https://orcid.org/0000-0001-9256-0651"},"institutions":[{"id":"https://openalex.org/I93860229","display_name":"Politecnico di Milano","ror":"https://ror.org/01nffqt88","country_code":"IT","type":"education","lineage":["https://openalex.org/I93860229"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Fabio Dercole","raw_affiliation_strings":["Department of Electronics, Information, and Bioengineering, Politecnico di Milano, 20133 Milano, Italy"],"affiliations":[{"raw_affiliation_string":"Department of Electronics, Information, and Bioengineering, Politecnico di Milano, 20133 Milano, Italy","institution_ids":["https://openalex.org/I93860229"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5064664899","display_name":"Giancarlo Storti Gajani","orcid":"https://orcid.org/0000-0001-8182-7891"},"institutions":[{"id":"https://openalex.org/I93860229","display_name":"Politecnico di Milano","ror":"https://ror.org/01nffqt88","country_code":"IT","type":"education","lineage":["https://openalex.org/I93860229"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Giancarlo Storti Gajani","raw_affiliation_strings":["Department of Electronics, Information, and Bioengineering, Politecnico di Milano, 20133 Milano, Italy"],"affiliations":[{"raw_affiliation_string":"Department of Electronics, Information, and Bioengineering, Politecnico di Milano, 20133 Milano, Italy","institution_ids":["https://openalex.org/I93860229"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5086093589"],"corresponding_institution_ids":["https://openalex.org/I93860229"],"apc_list":null,"apc_paid":null,"fwci":0.2253,"has_fulltext":true,"cited_by_count":5,"citation_normalized_percentile":{"value":0.5578952,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"15","issue":"1","first_page":"84","last_page":"105"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11261","display_name":"Quantum chaos and dynamical systems","score":0.9954000115394592,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11261","display_name":"Quantum chaos and dynamical systems","score":0.9954000115394592,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12396","display_name":"Advanced Differential Equations and Dynamical Systems","score":0.9952999949455261,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11187","display_name":"Nonlinear Dynamics and Pattern Formation","score":0.9904999732971191,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/singularity","display_name":"Singularity","score":0.6584494709968567},{"id":"https://openalex.org/keywords/lyapunov-exponent","display_name":"Lyapunov exponent","score":0.6190618276596069},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5996703505516052},{"id":"https://openalex.org/keywords/computation","display_name":"Computation","score":0.5793790221214294},{"id":"https://openalex.org/keywords/trajectory","display_name":"Trajectory","score":0.546178936958313},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.4904129207134247},{"id":"https://openalex.org/keywords/lyapunov-function","display_name":"Lyapunov function","score":0.45998257398605347},{"id":"https://openalex.org/keywords/control-theory","display_name":"Control theory (sociology)","score":0.421960711479187},{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.41949212551116943},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4059640169143677},{"id":"https://openalex.org/keywords/chaotic","display_name":"Chaotic","score":0.3265415132045746},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3017282485961914},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.26910215616226196},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.13082244992256165},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.12968748807907104},{"id":"https://openalex.org/keywords/control","display_name":"Control (management)","score":0.08244302868843079}],"concepts":[{"id":"https://openalex.org/C16171025","wikidata":"https://www.wikidata.org/wiki/Q863349","display_name":"Singularity","level":2,"score":0.6584494709968567},{"id":"https://openalex.org/C191544260","wikidata":"https://www.wikidata.org/wiki/Q1238630","display_name":"Lyapunov exponent","level":3,"score":0.6190618276596069},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5996703505516052},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.5793790221214294},{"id":"https://openalex.org/C13662910","wikidata":"https://www.wikidata.org/wiki/Q193139","display_name":"Trajectory","level":2,"score":0.546178936958313},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.4904129207134247},{"id":"https://openalex.org/C60640748","wikidata":"https://www.wikidata.org/wiki/Q2337858","display_name":"Lyapunov function","level":3,"score":0.45998257398605347},{"id":"https://openalex.org/C47446073","wikidata":"https://www.wikidata.org/wiki/Q5165890","display_name":"Control theory (sociology)","level":3,"score":0.421960711479187},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.41949212551116943},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4059640169143677},{"id":"https://openalex.org/C2777052490","wikidata":"https://www.wikidata.org/wiki/Q5072826","display_name":"Chaotic","level":2,"score":0.3265415132045746},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3017282485961914},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.26910215616226196},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.13082244992256165},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.12968748807907104},{"id":"https://openalex.org/C2775924081","wikidata":"https://www.wikidata.org/wiki/Q55608371","display_name":"Control (management)","level":2,"score":0.08244302868843079},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0},{"id":"https://openalex.org/C1276947","wikidata":"https://www.wikidata.org/wiki/Q333","display_name":"Astronomy","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/140959031","is_oa":false,"landing_page_url":"https://doi.org/10.1137/140959031","pdf_url":null,"source":{"id":"https://openalex.org/S161160597","display_name":"SIAM Journal on Applied Dynamical Systems","issn_l":"1536-0040","issn":["1536-0040"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Applied Dynamical Systems","raw_type":"journal-article"},{"id":"pmh:oai:re.public.polimi.it:11311/995134","is_oa":true,"landing_page_url":"http://hdl.handle.net/11311/995134","pdf_url":"http://hdl.handle.net/11311/995134","source":{"id":"https://openalex.org/S4306400312","display_name":"Virtual Community of Pathological Anatomy (University of Castilla La Mancha)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I79189158","host_organization_name":"University of Castilla-La Mancha","host_organization_lineage":["https://openalex.org/I79189158"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":{"id":"pmh:oai:re.public.polimi.it:11311/995134","is_oa":true,"landing_page_url":"http://hdl.handle.net/11311/995134","pdf_url":"http://hdl.handle.net/11311/995134","source":{"id":"https://openalex.org/S4306400312","display_name":"Virtual Community of Pathological Anatomy (University of Castilla La Mancha)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I79189158","host_organization_name":"University of Castilla-La Mancha","host_organization_lineage":["https://openalex.org/I79189158"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"info:eu-repo/semantics/article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2259214864.pdf","grobid_xml":"https://content.openalex.org/works/W2259214864.grobid-xml"},"referenced_works_count":38,"referenced_works":["https://openalex.org/W1980003568","https://openalex.org/W1981878019","https://openalex.org/W1990042683","https://openalex.org/W1997377177","https://openalex.org/W1998369000","https://openalex.org/W2002895823","https://openalex.org/W2011227277","https://openalex.org/W2022666585","https://openalex.org/W2023599497","https://openalex.org/W2030439623","https://openalex.org/W2030450697","https://openalex.org/W2035325221","https://openalex.org/W2048453783","https://openalex.org/W2055802137","https://openalex.org/W2062908485","https://openalex.org/W2064564945","https://openalex.org/W2065253969","https://openalex.org/W2066213568","https://openalex.org/W2067701663","https://openalex.org/W2072998188","https://openalex.org/W2076535316","https://openalex.org/W2080041960","https://openalex.org/W2080673725","https://openalex.org/W2085857709","https://openalex.org/W2091628354","https://openalex.org/W2094407151","https://openalex.org/W2105794832","https://openalex.org/W2113118965","https://openalex.org/W2120666720","https://openalex.org/W2125161896","https://openalex.org/W2135298211","https://openalex.org/W2140341464","https://openalex.org/W2141960683","https://openalex.org/W2152254020","https://openalex.org/W2162460266","https://openalex.org/W2165175255","https://openalex.org/W2167401140","https://openalex.org/W2327150249"],"related_works":["https://openalex.org/W1979229270","https://openalex.org/W2374600121","https://openalex.org/W125502673","https://openalex.org/W2384948345","https://openalex.org/W2027784897","https://openalex.org/W2361515791","https://openalex.org/W2375956839","https://openalex.org/W1982767765","https://openalex.org/W1979419366","https://openalex.org/W1983507234"],"abstract_inverted_index":{"In":[0,42,113],"discontinuous":[1,62],"systems,":[2],"the":[3,8,48,55,90,108,118,121,142,154,158,163,172,175,189],"fundamental":[4,56,122],"solution":[5,57,123],"matrix":[6,58,124],"of":[7,59,75,110,115,120,130,144,157,174,188],"linearized":[9],"dynamics":[10],"about":[11],"a":[12,32,60,73,95,111,136,167,181],"reference":[13,159],"trajectory":[14,160],"can":[15,20],"be":[16,21],"noninvertible.":[17],"This":[18,93,139],"feature":[19],"exploited,":[22],"for":[23,89,162],"instance,":[24],"to":[25,30,35,68,98,103,180],"design":[26,109],"robust":[27],"control":[28],"algorithms,":[29],"synchronize":[31],"network,":[33],"or":[34,39,80,102],"stabilize":[36],"otherwise":[37],"unstable":[38],"chaotic":[40],"dynamics.":[41],"this":[43],"paper":[44],"we":[45,83],"classify":[46],"all":[47],"phenomena":[49,67],"that":[50,126],"cause":[51],"rank":[52,91],"defect":[53],"in":[54,107,193],"generic":[61],"system.":[63,112,185],"We":[64,165],"relate":[65],"these":[66],"simple":[69],"geometric":[70],"conditions":[71,88],"at":[72],"point":[74],"vector":[76],"field":[77],"switching,":[78],"sliding,":[79],"impact,":[81],"and":[82,86],"derive":[84],"necessary":[85],"sufficient":[87],"defect.":[92],"constitutes":[94],"valuable":[96],"tool":[97],"detect":[99],"flow":[100],"noninvertibility":[101],"purposefully":[104],"include":[105],"it":[106],"terms":[114],"Lyapunov":[116,176],"exponents,":[117],"singularity":[119],"means":[125],"an":[127],"infinitesimal":[128],"sphere":[129],"initial":[131],"perturbations":[132],"is":[133,191],"mapped":[134],"onto":[135],"lower-dimensional":[137],"ellipsoid.":[138],"consequently":[140],"reduces":[141],"number":[143],"finite":[145],"exponents":[146,177],"and,":[147],"most":[148],"remarkably,":[149],"makes":[150],"them":[151],"depend":[152],"on":[153],"full":[155],"history":[156],"used":[161],"computation.":[164],"introduce":[166],"numerical":[168],"procedure":[169],"which":[170],"allows":[171],"computation":[173],"by":[178],"resorting":[179],"properly":[182],"reduced":[183],"variational":[184],"The":[186],"validity":[187],"approach":[190],"verified":[192],"two":[194],"examples.":[195]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":2},{"year":2018,"cited_by_count":1}],"updated_date":"2026-03-10T16:38:18.471706","created_date":"2025-10-10T00:00:00"}
