According to astronomical theory, ice ages are caused by variations in the Earth's orbit. However, ice core data shows strong fluctuations in ice volume at a low frequency not significantly present in orbital variations. To understand how this might occur, the dynamics of a two dimensional nonlinear differential equation representing glacier/temperature interaction of an idealized climate was studied. Self sustained oscillation of the autonomous equation was used to model the internal mechanisms that could produce these fluctuations. Periodic parametric modulation of a damped internal oscillation was used to model periodic climate response at double the external modulation period. Both phenomena rely on bounded, structurally stable invariant manifolds that occur when a constant equilibrium solution becomes unstable. For the autonomous formulation, asymptotic analysis was performed to obtain analytic approximations. An outflowing manifold of a second saddle equilibrium formed a heteroclinic connection to the small amplitude periodic orbit of the self sustained oscillation. This connection bifurcated to a homoclinic orbit when the periodic orbit intersected the saddle equilibrium. For periodic parametric modulations, internal frequencies that give rise to the period doubling phenomena were identified. The Poincare map showed cases where the bounded outflowing manifold intersects transversally with the unbounded inflowing manifold, a geometry indicative of chaotic dynamics.
","abstract_html":"<p>According to astronomical theory, ice ages are caused by variations in the Earth's orbit. However, ice core data shows strong fluctuations in ice volume at a low frequency not significantly present in orbital variations. To understand how this might occur, the dynamics of a two dimensional nonlinear differential equation representing glacier/temperature interaction of an idealized climate was studied. Self sustained oscillation of the autonomous equation was used to model the internal mechanisms that could produce these fluctuations. Periodic parametric modulation of a damped internal oscillation was used to model periodic climate response at double the external modulation period. Both phenomena rely on bounded, structurally stable invariant manifolds that occur when a constant equilibrium solution becomes unstable. For the autonomous formulation, asymptotic analysis was performed to obtain analytic approximations. An outflowing manifold of a second saddle equilibrium formed a heteroclinic connection to the small amplitude periodic orbit of the self sustained oscillation. This connection bifurcated to a homoclinic orbit when the periodic orbit intersected the saddle equilibrium. For periodic parametric modulations, internal frequencies that give rise to the period doubling phenomena were identified. The Poincare map showed cases where the bounded outflowing manifold intersects transversally with the unbounded inflowing manifold, a geometry indicative of chaotic dynamics.</p>","abstract_has_math":false,"creators":["Toner, Michael"],"institution":null,"degree_name":"Doctor of Philosophy (PhD)","degree_level":"Dissertation","degree_discipline":"Mathematics & Statistics","degree_department":null,"school":null,"contributors":["A. D. Kirwan, Jr.","John E. Kroll","John Tweed","John Adam","John J. Swetits"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":1994,"date_issued":"1994-01-01T08:00:00Z","date_published":"1994-01-01T08:00:00Z","updated_at":"2026-07-24T03:35:00Z","subjects":["Poincare map","Glaciers","Ice ages","Atmospheric Sciences","Climate","Geophysics and Seismology","Mathematics"],"languages":[],"rights":["In Copyright. URI: http://rightsstatements.org/vocab/InC/1.0/ This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).
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"]},{"key":"dc:title","label":"Title","values":["Invariant Manifolds of a Toy Climate Model"]}]}],"canonical_facts":{"dc:contributor":["A. D. Kirwan, Jr.","John E. Kroll","John Tweed","John Adam","John J. Swetits"],"dc:creator":["Toner, Michael"],"dc:date.available":["2019-06-13T07:00:00Z"],"dc:description.abstract":["According to astronomical theory, ice ages are caused by variations in the Earth's orbit. However, ice core data shows strong fluctuations in ice volume at a low frequency not significantly present in orbital variations. To understand how this might occur, the dynamics of a two dimensional nonlinear differential equation representing glacier/temperature interaction of an idealized climate was studied. Self sustained oscillation of the autonomous equation was used to model the internal mechanisms that could produce these fluctuations. Periodic parametric modulation of a damped internal oscillation was used to model periodic climate response at double the external modulation period. Both phenomena rely on bounded, structurally stable invariant manifolds that occur when a constant equilibrium solution becomes unstable. For the autonomous formulation, asymptotic analysis was performed to obtain analytic approximations. An outflowing manifold of a second saddle equilibrium formed a heteroclinic connection to the small amplitude periodic orbit of the self sustained oscillation. This connection bifurcated to a homoclinic orbit when the periodic orbit intersected the saddle equilibrium. For periodic parametric modulations, internal frequencies that give rise to the period doubling phenomena were identified. The Poincare map showed cases where the bounded outflowing manifold intersects transversally with the unbounded inflowing manifold, a geometry indicative of chaotic dynamics.
"],"dc:identifier":["https://digitalcommons.odu.edu/mathstat_etds/63"],"dc:rights":["In Copyright. URI: http://rightsstatements.org/vocab/InC/1.0/ This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).
"],"dc:subject":["Poincare map","Glaciers","Ice ages","Atmospheric Sciences","Climate","Geophysics and Seismology","Mathematics"],"dc:title":["Invariant Manifolds of a Toy Climate Model"],"thesis:degree_discipline":["Mathematics & Statistics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-24T03:35:00Z"}