{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/43918"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/43918","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"A Newton Method For The Continuation Of Invariant Tori","abstract":"This thesis proposes a novel method for locating a p-dimensional invariant torus of an n-dimensional map. A set of non-linear equations is formulated and solved using the Newton-Raphson scheme. The method requires a set of sampled points on a guess invariant torus. An interpolant is passed through these points to compute the pointwise shift on the invariant torus, which is used to formulate the equation of invariance for the torus under the given map. The principal application of this method is to locate invariant tori of continuous systems. These tori occur for continuous dynamical systems having quasiperiodic orbits in state space. The discretization of the continuous system in terms of a map is accomplished in terms of its flow function. Results for one-dimensional invariant tori in two and three-dimensional state space and for two-dimensional invariant tori in three and four-dimensional maps are presented.","abstract_html":"This thesis proposes a novel method for locating a p-dimensional invariant torus of an n-dimensional map. A set of non-linear equations is formulated and solved using the Newton-Raphson scheme. The method requires a set of sampled points on a guess invariant torus. An interpolant is passed through these points to compute the pointwise shift on the invariant torus, which is used to formulate the equation of invariance for the torus under the given map. The principal application of this method is to locate invariant tori of continuous systems. These tori occur for continuous dynamical systems having quasiperiodic orbits in state space. The discretization of the continuous system in terms of a map is accomplished in terms of its flow function. Results for one-dimensional invariant tori in two and three-dimensional state space and for two-dimensional invariant tori in three and four-dimensional maps are presented.","abstract_has_math":false,"creators":["Thakur, Gunjan Singh"],"institution":"Virginia Tech","degree_name":"Master of Science","degree_level":"masters","degree_discipline":"Engineering Science and Mechanics","degree_department":"Engineering Science and Mechanics","school":null,"contributors":[],"advisors":[],"committee_chairs":["Dankowicz, Harry J."],"committee_members":["Kachroo, Pushkin","Hendricks, Scott L."],"year":2004,"date_issued":"2004-07-20","date_published":"2004-07-20","updated_at":"2026-07-22T22:20:07Z","subjects":["Dynamical system","Continuation","Nonlinear system","Invarinat tori","Newton method"],"languages":[],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["etd-07282004-141952"],"render_values":[{"text":"etd-07282004-141952","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/10919/43918","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeechair","label":"Committee Chair","values":["Dankowicz, Harry J."]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Kachroo, Pushkin","Hendricks, Scott L."]},{"key":"dc:contributor.department","label":"Department","values":["Engineering Science and Mechanics"]},{"key":"dc:creator","label":"Author","values":["Thakur, Gunjan Singh"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2014-03-14T21:41:12Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2014-03-14T21:41:12Z","2006-11-05"]},{"key":"dc:date.issued","label":"Date","values":["2004-07-20"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Tech"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Engineering Science and Mechanics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["masters"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Virginia Polytechnic Institute and State University"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Dynamical system","Continuation","Nonlinear system","Invarinat tori","Newton method"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["In Copyright"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["etd-07282004-141952"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10919/43918"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["This thesis proposes a novel method for locating a p-dimensional invariant torus of an n-dimensional map. A set of non-linear equations is formulated and solved using the Newton-Raphson scheme. The method requires a set of sampled points on a guess invariant torus. An interpolant is passed through these points to compute the pointwise shift on the invariant torus, which is used to formulate the equation of invariance for the torus under the given map. The principal application of this method is to locate invariant tori of continuous systems. These tori occur for continuous dynamical systems having quasiperiodic orbits in state space. The discretization of the continuous system in terms of a map is accomplished in terms of its flow function. Results for one-dimensional invariant tori in two and three-dimensional state space and for two-dimensional invariant tori in three and four-dimensional maps are presented."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Master of Science"]},{"key":"dc:title","label":"Title","values":["A Newton Method For The Continuation Of Invariant Tori"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Dankowicz, Harry J."],"dc:contributor.committeemember":["Kachroo, Pushkin","Hendricks, Scott L."],"dc:contributor.department":["Engineering Science and Mechanics"],"dc:creator":["Thakur, Gunjan Singh"],"dc:date.accessioned":["2014-03-14T21:41:12Z"],"dc:date.available":["2014-03-14T21:41:12Z","2006-11-05"],"dc:date.issued":["2004-07-20"],"dc:description.abstract":["This thesis proposes a novel method for locating a p-dimensional invariant torus of an n-dimensional map. A set of non-linear equations is formulated and solved using the Newton-Raphson scheme. The method requires a set of sampled points on a guess invariant torus. An interpolant is passed through these points to compute the pointwise shift on the invariant torus, which is used to formulate the equation of invariance for the torus under the given map. The principal application of this method is to locate invariant tori of continuous systems. These tori occur for continuous dynamical systems having quasiperiodic orbits in state space. The discretization of the continuous system in terms of a map is accomplished in terms of its flow function. 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