#### Date of Award

Winter 1994

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics and Statistics

#### Program/Concentration

Computational and Applied Mathematics

#### Committee Director

A. D. Kirwan, Jr.

#### Committee Director

John E. Kroll

#### Committee Member

John Tweed

#### Committee Member

John Adam

#### Committee Member

John J. Swetits

#### 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.

#### DOI

10.25777/w41a-bm34

#### Recommended Citation

Toner, Michael.
"Invariant Manifolds of a Toy Climate Model"
(1994). Doctor of Philosophy (PhD), dissertation, Mathematics and Statistics, Old Dominion University, DOI: 10.25777/w41a-bm34

https://digitalcommons.odu.edu/mathstat_etds/63

#### Included in

Atmospheric Sciences Commons, Climate Commons, Geophysics and Seismology Commons, Mathematics Commons