{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/86804"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/86804","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"A Problem From Hamiltonian Mechanics With Time -Periodic Coefficients, Small Noise, and Degeneracy","abstract":"We consider a two-dimensional weakly dissipative dynamical system with time-periodic co-efficients. Their time average is governed by a degenerate Hamiltonian whose set of critical points has an interior. The dynamics of the system is studied in the presence of three time scales. The periodic fluctuations of the coefficients occur on the time scale of order 1/epsilon2, the effect of the drift is visible on the time scale of order 1/epsilon, and the diffusion coefficients are of order 1. Using the martingale-problem approach and separating the time scales, we average the system to show convergence to a Markov process on a stratified space. The averaging combines the deterministic time averaging of periodic coefficient, and the stochastic averaging of the resulting system. The corresponding strata of the reduced space are a two-sphere, a point and a line segment. Special attention is given to the description of the domain of the limiting generator, including the analysis of the gluing conditions at the point where the strata meet. These gluing conditions, resulting from the effects of the hierarchy of the time scales, are similar to the conditions on the domain of skew Brownian motion, and are related to the description of spider martingales. The dynamics of the reduced process can also be understood through the Fokker-Planck equation, and the gluing coefficients arise in the corresponding conservation of flux and in the continuity equation.","abstract_html":"We consider a two-dimensional weakly dissipative dynamical system with time-periodic co-efficients. Their time average is governed by a degenerate Hamiltonian whose set of critical points has an interior. The dynamics of the system is studied in the presence of three time scales. The periodic fluctuations of the coefficients occur on the time scale of order 1/epsilon2, the effect of the drift is visible on the time scale of order 1/epsilon, and the diffusion coefficients are of order 1. Using the martingale-problem approach and separating the time scales, we average the system to show convergence to a Markov process on a stratified space. The averaging combines the deterministic time averaging of periodic coefficient, and the stochastic averaging of the resulting system. The corresponding strata of the reduced space are a two-sphere, a point and a line segment. Special attention is given to the description of the domain of the limiting generator, including the analysis of the gluing conditions at the point where the strata meet. These gluing conditions, resulting from the effects of the hierarchy of the time scales, are similar to the conditions on the domain of skew Brownian motion, and are related to the description of spider martingales. The dynamics of the reduced process can also be understood through the Fokker-Planck equation, and the gluing coefficients arise in the corresponding conservation of flux and in the continuity equation.","abstract_has_math":false,"creators":["O'Bryant, Natella V."],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Sowers, Richard B."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-09-28T15:19:37Z","date_published":"2015-09-28T15:19:37Z","updated_at":"2026-07-22T22:26:28Z","subjects":["Mathematics"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(MiAaPQ)AAI3070399"],"render_values":[{"text":"(MiAaPQ)AAI3070399","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/86804","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Sowers, Richard B."]},{"key":"dc:creator","label":"Author","values":["O'Bryant, Natella V."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2015-09-28T15:19:37Z","10000-01-01","2002"]},{"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"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/86804","(MiAaPQ)AAI3070399"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["We consider a two-dimensional weakly dissipative dynamical system with time-periodic co-efficients. Their time average is governed by a degenerate Hamiltonian whose set of critical points has an interior. The dynamics of the system is studied in the presence of three time scales. The periodic fluctuations of the coefficients occur on the time scale of order 1/epsilon2, the effect of the drift is visible on the time scale of order 1/epsilon, and the diffusion coefficients are of order 1. Using the martingale-problem approach and separating the time scales, we average the system to show convergence to a Markov process on a stratified space. The averaging combines the deterministic time averaging of periodic coefficient, and the stochastic averaging of the resulting system. The corresponding strata of the reduced space are a two-sphere, a point and a line segment. Special attention is given to the description of the domain of the limiting generator, including the analysis of the gluing conditions at the point where the strata meet. These gluing conditions, resulting from the effects of the hierarchy of the time scales, are similar to the conditions on the domain of skew Brownian motion, and are related to the description of spider martingales. The dynamics of the reduced process can also be understood through the Fokker-Planck equation, and the gluing coefficients arise in the corresponding conservation of flux and in the continuity equation.","Made available in DSpace on 2015-09-28T15:19:37Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 3070399.pdf: 3240058 bytes, checksum: 9254aaa2f12ae631c5c5189bef06d308 (MD5) Previous issue date: 2002","Embargo set by: Seth Robbins for item 88085 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","104 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2002."]},{"key":"dc:title","label":"Title","values":["A Problem From Hamiltonian Mechanics With Time -Periodic Coefficients, Small Noise, and Degeneracy"]}]}],"canonical_facts":{"dc:contributor":["Sowers, Richard B."],"dc:creator":["O'Bryant, Natella V."],"dc:date":["2015-09-28T15:19:37Z","10000-01-01","2002"],"dc:description":["We consider a two-dimensional weakly dissipative dynamical system with time-periodic co-efficients. Their time average is governed by a degenerate Hamiltonian whose set of critical points has an interior. The dynamics of the system is studied in the presence of three time scales. The periodic fluctuations of the coefficients occur on the time scale of order 1/epsilon2, the effect of the drift is visible on the time scale of order 1/epsilon, and the diffusion coefficients are of order 1. Using the martingale-problem approach and separating the time scales, we average the system to show convergence to a Markov process on a stratified space. The averaging combines the deterministic time averaging of periodic coefficient, and the stochastic averaging of the resulting system. The corresponding strata of the reduced space are a two-sphere, a point and a line segment. Special attention is given to the description of the domain of the limiting generator, including the analysis of the gluing conditions at the point where the strata meet. These gluing conditions, resulting from the effects of the hierarchy of the time scales, are similar to the conditions on the domain of skew Brownian motion, and are related to the description of spider martingales. The dynamics of the reduced process can also be understood through the Fokker-Planck equation, and the gluing coefficients arise in the corresponding conservation of flux and in the continuity equation.","Made available in DSpace on 2015-09-28T15:19:37Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 3070399.pdf: 3240058 bytes, checksum: 9254aaa2f12ae631c5c5189bef06d308 (MD5) Previous issue date: 2002","Embargo set by: Seth Robbins for item 88085 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","104 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2002."],"dc:identifier":["http://hdl.handle.net/2142/86804","(MiAaPQ)AAI3070399"],"dc:language":["eng"],"dc:subject":["Mathematics"],"dc:title":["A Problem From Hamiltonian Mechanics With Time -Periodic Coefficients, Small Noise, and Degeneracy"],"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":"2026-07-22T22:26:28Z"}