Nonlinear Oscillator Models for Long-Timescale Climate Variability

Abstract

This thesis is motivated by study of long timescale variability of the climate system. We focus on two models of nonlinear behaviour that are used in climate modelling. The first of these models is the forced van der Pol oscillator, motivated by examination of the Pleistocene ice age oscillations forced by astronomical orbital variations. The second of these is the long timescale carbon cycle model of Rothman [1]. In Chapters 2-4, we discuss unforced and forced van der Pol oscillators, following the analysis of Guckenheimer et al. [2] for periodically cases. We use a geometric singular perturbation theory (GSPT) approach of [2] to reduce to the dynamics of the return map and extend to their work to construct return maps for quasiperiodically forced cases. We note this return map can be noninvertible in various values to the parameters. In the remaining chapters, we study the dynamics of a recent model of Rothman for long timescale carbon cycle. We reproduce and extend various results of the Rothman model. In particular, we numerically find normal forms of Bautin bifurcations to confirm their criticality. We also extend the analysis of the normal form coefficients to identify where the fold limit cycle bifurcation occurs.