{"id":"https://openalex.org/W2553815124","doi":"https://doi.org/10.1142/s0218127417300014","title":"A Theoretical Framework for Lagrangian Descriptors","display_name":"A Theoretical Framework for Lagrangian Descriptors","publication_year":2017,"publication_date":"2017-01-01","ids":{"openalex":"https://openalex.org/W2553815124","doi":"https://doi.org/10.1142/s0218127417300014","mag":"2553815124"},"language":"en","primary_location":{"id":"doi:10.1142/s0218127417300014","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127417300014","pdf_url":null,"source":{"id":"https://openalex.org/S81011612","display_name":"International Journal of Bifurcation and Chaos","issn_l":"0218-1274","issn":["0218-1274","1793-6551"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Bifurcation and Chaos","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://hdl.handle.net/1983/f4a66b83-2974-43e0-ade5-cd20e288a382","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":null,"display_name":"C. Lopesino","orcid":null},"institutions":[{"id":"https://openalex.org/I4210157032","display_name":"Institute of Mathematical Sciences","ror":"https://ror.org/05e9bn444","country_code":"ES","type":"facility","lineage":["https://openalex.org/I4210157032"]},{"id":"https://openalex.org/I63634437","display_name":"Universidad Aut\u00f3noma de Madrid","ror":"https://ror.org/01cby8j38","country_code":"ES","type":"education","lineage":["https://openalex.org/I63634437"]},{"id":"https://openalex.org/I50357001","display_name":"Universidad Carlos III de Madrid","ror":"https://ror.org/03ths8210","country_code":"ES","type":"education","lineage":["https://openalex.org/I50357001"]}],"countries":["ES"],"is_corresponding":true,"raw_author_name":"C. Lopesino","raw_affiliation_strings":["Instituto de Ciencias Matem\u00e1ticas, CSIC-UAM-UC3M-UCM, C/Nicol\u00e1s Cabrera 15, Campus Cantoblanco UAM, 28049 Madrid, Spain"],"affiliations":[{"raw_affiliation_string":"Instituto de Ciencias Matem\u00e1ticas, CSIC-UAM-UC3M-UCM, C/Nicol\u00e1s Cabrera 15, Campus Cantoblanco UAM, 28049 Madrid, Spain","institution_ids":["https://openalex.org/I4210157032","https://openalex.org/I50357001","https://openalex.org/I63634437"]}]},{"author_position":"middle","author":{"id":null,"display_name":"F. Balibrea-Iniesta","orcid":null},"institutions":[{"id":"https://openalex.org/I4210157032","display_name":"Institute of Mathematical Sciences","ror":"https://ror.org/05e9bn444","country_code":"ES","type":"facility","lineage":["https://openalex.org/I4210157032"]},{"id":"https://openalex.org/I63634437","display_name":"Universidad Aut\u00f3noma de Madrid","ror":"https://ror.org/01cby8j38","country_code":"ES","type":"education","lineage":["https://openalex.org/I63634437"]},{"id":"https://openalex.org/I50357001","display_name":"Universidad Carlos III de Madrid","ror":"https://ror.org/03ths8210","country_code":"ES","type":"education","lineage":["https://openalex.org/I50357001"]}],"countries":["ES"],"is_corresponding":false,"raw_author_name":"F. Balibrea-Iniesta","raw_affiliation_strings":["Instituto de Ciencias Matem\u00e1ticas, CSIC-UAM-UC3M-UCM, C/Nicol\u00e1s Cabrera 15, Campus Cantoblanco UAM, 28049 Madrid, Spain"],"affiliations":[{"raw_affiliation_string":"Instituto de Ciencias Matem\u00e1ticas, CSIC-UAM-UC3M-UCM, C/Nicol\u00e1s Cabrera 15, Campus Cantoblanco UAM, 28049 Madrid, Spain","institution_ids":["https://openalex.org/I4210157032","https://openalex.org/I50357001","https://openalex.org/I63634437"]}]},{"author_position":"middle","author":{"id":null,"display_name":"V. J. Garc\u00eda-Garrido","orcid":null},"institutions":[{"id":"https://openalex.org/I4210157032","display_name":"Institute of Mathematical Sciences","ror":"https://ror.org/05e9bn444","country_code":"ES","type":"facility","lineage":["https://openalex.org/I4210157032"]},{"id":"https://openalex.org/I50357001","display_name":"Universidad Carlos III de Madrid","ror":"https://ror.org/03ths8210","country_code":"ES","type":"education","lineage":["https://openalex.org/I50357001"]},{"id":"https://openalex.org/I63634437","display_name":"Universidad Aut\u00f3noma