{"id":{"repo_id":"gatech","oai_identifier":"oai:repository.gatech.edu:1853/4833"},"canonical_url":"https://search.dev.ndltd.org/etd/gatech/oai:repository.gatech.edu:1853/4833","repository":{"repo_id":"gatech","name":"Georgia Tech","base_url":"https://repository.gatech.edu/server/oai/request"},"display":{"title":"Dynamical systems approach to one-dimensional spatiotemporal chaos -- A cyclist's view","abstract":"We propose a dynamical systems approach to the study of weak turbulence(spatiotemporal chaos) based on the periodic orbit theory, emphasizing the role of recurrent patterns and coherent structures. After a brief review of the periodic orbit theory and its application to low-dimensional dynamics, we discuss its possible extension to study dynamics of spatially extended systems. The discussion is three-fold. First, we introduce a novel variational scheme for finding periodic orbits in high-dimensional systems. Second, we prove rigorously the existence of periodic structures (modulated amplitude waves) near the first instability of the complex Ginzburg-Landau equation, and check their role in pattern formation. Third, we present the extensive numerical exploration of the Kuramoto-Sivashinsky system in the chaotic regime: structure of the equilibrium solutions, our search for the shortest periodic orbits, description of the chaotic invariant set in terms of intrinsic coordinates and return maps on the Poincare section.","abstract_html":"We propose a dynamical systems approach to the study of weak turbulence(spatiotemporal chaos) based on the periodic orbit theory, emphasizing the role of recurrent patterns and coherent structures. After a brief review of the periodic orbit theory and its application to low-dimensional dynamics, we discuss its possible extension to study dynamics of spatially extended systems. The discussion is three-fold. First, we introduce a novel variational scheme for finding periodic orbits in high-dimensional systems. Second, we prove rigorously the existence of periodic structures (modulated amplitude waves) near the first instability of the complex Ginzburg-Landau equation, and check their role in pattern formation. Third, we present the extensive numerical exploration of the Kuramoto-Sivashinsky system in the chaotic regime: structure of the equilibrium solutions, our search for the shortest periodic orbits, description of the chaotic invariant set in terms of intrinsic coordinates and return maps on the Poincare section.","abstract_has_math":false,"creators":["Lan, Yueheng"],"institution":"Georgia Institute of Technology","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":"Physics","school":null,"contributors":[],"advisors":["Cvitanović, Predrag"],"committee_chairs":[],"committee_members":["Jean Bellissard","Konstantin Mischaikow","Roman Grigoriev","Uzer, Turgay"],"year":2004,"date_issued":"2004-11-19","date_published":"2004-11-19","updated_at":"2026-07-27T19:48:57Z","subjects":["Periodic orbit theory","Spatio-temporal chaos","Pattern formation"],"languages":["en_US"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/1853/4833","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Cvitanović, Predrag"]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Jean Bellissard","Konstantin Mischaikow","Roman Grigoriev","Uzer, Turgay"]},{"key":"dc:contributor.department","label":"Department","values":["Physics"]},{"key":"dc:creator","label":"Author","values":["Lan, Yueheng"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2005-03-01T19:30:32Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2005-03-01T19:30:32Z"]},{"key":"dc:date.issued","label":"Date","values":["2004-11-19"]},{"key":"dc:publisher","label":"Institution","values":["Georgia Institute of Technology"]},{"key":"dc:type","label":"Dc Type","values":["Text"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Periodic orbit theory","Spatio-temporal chaos","Pattern formation"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en_US"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/1853/4833"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["We propose a dynamical systems approach to the study of weak turbulence(spatiotemporal chaos) based on the periodic orbit theory, emphasizing the role of recurrent patterns and coherent structures. After a brief review of the periodic orbit theory and its application to low-dimensional dynamics, we discuss its possible extension to study dynamics of spatially extended systems. The discussion is three-fold. First, we introduce a novel variational scheme for finding periodic orbits in high-dimensional systems. Second, we prove rigorously the existence of periodic structures (modulated amplitude waves) near the first instability of the complex Ginzburg-Landau equation, and check their role in pattern formation. Third, we present the extensive numerical exploration of the Kuramoto-Sivashinsky system in the chaotic regime: structure of the equilibrium solutions, our search for the shortest periodic orbits, description of the chaotic invariant set in terms of intrinsic coordinates and return maps on the Poincare section."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph.D."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Dynamical systems approach to one-dimensional spatiotemporal chaos -- A cyclist's view"]}]}],"canonical_facts":{"dc:contributor.advisor":["Cvitanović, Predrag"],"dc:contributor.committeemember":["Jean Bellissard","Konstantin Mischaikow","Roman Grigoriev","Uzer, Turgay"],"dc:contributor.department":["Physics"],"dc:creator":["Lan, Yueheng"],"dc:date.accessioned":["2005-03-01T19:30:32Z"],"dc:date.available":["2005-03-01T19:30:32Z"],"dc:date.issued":["2004-11-19"],"dc:description.abstract":["We propose a dynamical systems approach to the study of weak turbulence(spatiotemporal chaos) based on the periodic orbit theory, emphasizing the role of recurrent patterns and coherent structures. After a brief review of the periodic orbit theory and its application to low-dimensional dynamics, we discuss its possible extension to study dynamics of spatially extended systems. The discussion is three-fold. First, we introduce a novel variational scheme for finding periodic orbits in high-dimensional systems. Second, we prove rigorously the existence of periodic structures (modulated amplitude waves) near the first instability of the complex Ginzburg-Landau equation, and check their role in pattern formation. 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