{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/102392"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/102392","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Multidimensional continuation of families of periodic orbits","abstract":"In this work, we develop an atlas algorithm for continuation of piecewise polynomial discretizations of periodic orbits of ordinary differential equations. Such an algorithm generates a discretized representation of a manifold of such orbits embedded in a larger variable space. Each chart associated with the discretized atlas is defined in terms of a base point on the manifold and a basis for the local tangent space. The goal of any such algorithm is to cover all parts of the manifold without leaving any holes behind, and to do so efficiently without covering areas more than once. The current implementation of atlas algorithms in the continuation package COCO fails in both regards when applied to continuation of solutions to general periodic boundary value problems. This failure arrises due to the fact that COCO treats the variable space in which the manifold is embedded as Euclidean space, e.g., the distance between charts is calculated in terms of the Euclidean norm of the vector between the charts’ base points. For two charts with base points corresponding to the same periodic orbits with two different phases, the result is a non-zero distance even as the intent may be to treat them as the same orbit. Since such distances are used to calculate suitable directions of continuation at each step of the algorithm, an incorrectly computed distance may result in continuation along an inappropriate direction. In this thesis, we overcome this problem by projecting the representation of individual charts to a phase-invariant Fourier representation in which suitable directions of continuation may be identified. We use two examples to illustrate our methodology: continuation along 1- and 2-dimensional manifolds of periodic orbits of two nonlinear dynamical systems. It is observed that the manifolds generated using the proposed algorithm are well-organized and repetitive covering is minimized.","abstract_html":"In this work, we develop an atlas algorithm for continuation of piecewise polynomial discretizations of periodic orbits of ordinary differential equations. Such an algorithm generates a discretized representation of a manifold of such orbits embedded in a larger variable space. Each chart associated with the discretized atlas is defined in terms of a base point on the manifold and a basis for the local tangent space. The goal of any such algorithm is to cover all parts of the manifold without leaving any holes behind, and to do so efficiently without covering areas more than once. The current implementation of atlas algorithms in the continuation package COCO fails in both regards when applied to continuation of solutions to general periodic boundary value problems. This failure arrises due to the fact that COCO treats the variable space in which the manifold is embedded as Euclidean space, e.g., the distance between charts is calculated in terms of the Euclidean norm of the vector between the charts’ base points. For two charts with base points corresponding to the same periodic orbits with two different phases, the result is a non-zero distance even as the intent may be to treat them as the same orbit. Since such distances are used to calculate suitable directions of continuation at each step of the algorithm, an incorrectly computed distance may result in continuation along an inappropriate direction. In this thesis, we overcome this problem by projecting the representation of individual charts to a phase-invariant Fourier representation in which suitable directions of continuation may be identified. We use two examples to illustrate our methodology: continuation along 1- and 2-dimensional manifolds of periodic orbits of two nonlinear dynamical systems. It is observed that the manifolds generated using the proposed algorithm are well-organized and repetitive covering is minimized.","abstract_has_math":false,"creators":["Wang, Yuqing"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"M.S.","degree_level":"Thesis","degree_discipline":"Mechanical Engineering","degree_department":null,"school":null,"contributors":["Dankowicz, Harry"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2018,"date_issued":"2018-08-28","date_published":"2018-08-28","updated_at":"2026-07-22T22:24:40Z","subjects":["Numerical continuation, ordinary differential equations, periodic orbits, multidimensional manifolds."],"languages":["en"],"rights":["Copyright 2018 Yuqing Wang"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/102392","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Dankowicz, Harry"]},{"key":"dc:creator","label":"Author","values":["Wang, Yuqing"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2018-08-28","2018-12","2019-02-06T19:32:38Z"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mechanical Engineering"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Thesis"]},{"key":"thesis:degree_name","label":"Degree Name","values":["M.S."]},{"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":["Numerical continuation, ordinary differential equations, periodic orbits, multidimensional manifolds."]