de Madrid","ror":"https://ror.org/01cby8j38","country_code":"ES","type":"education","lineage":["https://openalex.org/I63634437"]}],"countries":["ES"],"is_corresponding":false,"raw_author_name":"V. J. Garc\u00eda-Garrido","raw_affiliation_strings":["Instituto de Ciencias Matem\u00e1ticas, CSIC-UAM-UC3M-UCM, C/Nicol\u00e1s Cabrera 15, Campus Cantoblanco UAM, 28049 Madrid, Spain"],"affiliations":[{"raw_affiliation_string":"Instituto de Ciencias Matem\u00e1ticas, CSIC-UAM-UC3M-UCM, C/Nicol\u00e1s Cabrera 15, Campus Cantoblanco UAM, 28049 Madrid, Spain","institution_ids":["https://openalex.org/I4210157032","https://openalex.org/I50357001","https://openalex.org/I63634437"]}]},{"author_position":"middle","author":{"id":null,"display_name":"S. Wiggins","orcid":null},"institutions":[{"id":"https://openalex.org/I36234482","display_name":"University of Bristol","ror":"https://ror.org/0524sp257","country_code":"GB","type":"education","lineage":["https://openalex.org/I36234482"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"S. Wiggins","raw_affiliation_strings":["School of Mathematics, University of Bristol, Bristol BS8 1TW, UK"],"affiliations":[{"raw_affiliation_string":"School of Mathematics, University of Bristol, Bristol BS8 1TW, UK","institution_ids":["https://openalex.org/I36234482"]}]},{"author_position":"last","author":{"id":null,"display_name":"A. M. Mancho","orcid":null},"institutions":[{"id":"https://openalex.org/I50357001","display_name":"Universidad Carlos III de Madrid","ror":"https://ror.org/03ths8210","country_code":"ES","type":"education","lineage":["https://openalex.org/I50357001"]},{"id":"https://openalex.org/I4210157032","display_name":"Institute of Mathematical Sciences","ror":"https://ror.org/05e9bn444","country_code":"ES","type":"facility","lineage":["https://openalex.org/I4210157032"]},{"id":"https://openalex.org/I63634437","display_name":"Universidad Aut\u00f3noma de Madrid","ror":"https://ror.org/01cby8j38","country_code":"ES","type":"education","lineage":["https://openalex.org/I63634437"]}],"countries":["ES"],"is_corresponding":false,"raw_author_name":"A. M. Mancho","raw_affiliation_strings":["Instituto de Ciencias Matem\u00e1ticas, CSIC-UAM-UC3M-UCM, C/Nicol\u00e1s Cabrera 15, Campus Cantoblanco UAM, 28049 Madrid, Spain"],"affiliations":[{"raw_affiliation_string":"Instituto de Ciencias Matem\u00e1ticas, CSIC-UAM-UC3M-UCM, C/Nicol\u00e1s Cabrera 15, Campus Cantoblanco UAM, 28049 Madrid, Spain","institution_ids":["https://openalex.org/I4210157032","https://openalex.org/I50357001","https://openalex.org/I63634437"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":5,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I4210157032","https://openalex.org/I50357001","https://openalex.org/I63634437"],"apc_list":null,"apc_paid":null,"fwci":5.026,"has_fulltext":true,"cited_by_count":89,"citation_normalized_percentile":{"value":0.95774502,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":96,"max":100},"biblio":{"volume":"27","issue":"01","first_page":"1730001","last_page":"1730001"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11261","display_name":"Quantum chaos and dynamical systems","score":0.697700023651123,"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":0.697700023651123,"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/T10244","display_name":"Chaos control and synchronization","score":0.052299998700618744,"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/T11187","display_name":"Nonlinear Dynamics and Pattern Formation","score":0.031300000846385956,"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/saddle-point","display_name":"Saddle point","score":0.6060000061988831},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.5730999708175659},{"id":"https://openalex.org/keywords/invariant","display_name":"Invariant (physics)","score":0.524399995803833},{"id":"https://openalex.org/keywords/ergodic-theory","display_name":"Ergodic theory","score":0.5074999928474426},{"id":"https://openalex.org/keywords/hamiltonian-system","display_name":"Hamiltonian