Forced oscillators with Dynamic Hopf bifurcations and applications to paleoclimate

Abstract

Mathematical 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 &delta;<super>18</super>O 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 &delta;<super>18</super>O data and of dynamic Hopf bifurcations increases, the Hopf bifurcation models become less viable choices for modeling the mid-Pleistocene transition.