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2018 Yuqing Wang"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/102392"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["In this work, we develop an atlas algorithm for continuation of piecewise polynomial discretizations of periodic orbits of ordinary differential equations. Such an algorithm generates a discretized representation of a manifold of such orbits embedded in a larger variable space. Each chart associated with the discretized atlas is defined in terms of a base point on the manifold and a basis for the local tangent space. The goal of any such algorithm is to cover all parts of the manifold without leaving any holes behind, and to do so efficiently without covering areas more than once. The current implementation of atlas algorithms in the continuation package COCO fails in both regards when applied to continuation of solutions to general periodic boundary value problems. This failure arrises due to the fact that COCO treats the variable space in which the manifold is embedded as Euclidean space, e.g., the distance between charts is calculated in terms of the Euclidean norm of the vector between the charts’ base points. For two charts with base points corresponding to the same periodic orbits with two different phases, the result is a non-zero distance even as the intent may be to treat them as the same orbit. Since such distances are used to calculate suitable directions of continuation at each step of the algorithm, an incorrectly computed distance may result in continuation along an inappropriate direction. In this thesis, we overcome this problem by projecting the representation of individual charts to a phase-invariant Fourier representation in which suitable directions of continuation may be identified. We use two examples to illustrate our methodology: continuation along 1- and 2-dimensional manifolds of periodic orbits of two nonlinear dynamical systems. It is observed that the manifolds generated using the proposed algorithm are well-organized and repetitive covering is minimized.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2019-02-05 without embargo terms","The student, Yuqing Wang, accepted the attached license on 2018-08-24 at 17:00.","The student, Yuqing Wang, submitted this Thesis for approval on 2018-08-24 at 17:08.","This Thesis was approved for publication on 2018-08-28 at 11:17.","DSpace SAF Submission Ingestion Package generated from Vireo submission #12980 on 2019-02-05 at 11:07:39","Made available in DSpace on 2019-02-06T19:32:38Z (GMT). No. of bitstreams: 2 WANG-THESIS-2018.pdf: 2503371 bytes, checksum: 4d95b27296a29cca8a0deb721487fd99 (MD5) LICENSE.txt: 4208 bytes, checksum: 706ee768bfc3eb4754ec67875731c1dc (MD5) Previous issue date: 2018-08-28"]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Multidimensional continuation of families of periodic orbits"]}]}],"canonical_facts":{"dc:contributor":["Dankowicz, Harry"],"dc:creator":["Wang, Yuqing"],"dc:date":["2018-08-28","2018-12","2019-02-06T19:32:38Z"],"dc:description":["In this work, we develop an atlas algorithm for continuation of piecewise polynomial discretizations of periodic orbits of ordinary differential equations. Such an algorithm generates a discretized representation of a manifold of such orbits embedded in a larger variable space. Each chart associated with the discretized atlas is defined in terms of a base point on the manifold and a basis for the local tangent space. The goal of any such algorithm is to cover all parts of the manifold without leaving any holes behind, and to do so efficiently without covering areas more than once. The current implementation of atlas algorithms in the continuation package COCO fails in both regards when applied to continuation of solutions to general periodic boundary value problems. This failure arrises due to the fact that COCO treats the variable space in which the manifold is embedded as Euclidean space, e.g., the distance between charts is calculated in terms of the Euclidean norm of the vector between the charts’ base points. For two charts with base points corresponding to the same periodic orbits with two different phases, the result is a non-zero distance even as the intent may be to treat them as the same orbit. Since such distances are used to calculate suitable directions of continuation at each step of the algorithm, an incorrectly computed distance may result in continuation along an inappropriate direction. In this thesis, we overcome this problem by projecting the representation of individual charts to a phase-invariant Fourier representation in which suitable directions of continuation may be identified. We use two examples to illustrate our methodology: continuation along 1- and 2-dimensional manifolds of periodic orbits of two nonlinear dynamical systems. It is observed that the manifolds generated using the proposed algorithm are well-organized and repetitive covering is minimized.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2019-02-05 without embargo terms","The student, Yuqing Wang, accepted the attached license on 2018-08-24 at 17:00.","The student, Yuqing Wang, submitted this Thesis for approval on 2018-08-24 at 17:08.","This Thesis was approved for publication on 2018-08-28 at 11:17.","DSpace SAF Submission Ingestion Package generated from Vireo submission #12980 on 2019-02-05 at 11:07:39","Made available in DSpace on 2019-02-06T19:32:38Z (GMT). No. of bitstreams: 2 WANG-THESIS-2018.pdf: 2503371 bytes, checksum: 4d95b27296a29cca8a0deb721487fd99 (MD5) LICENSE.txt: 4208 bytes, checksum: 706ee768bfc3eb4754ec67875731c1dc (MD5) Previous issue date: 2018-08-28"],"dc:format":["application/pdf"],"dc:identifier":["http://hdl.handle.net/2142/102392"],"dc:language":["en"],"dc:rights":["Copyright 2018 Yuqing Wang"],"dc:subject":["Numerical continuation, ordinary differential equations, periodic orbits, multidimensional manifolds."],"dc:title":["Multidimensional continuation of families of periodic orbits"],"dc:type":["text"],"thesis:degree_discipline":["Mechanical Engineering"],"thesis:degree_level":["Thesis"],"thesis:degree_name":["M.S."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:24:40Z"}