{"id":{"repo_id":"uoit","oai_identifier":"oai:ontariotechu.scholaris.ca:10155/1764"},"canonical_url":"https://search.dev.ndltd.org/etd/uoit/oai:ontariotechu.scholaris.ca:10155/1764","repository":{"repo_id":"uoit","name":"Ontario Institute of Technology","base_url":"https://ontariotechu.scholaris.ca/server/oai/request"},"display":{"title":"Equivariant Kolmogorov-Arnold-Moser theory","abstract":"KAM theory is a collection of theorems and approaches to describe the stability of certain invariant tori within nearly integrable Hamiltonian systems. The invariant tori are associated with a set of torus frequencies, and a key result in the theory is that these frequencies must satisfy a Diophantine condition, with no rational resonances amongst the frequencies. In this paper we investigate the additional property of equivariance with a discrete symmetry group, Γ. The setting then becomes Γ-Equivariant, nearly integrable, Hamiltonian, ODEs. In this setting the Diophantine condition is no longer valid as the symmetry constraints imposed by Γ, generically force the frequencies to be repeated. Utilizing equivariant-singularity and linear representation theory we develop an alternative Γ-Diophantine condition and show that by modifying the Diophantine assumption in this way KAM theory can be applied to this new setting. Moreover, we demonstrate that this condition can be utilized in the Hyperbolic, Γ- Equivariant case.","abstract_html":"KAM theory is a collection of theorems and approaches to describe the stability of certain invariant tori within nearly integrable Hamiltonian systems. The invariant tori are associated with a set of torus frequencies, and a key result in the theory is that these frequencies must satisfy a Diophantine condition, with no rational resonances amongst the frequencies. In this paper we investigate the additional property of equivariance with a discrete symmetry group, Γ. The setting then becomes Γ-Equivariant, nearly integrable, Hamiltonian, ODEs. In this setting the Diophantine condition is no longer valid as the symmetry constraints imposed by Γ, generically force the frequencies to be repeated. Utilizing equivariant-singularity and linear representation theory we develop an alternative Γ-Diophantine condition and show that by modifying the Diophantine assumption in this way KAM theory can be applied to this new setting. Moreover, we demonstrate that this condition can be utilized in the Hyperbolic, Γ- Equivariant case.","abstract_has_math":false,"creators":["Faulkner, Nicholas"],"institution":"University of Ontario Institute of Technology","degree_name":"Doctor of Philosophy (PhD)","degree_level":null,"degree_discipline":"Modelling and Computational Science","degree_department":null,"school":null,"contributors":[],"advisors":["van Veen, Lennaert","Buono, Pietro-Luciano"],"committee_chairs":[],"committee_members":[],"year":2024,"date_issued":"2024-03-01","date_published":"2024-03-01","updated_at":"2026-07-24T05:35:32Z","subjects":[],"languages":["en"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/10155/1764","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["van Veen, Lennaert","Buono, Pietro-Luciano"]},{"key":"dc:creator","label":"Author","values":["Faulkner, Nicholas"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2024-06-11T16:10:12Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2024-06-11T16:10:12Z"]},{"key":"dc:date.issued","label":"Date","values":["2024-03-01"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Modelling and Computational Science"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy (PhD)"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Ontario Institute of Technology"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/10155/1764"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["KAM theory is a collection of theorems and approaches to describe the stability of certain invariant tori within nearly integrable Hamiltonian systems. The invariant tori are associated with a set of torus frequencies, and a key result in the theory is that these frequencies must satisfy a Diophantine condition, with no rational resonances amongst the frequencies. In this paper we investigate the additional property of equivariance with a discrete symmetry group, Γ. The setting then becomes Γ-Equivariant, nearly integrable, Hamiltonian, ODEs. In this setting the Diophantine condition is no longer valid as the symmetry constraints imposed by Γ, generically force the frequencies to be repeated. Utilizing equivariant-singularity and linear representation theory we develop an alternative Γ-Diophantine condition and show that by modifying the Diophantine assumption in this way KAM theory can be applied to this new setting. Moreover, we demonstrate that this condition can be utilized in the Hyperbolic, Γ- Equivariant case."]},{"key":"dc:title","label":"Title","values":["Equivariant Kolmogorov-Arnold-Moser theory"]}]}],"canonical_facts":{"dc:contributor.advisor":["van Veen, Lennaert","Buono, Pietro-Luciano"],"dc:creator":["Faulkner, Nicholas"],"dc:date.accessioned":["2024-06-11T16:10:12Z"],"dc:date.available":["2024-06-11T16:10:12Z"],"dc:date.issued":["2024-03-01"],"dc:description.abstract":["KAM theory is a collection of theorems and approaches to describe the stability of certain invariant tori within nearly integrable Hamiltonian systems. The invariant tori are associated with a set of torus frequencies, and a key result in the theory is that these frequencies must satisfy a Diophantine condition, with no rational resonances amongst the frequencies. In this paper we investigate the additional property of equivariance with a discrete symmetry group, Γ. The setting then becomes Γ-Equivariant, nearly integrable, Hamiltonian, ODEs. In this setting the Diophantine condition is no longer valid as the symmetry constraints imposed by Γ, generically force the frequencies to be repeated. Utilizing equivariant-singularity and linear representation theory we develop an alternative Γ-Diophantine condition and show that by modifying the Diophantine assumption in this way KAM theory can be applied to this new setting. Moreover, we demonstrate that this condition can be utilized in the Hyperbolic, Γ- Equivariant case."],"dc:identifier.uri":["https://hdl.handle.net/10155/1764"],"dc:language.iso":["en"],"dc:title":["Equivariant Kolmogorov-Arnold-Moser theory"],"dc:type":["Dissertation"],"thesis:degree_discipline":["Modelling and Computational Science"],"thesis:degree_name":["Doctor of Philosophy (PhD)"],"thesis:institution_name":["University of Ontario Institute of Technology"]},"updated_at":"2026-07-24T05:35:32Z"}