{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/22869"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/22869","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"On the maximum number of limit cycles of certain polynomial Lienard equations","abstract":"We consider the Lienard equation of the form$$\\ddot x + \\epsilon f(x)\\dot x + x\\sp{2m+1} = 0,\\eqno(1)\\cr$$which is equivalent to the planar system$$\\eqalignno{\\dot x &= y&\\enspace\\cr \\dot{y} &= -x\\sp{2m+1} - \\epsilon f(x)y,&(2)\\cr}$$where m is a nonnegative integer, f is a real polynomial and $\\epsilon$ is a small parameter.","abstract_html":"We consider the Lienard equation of the form$<span class=\"etd-inline-math\">\\ddot x + &epsilon; f(x)\\dot x + x\\sp{2m+1} = 0,\\eqno(1)\\cr</span>$which is equivalent to the planar system$<span class=\"etd-inline-math\">\\eqalignno{\\dot x &amp;= y&amp;\\enspace\\cr \\dot{y} &amp;= -x\\sp{2m+1} - &epsilon; f(x)y,&amp;(2)\\cr}</span>$where m is a nonnegative integer, f is a real polynomial and $\\epsilon$ is a small parameter.","abstract_has_math":true,"creators":["Yao, Leummim"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Albrecht, Felix"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T13:54:13Z","date_published":"2011-05-07T13:54:13Z","updated_at":"2026-07-22T22:25:20Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1995 Yao, Leummim"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9624545","(UMI)AAI9624545"],"render_values":[{"text":"AAI9624545","href":null,"code":true},{"text":"(UMI)AAI9624545","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/22869","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Albrecht, Felix"]},{"key":"dc:creator","label":"Author","values":["Yao, Leummim"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T13:54:13Z","10000-01-01","1995"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1995 Yao, Leummim"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9624545","(UMI)AAI9624545","http://hdl.handle.net/2142/22869"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["We consider the Lienard equation of the form$$\\ddot x + \\epsilon f(x)\\dot x + x\\sp{2m+1} = 0,\\eqno(1)\\cr$$which is equivalent to the planar system$$\\eqalignno{\\dot x &= y&\\enspace\\cr \\dot{y} &= -x\\sp{2m+1} - \\epsilon f(x)y,&(2)\\cr}$$where m is a nonnegative integer, f is a real polynomial and $\\epsilon$ is a small parameter.","We establish a sharp upper bound for the number of limit cycles (nontrivial isolated periodic orbits of system (2) depending on m and the degree of f provided $\\epsilon$ is sufficiently small. This is done by investigating the fixed points of the Poincare return map associated with system (2).","This result is an important step in the study of planar polynomial vector fields in connection with the second part of Hilbert's Sixteenth Problem, which is still an open question.","Made available in DSpace on 2011-05-07T13:54:13Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9624545.pdf: 1576418 bytes, checksum: aa4db2535ed9c62557c8df23e86e8928 (MD5) Previous issue date: 1995","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T15:00:34Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:28:40-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"]},{"key":"dc:title","label":"Title","values":["On the maximum number of limit cycles of certain polynomial Lienard equations"]}]}],"canonical_facts":{"dc:contributor":["Albrecht, Felix"],"dc:creator":["Yao, Leummim"],"dc:date":["2011-05-07T13:54:13Z","10000-01-01","1995"],"dc:description":["We consider the Lienard equation of the form$$\\ddot x + \\epsilon f(x)\\dot x + x\\sp{2m+1} = 0,\\eqno(1)\\cr$$which is equivalent to the planar system$$\\eqalignno{\\dot x &= y&\\enspace\\cr \\dot{y} &= -x\\sp{2m+1} - \\epsilon f(x)y,&(2)\\cr}$$where m is a nonnegative integer, f is a real polynomial and $\\epsilon$ is a small parameter.","We establish a sharp upper bound for the number of limit cycles (nontrivial isolated periodic orbits of system (2) depending on m and the degree of f provided $\\epsilon$ is sufficiently small. This is done by investigating the fixed points of the Poincare return map associated with system (2).","This result is an important step in the study of planar polynomial vector fields in connection with the second part of Hilbert's Sixteenth Problem, which is still an open question.","Made available in DSpace on 2011-05-07T13:54:13Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9624545.pdf: 1576418 bytes, checksum: aa4db2535ed9c62557c8df23e86e8928 (MD5) Previous issue date: 1995","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T15:00:34Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:28:40-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"],"dc:identifier":["AAI9624545","(UMI)AAI9624545","http://hdl.handle.net/2142/22869"],"dc:language":["eng"],"dc:rights":["Copyright 1995 Yao, Leummim"],"dc:subject":["Mathematics"],"dc:title":["On the maximum number of limit cycles of certain polynomial Lienard equations"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:20Z"}