{"id":"https://openalex.org/W2094202995","doi":"https://doi.org/10.1137/130930224","title":"Evolution of the Tangent Vectors and Localization of the Stable and Unstable Manifolds of Hyperbolic Orbits by Fast Lyapunov Indicators","display_name":"Evolution of the Tangent Vectors and Localization of the Stable and Unstable Manifolds of Hyperbolic Orbits by Fast Lyapunov Indicators","publication_year":2014,"publication_date":"2014-01-01","ids":{"openalex":"https://openalex.org/W2094202995","doi":"https://doi.org/10.1137/130930224","mag":"2094202995"},"language":"en","primary_location":{"id":"doi:10.1137/130930224","is_oa":false,"landing_page_url":"https://doi.org/10.1137/130930224","pdf_url":null,"source":{"id":"https://openalex.org/S77200987","display_name":"SIAM Journal on Applied Mathematics","issn_l":"0036-1399","issn":["0036-1399","1095-712X"],"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 Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5108452259","display_name":"Massimiliano Guzzo","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Massimiliano Guzzo","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5076976542","display_name":"Elena Lega","orcid":"https://orcid.org/0000-0001-6086-0545"},"institutions":[{"id":"https://openalex.org/I4210113427","display_name":"Lagrange Laboratory","ror":"https://ror.org/02fdv8735","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I1294671590","https://openalex.org/I201841394","https://openalex.org/I201841394","https://openalex.org/I4210113427","https://openalex.org/I4210126779","https://openalex.org/I4210148025"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Elena Lega","raw_affiliation_strings":["LAGRANGE - Joseph Louis LAGRANGE (Boulevard de l'Observatoire B.P. 4229 06304 Nice Cedex 04 - France - France)"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"LAGRANGE - Joseph Louis LAGRANGE (Boulevard de l'Observatoire B.P. 4229 06304 Nice Cedex 04 - France - France)","institution_ids":["https://openalex.org/I4210113427"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":1.8801,"has_fulltext":false,"cited_by_count":26,"citation_normalized_percentile":{"value":0.86169293,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"74","issue":"4","first_page":"1058","last_page":"1086"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11261","display_name":"Quantum chaos and dynamical systems","score":0.9997000098228455,"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":0.9997000098228455,"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/T10325","display_name":"Astro and Planetary Science","score":0.9976999759674072,"subfield":{"id":"https://openalex.org/subfields/3103","display_name":"Astronomy and Astrophysics"},"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/T12158","display_name":"Guidance and Control Systems","score":0.9833999872207642,"subfield":{"id":"https://openalex.org/subfields/2202","display_name":"Aerospace Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/stable-manifold","display_name":"Stable manifold","score":0.7496863007545471},{"id":"https://openalex.org/keywords/lyapunov-function","display_name":"Lyapunov function","score":0.6414859294891357},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6244999766349792},{"id":"https://openalex.org/keywords/tangent-space","display_name":"Tangent space","score":0.6163535118103027},{"id":"https://openalex.org/keywords/lyapunov-exponent","display_name":"Lyapunov exponent","score":0.6104777455329895},{"id":"https://openalex.org/keywords/center-manifold","display_name":"Center manifold","score":0.6045970320701599},{"id":"https://openalex.org/keywords/hyperbolic-equilibrium-point","display_name":"Hyperbolic equilibrium point","score":0.5678996443748474},{"id":"https://openalex.org/keywords/manifold","display_name":"Manifold (fluid mechanics)","score":0.5594809055328369},{"id":"https://openalex.org/keywords/phase-space","display_name":"Phase space","score":0.5248251557350159},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.49244531989097595},{"id":"https://openalex.org/keywords/saddle-point","display_name":"Saddle point","score":0.48953089118003845},{"id":"https://openalex.org/keywords/invariant-manifold","display_name":"Invariant manifold","score":0.4847862124443054},{"id":"https://openalex.org/keywords/invariant","display_name":"Invariant (physics)","score":0.46712949872016907},{"id":"https://openalex.org/keywords/tangent","display_name":"Tangent","score":0.44414424896240234},{"id":"https://openalex.org/keywords/fixed-point","display_name":"Fixed point","score":0.4388720989227295},{"id":"https://openalex.org/keywords/tangent-vector","display_name":"Tangent vector","score":0.4259647727012634},{"id":"https://openalex.org/keywords/attractor","display_name":"Attractor","score":0.41999197006225586},{"id":"https://openalex.org/keywords/equilibrium-point","display_name":"Equilibrium point","score":0.4185490608215332},{"id":"https://openalex.org/keywords/lyapunov-equation","display_name":"Lyapunov