Browsing by Subject "Phase angle"
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Item Forced oscillators with Dynamic Hopf bifurcations and applications to paleoclimate(2014-05) Oestreicher, Samantha MeganMathematical modeling is an important tool for understanding historic and future climate. The 100,000 year problem, or the mid-Pleistocene transition, has generated a variety of models to understand Earth's climate. In this work a collection of dynamic Hopf bifurcation models are analyzed to isolate the problems and challenges of this type of model. A classic Maasch and Saltzman model is shown to be insufficient. Hopf bifurcation is generalized to the McGehee and Peckham model to conclude that, in certain idealized situations, the phase of a numerical solution can be predicted. However, when small stochastic noise is added to the system, all structure is lost. Dynamic Hopf bifurcation models do not reproduce the phase correlations which δ18O has with obliquity and eccentricity. Some directions for future mathematical research are described and several oddities about quasi-periodic forcing of a Hopf bifurcation model are presented. Finally a discussion of discontinuing the use of dynamic Hopf bifurcations in mid-Pleistocene transition research is presented. Dynamic Hopf bifurcations provide a rich field for mathematical inquiry. However, as the understanding of the δ18O data and of dynamic Hopf bifurcations increases, the Hopf bifurcation models become less viable choices for modeling the mid-Pleistocene transition.