system","score":0.4790000021457672},{"id":"https://openalex.org/keywords/saddle","display_name":"Saddle","score":0.46889999508857727},{"id":"https://openalex.org/keywords/hyperbolic-equilibrium-point","display_name":"Hyperbolic equilibrium point","score":0.40400001406669617},{"id":"https://openalex.org/keywords/mathematical-proof","display_name":"Mathematical proof","score":0.39149999618530273},{"id":"https://openalex.org/keywords/vector-field","display_name":"Vector field","score":0.35929998755455017}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7382000088691711},{"id":"https://openalex.org/C2681867","wikidata":"https://www.wikidata.org/wiki/Q690935","display_name":"Saddle point","level":2,"score":0.6060000061988831},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5730999708175659},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.524399995803833},{"id":"https://openalex.org/C122044880","wikidata":"https://www.wikidata.org/wiki/Q5498822","display_name":"Ergodic theory","level":2,"score":0.5074999928474426},{"id":"https://openalex.org/C121770821","wikidata":"https://www.wikidata.org/wiki/Q2072471","display_name":"Hamiltonian system","level":2,"score":0.4790000021457672},{"id":"https://openalex.org/C2777127463","wikidata":"https://www.wikidata.org/wiki/Q10862618","display_name":"Saddle","level":2,"score":0.46889999508857727},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4140999913215637},{"id":"https://openalex.org/C51691630","wikidata":"https://www.wikidata.org/wiki/Q3925833","display_name":"Hyperbolic equilibrium point","level":4,"score":0.40400001406669617},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.398499995470047},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.39149999618530273},{"id":"https://openalex.org/C91188154","wikidata":"https://www.wikidata.org/wiki/Q186247","display_name":"Vector field","level":2,"score":0.35929998755455017},{"id":"https://openalex.org/C151342819","wikidata":"https://www.wikidata.org/wiki/Q62542","display_name":"Phase space","level":2,"score":0.35499998927116394},{"id":"https://openalex.org/C191399111","wikidata":"https://www.wikidata.org/wiki/Q64861","display_name":"Transitive relation","level":2,"score":0.34040001034736633},{"id":"https://openalex.org/C134786449","wikidata":"https://www.wikidata.org/wiki/Q3391255","display_name":"Planar","level":2,"score":0.3319999873638153},{"id":"https://openalex.org/C136643341","wikidata":"https://www.wikidata.org/wiki/Q1361526","display_name":"Reachability","level":2,"score":0.3280999958515167},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.31839999556541443},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.30410000681877136},{"id":"https://openalex.org/C50429861","wikidata":"https://www.wikidata.org/wiki/Q3554830","display_name":"Stable manifold","level":2,"score":0.29899999499320984},{"id":"https://openalex.org/C40934718","wikidata":"https://www.wikidata.org/wiki/Q190070","display_name":"Harmonic oscillator","level":2,"score":0.2985000014305115},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.2922999858856201},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.2856999933719635},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.2745000123977661},{"id":"https://openalex.org/C106301342","wikidata":"https://www.wikidata.org/wiki/Q4117933","display_name":"Entropy (arrow of time)","level":2,"score":0.26409998536109924},{"id":"https://openalex.org/C2778701210","wikidata":"https://www.wikidata.org/wiki/Q28130034","display_name":"Constructive","level":3,"score":0.2630000114440918},{"id":"https://openalex.org/C3445786","wikidata":"https://www.wikidata.org/wiki/Q205805","display_name":"Conservation law","level":2,"score":0.2621999979019165},{"id":"https://openalex.org/C61445026","wikidata":"https://www.wikidata.org/wiki/Q217608","display_name":"Fixed point","level":2,"score":0.260699987411499},{"id":"https://openalex.org/C200581526","wikidata":"https://www.wikidata.org/wiki/Q690494","display_name":"Homoclinic orbit","level":4,"score":0.2605000138282776},{"id":"https://openalex.org/C189950617","wikidata":"https://www.wikidata.org/wiki/Q937228","display_name":"Property (philosophy)","level":2,"score":0.2578999996185303}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1142/s0218127417300014","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127417300014","pdf_url":null,"source":{"id":"https://openalex.org/S81011612","display_name":"International Journal of Bifurcation and Chaos","issn_l":"0218-1274","issn":["0218-1274","1793-6551"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Bifurcation and Chaos","raw_type":"journal-article"},{"id":"pmh:oai:research-information.bris.ac.uk:openaire_cris_publications/f4a66b83-2974-43e0-ade5-cd20e288a382","is_oa":true,"landing_page_url":"https://hdl.handle.net/1983/f4a66b83-2974-43e0-ade5-cd20e288a382","pdf_url":null,"source":{"id":"https://openalex.org/S4306400895","display_name":"Bristol Research (University of Bristol)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I36234482","host_organization_name":"University of Bristol","host_organization_lineage":["https://openalex.org/I36234482"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":"Lopesino, C, Balibrea, F, Garcia-Garrido, V, Wiggins, S & Mancho, A M 2017, 'A theoretical framework for Lagrangian descriptors', International Journal of Bifurcation and Chaos, vol. 27, no. 1. https://doi.org/10.1142/S0218127417300014","raw_type":"info:eu-repo/semantics/publishedVersion"},{"id":"pmh:oai:arXiv.org:1705.10213","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1705.10213","pdf_url":"https://arxiv.org/pdf/1705.10213","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"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":"text"},{"id":"pmh:oai:research-information.bris.ac.uk:publications/f4a66b83-2974-43e0-ade5-cd20e288a382","is_oa":true,"landing_page_url":"https://research-information.bris.ac.uk/en/publications/f4a66b83-2974-43e0-ade5-cd20e288a382","pdf_url":"https://research-information.bris.ac.uk/ws/files/89443872/submission.pdf","source":{"id":"https://openalex.org/S4306400895","display_name":"Bristol Research (University of Bristol)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I36234482","host_organization_name":"University of Bristol","host_organization_lineage":["https://openalex.org/I36234482"],"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":"Lopesino, C, Balibrea, F, Garcia-Garrido, V, Wiggins, S & Mancho, A M 2017, 'A theoretical framework for Lagrangian descriptors', International Journal of Bifurcation and Chaos, vol. 27, no. 1. https://doi.org/10.1142/S0218127417300014","raw_type":"info:eu-repo/semantics/publishedVersion"}],"best_oa_location":{"id":"pmh:oai:research-information.bris.ac.uk:openaire_cris_publications/f4a66b83-2974-43e0-ade5-cd20e288a382","is_oa":true,"landing_page_url":"https://hdl.handle.net/1983/f4a66b83-2974-43e0-ade5-cd20e288a382","pdf_url":null,"source":{"id":"https://openalex.org/S4306400895","display_name":"Bristol Research (University of Bristol)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I36234482","host_organization_name":"University of Bristol","host_organization_lineage":["https://openalex.org/I36234482"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":"Lopesino, C, Balibrea, F, Garcia-Garrido, V, Wiggins, S & Mancho, A M 2017, 'A theoretical framework for Lagrangian descriptors', International Journal of Bifurcation and Chaos, vol. 27, no. 1. https://doi.org/10.1142/S0218127417300014","raw_type":"info:eu-repo/semantics/publishedVersion"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G7961048076","display_name":null,"funder_award_id":"N00014-01-1-0769","funder_id":"https://openalex.org/F4320337345","funder_display_name":"Office of Naval Research"}],"funders":[{"id":"https://openalex.org/F4320337345","display_name":"Office of Naval