equation","score":0.41138774156570435},{"id":"https://openalex.org/keywords/hyperbolic-function","display_name":"Hyperbolic function","score":0.23667937517166138},{"id":"https://openalex.org/keywords/hyperbolic-manifold","display_name":"Hyperbolic manifold","score":0.1913510262966156},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.1550273895263672},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.14561983942985535},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.1445147693157196},{"id":"https://openalex.org/keywords/bifurcation","display_name":"Bifurcation","score":0.11384901404380798},{"id":"https://openalex.org/keywords/hopf-bifurcation","display_name":"Hopf bifurcation","score":0.10759267210960388},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.09690853953361511}],"concepts":[{"id":"https://openalex.org/C50429861","wikidata":"https://www.wikidata.org/wiki/Q3554830","display_name":"Stable manifold","level":2,"score":0.7496863007545471},{"id":"https://openalex.org/C60640748","wikidata":"https://www.wikidata.org/wiki/Q2337858","display_name":"Lyapunov function","level":3,"score":0.6414859294891357},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6244999766349792},{"id":"https://openalex.org/C157157409","wikidata":"https://www.wikidata.org/wiki/Q909601","display_name":"Tangent space","level":2,"score":0.6163535118103027},{"id":"https://openalex.org/C191544260","wikidata":"https://www.wikidata.org/wiki/Q1238630","display_name":"Lyapunov exponent","level":3,"score":0.6104777455329895},{"id":"https://openalex.org/C74774103","wikidata":"https://www.wikidata.org/wiki/Q4504202","display_name":"Center manifold","level":5,"score":0.6045970320701599},{"id":"https://openalex.org/C51691630","wikidata":"https://www.wikidata.org/wiki/Q3925833","display_name":"Hyperbolic equilibrium point","level":4,"score":0.5678996443748474},{"id":"https://openalex.org/C529865628","wikidata":"https://www.wikidata.org/wiki/Q1790740","display_name":"Manifold (fluid mechanics)","level":2,"score":0.5594809055328369},{"id":"https://openalex.org/C151342819","wikidata":"https://www.wikidata.org/wiki/Q62542","display_name":"Phase space","level":2,"score":0.5248251557350159},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.49244531989097595},{"id":"https://openalex.org/C2681867","wikidata":"https://www.wikidata.org/wiki/Q690935","display_name":"Saddle point","level":2,"score":0.48953089118003845},{"id":"https://openalex.org/C20483540","wikidata":"https://www.wikidata.org/wiki/Q4200516","display_name":"Invariant manifold","level":2,"score":0.4847862124443054},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.46712949872016907},{"id":"https://openalex.org/C138187205","wikidata":"https://www.wikidata.org/wiki/Q131251","display_name":"Tangent","level":2,"score":0.44414424896240234},{"id":"https://openalex.org/C61445026","wikidata":"https://www.wikidata.org/wiki/Q217608","display_name":"Fixed point","level":2,"score":0.4388720989227295},{"id":"https://openalex.org/C47890412","wikidata":"https://www.wikidata.org/wiki/Q1179296","display_name":"Tangent vector","level":3,"score":0.4259647727012634},{"id":"https://openalex.org/C164380108","wikidata":"https://www.wikidata.org/wiki/Q507187","display_name":"Attractor","level":2,"score":0.41999197006225586},{"id":"https://openalex.org/C94766913","wikidata":"https://www.wikidata.org/wiki/Q1530271","display_name":"Equilibrium point","level":3,"score":0.4185490608215332},{"id":"https://openalex.org/C25854792","wikidata":"https://www.wikidata.org/wiki/Q1028945","display_name":"Lyapunov equation","level":4,"score":0.41138774156570435},{"id":"https://openalex.org/C92047909","wikidata":"https://www.wikidata.org/wiki/Q204034","display_name":"Hyperbolic function","level":2,"score":0.23667937517166138},{"id":"https://openalex.org/C70984080","wikidata":"https://www.wikidata.org/wiki/Q2619102","display_name":"Hyperbolic manifold","level":3,"score":0.1913510262966156},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.1550273895263672},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.14561983942985535},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.1445147693157196},{"id":"https://openalex.org/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.11384901404380798},{"id":"https://openalex.org/C172435161","wikidata":"https://www.wikidata.org/wiki/Q531126","display_name":"Hopf bifurcation","level":4,"score":0.10759267210960388},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.09690853953361511},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0},{"id":"https://openalex.org/C78519656","wikidata":"https://www.wikidata.org/wiki/Q101333","display_name":"Mechanical engineering","level":1,"score":0.0},{"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/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1137/130930224","is_oa":false,"landing_page_url":"https://doi.org/10.1137/130930224","pdf_url":null,"source":{"id":"https://openalex.org/S77200987","display_name":"SIAM Journal on Applied Mathematics","issn_l":"0036-1399","issn":["0036-1399","1095-712X"],"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 Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:HAL:hal-03025586v1","is_oa":false,"landing_page_url":"https://hal.science/hal-03025586","pdf_url":null,"source":{"id":"https://openalex.org/S4306402512","display_name":"HAL (Le Centre pour la Communication Scientifique Directe)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1294671590","host_organization_name":"Centre National de la Recherche Scientifique","host_organization_lineage":["https://openalex.org/I1294671590"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"SIAM Journal on Applied Mathematics, 2014, 74 (4), pp.1058-1086. ⟨10.1137/130930224⟩","raw_type":"Journal articles"},{"id":"pmh:oai:www.research.unipd.it:11577/2887700","is_oa":false,"landing_page_url":"http://hdl.handle.net/11577/2887700","pdf_url":null,"source":{"id":"https://openalex.org/S4377196283","display_name":"Research Padua  Archive (University of Padua)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I138689650","host_organization_name":"University of Padua","host_organization_lineage":["https://openalex.org/I138689650"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":29,"referenced_works":["https://openalex.org/W110826557","https://openalex.org/W173023417","https://openalex.org/W1821355183","https://openalex.org/W1916606379","https://openalex.org/W1968389978","https://openalex.org/W1968536505","https://openalex.org/W1970738437","https://openalex.org/W1991978029","https://openalex.org/W1994006347","https://openalex.org/W2004900586","https://openalex.org/W2016595627","https://openalex.org/W2022964354","https://openalex.org/W2024621680","https://openalex.org/W2029889371","https://openalex.org/W2032775939","https://openalex.org/W2033167869","https://openalex.org/W2041367231","https://openalex.org/W2046210386","https://openalex.org/W2052556616","https://openalex.org/W2065890128","https://openalex.org/W2071248198","https://openalex.org/W2074216199","https://openalex.org/W2081499981","https://openalex.org/W2088017846","https://openalex.org/W2090402096","https://openalex.org/W2136808562","https://openalex.org/W2137208105","https://openalex.org/W2151358954","https://openalex.org/W4231317193"],"related_works":["https://openalex.org/W2243875171","https://openalex.org/W3021060928","https://openalex.org/W2087723889","https://openalex.org/W2077286646","https://openalex.org/W1983459863","https://openalex.org/W2129195111","https://openalex.org/W2894196946","https://openalex.org/W3047932213","https://openalex.org/W3152370143","https://openalex.org/W2350657230"],"abstract_inverted_index":{"The":[0,94,136],"fast":[1,99,124],"Lyapunov":[2,100,125,178],"indicators":[3],"are":[4,84],"functions":[5],"defined":[6],"on":[7,165],"the":[8,12,29,32,45,48,54,73,87,98,103,116,133,144,158,163,166,172,177,183,191,197],"tangent":[9,76],"fiber":[10],"of":[11,14,26,28,59,72,75,90,106,143,149,169,171,176,180,190,196,200],"phase-space":[13,49],"a":[15,23,112,121,141,147,201],"discrete":[16],"(or":[17],"continuous)":[18],"dynamical":[19],"system":[20],"by":[21],"using":[22],"finite":[24],"number":[25,148],"iterations":[27],"dynamics.":[30],"In":[31,64],"last":[33],"decade,":[34],"they":[35],"have":[36],"been":[37],"largely":[38],"used":[39,130],"in":[40,47,182],"numerical":[41],"computations":[42],"to":[43,86,131,157],"localize":[44],"resonances":[46],"and,":[50],"more":[51],"recently,":[52],"also":[53],"stable":[55,198],"and":[56,194],"unstable":[57],"manifolds":[58,89,105,145,175],"normally":[60],"hyperbolic":[61,91],"invariant":[62],"manifolds.":[63,135],"this":[65],"paper,":[66],"we":[67],"provide":[68],"an":[69],"analytic":[70],"description":[71],"growth":[74],"vectors":[77],"for":[78,140,205],"orbits":[79,179],"with":[80,146,155],"initial":[81],"conditions":[82],"which":[83,110,152],"close":[85],"stable-unstable":[88,104,134],"saddle":[92,202],"points.":[93],"representation":[95],"explains":[96],"why":[97],"indicator":[101,126],"detects":[102],"all":[107],"fixed":[108],"points":[109],"satisfy":[111],"certain":[113],"condition.":[114],"If":[115],"condition":[117],"is":[118],"not":[119],"satisfied,":[120],"suitably":[122],"modified":[123],"can":[127],"be":[128],"still":[129],"detect":[132],"new":[137],"method":[138,164],"allows":[139],"detection":[142,170,189,195],"precision":[150],"digits":[151],"increases":[153],"linearly":[154],"respect":[156],"integration":[159],"time.":[160],"We":[161],"illustrate":[162],"critical":[167],"problems":[168],"so-called":[173],"tube":[174],"$L_1,L_2$":[181],"planar":[184],"circular":[185],"restricted":[186],"three-body":[187],"problem;":[188],"Lorenz":[192],"manifold;":[193],"manifold":[199],"equilibrium":[203],"point":[204],"two":[206],"strongly":[207],"coupled":[208],"pendula.":[209]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":4},{"year":2022,"cited_by_count":4},{"year":2021,"cited_by_count":4},{"year":2020,"cited_by_count":4},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":4},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":1}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}