Research","ror":"https://ror.org/00rk2pe57"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":53,"referenced_works":["https://openalex.org/W601869946","https://openalex.org/W1579858487","https://openalex.org/W1595575758","https://openalex.org/W1600374794","https://openalex.org/W1931967839","https://openalex.org/W1933895483","https://openalex.org/W1950423294","https://openalex.org/W1967994787","https://openalex.org/W1976146601","https://openalex.org/W1981214426","https://openalex.org/W1984607137","https://openalex.org/W1986103149","https://openalex.org/W1997260956","https://openalex.org/W2001786442","https://openalex.org/W2002087733","https://openalex.org/W2004429826","https://openalex.org/W2007261951","https://openalex.org/W2009897647","https://openalex.org/W2024214559","https://openalex.org/W2040576222","https://openalex.org/W2041433508","https://openalex.org/W2042508623","https://openalex.org/W2045739193","https://openalex.org/W2051058112","https://openalex.org/W2055181122","https://openalex.org/W2074224167","https://openalex.org/W2084430802","https://openalex.org/W2092903609","https://openalex.org/W2094853809","https://openalex.org/W2096577530","https://openalex.org/W2114386955","https://openalex.org/W2119921851","https://openalex.org/W2121010584","https://openalex.org/W2122424119","https://openalex.org/W2123739111","https://openalex.org/W2123967129","https://openalex.org/W2124472797","https://openalex.org/W2130454294","https://openalex.org/W2135038094","https://openalex.org/W2138212611","https://openalex.org/W2152481940","https://openalex.org/W2158082510","https://openalex.org/W2159495114","https://openalex.org/W2170184529","https://openalex.org/W2174282106","https://openalex.org/W2195757964","https://openalex.org/W2231794071","https://openalex.org/W2402813293","https://openalex.org/W2508817638","https://openalex.org/W2544879524","https://openalex.org/W2952134324","https://openalex.org/W4212958445","https://openalex.org/W4300262496"],"related_works":[],"abstract_inverted_index":{"This":[0],"paper":[1],"provides":[2],"a":[3,69,77,85,94,128,163],"theoretical":[4],"background":[5],"for":[6,33,39,73,81,89,98,174,185],"Lagrangian":[7,186],"Descriptors":[8],"(LDs).":[9],"The":[10,35],"goal":[11],"of":[12,20,61,112,131,151,169],"achieving":[13],"rigorous":[14,106],"proofs":[15],"that":[16,52],"justify":[17],"the":[18,56,110,132,148,152,156,167,170],"ability":[19,111],"LDs":[21,113],"to":[22,114,143,177],"detect":[23],"invariant":[24],"manifolds":[25,60],"is":[26,37],"simplified":[27],"by":[28,139],"introducing":[29],"an":[30],"alternative":[31],"definition":[32,36],"LDs.":[34],"stated":[38],"[Formula:":[40,121],"see":[41,122],"text]-dimensional":[42],"systems":[43,92],"with":[44],"general":[45],"time":[46],"dependence,":[47],"however":[48],"we":[49,137,161],"rigorously":[50],"prove":[51],"this":[53,141],"method":[54],"reveals":[55],"stable":[57],"and":[58,93,155,182],"unstable":[59],"hyperbolic":[62,70,78,86,95],"points":[63],"in":[64],"four":[65],"particular":[66],"2D":[67],"cases:":[68],"saddle":[71,79,87,96],"point":[72,80,88,97],"linear":[74,90],"autonomous":[75,83],"systems,":[76,84],"nonlinear":[82,99],"nonautonomous":[91,100],"systems.":[101],"We":[102],"also":[103],"discuss":[104],"further":[105],"results":[107,125],"which":[108,136],"show":[109],"highlight":[115],"additional":[116],"invariants":[117],"sets,":[118],"such":[119,146],"as":[120,147],"text]-tori.":[123],"These":[124],"are":[126],"just":[127],"simple":[129],"extension":[130],"ergodic":[133],"partition":[134],"theory":[135],"illustrate":[138],"applying":[140],"methodology":[142],"well-known":[144],"examples,":[145],"planar":[149],"field":[150],"harmonic":[153],"oscillator":[154],"3D":[157],"ABC":[158],"flow.":[159],"Finally,":[160],"provide":[162],"thorough":[164],"discussion":[165],"on":[166],"requirement":[168],"objectivity":[171],"(frame-invariance)":[172],"property":[173],"tools":[175],"designed":[176],"reveal":[178],"phase":[179],"space":[180],"structures":[181],"their":[183],"implications":[184],"descriptors.":[187]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":4},{"year":2024,"cited_by_count":13},{"year":2023,"cited_by_count":11},{"year":2022,"cited_by_count":12},{"year":2021,"cited_by_count":18},{"year":2020,"cited_by_count":17},{"year":2019,"cited_by_count":5},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":4}],"updated_date":"2026-04-09T08:11:56.329763","created_date":"2016-11-30T00:00:00"}