Human Impact on Planetary Temperature and Glacial Volume: Extending a Toy Climate Model to a New Millennium

Abstract

Starting with a toy climate model from the literature, we employ a system of two nonlinear differential equations to model the reciprocal effects of the average temperature and the percentage of glacial volume on Earth. In the literature, this model is used to demonstrate the potential for a stable periodic orbit over a long time span in the form of an attracting limit cycle. In the roughly twenty five years since this model appeared in the literature, the effects of global warming and human-impacted climate change have become much more well known and apparent. We demonstrate modification of initial conditions to understand how human activity could affect the model results. Although we too see the attracting limit cycle that yields a periodic orbit, we demonstrate that small perturbations in initial conditions can lead to extreme outcomes due to the presence of a nearby saddle point. We simulate the results over time to to highlight the critical nature of perturbations that in effect change the initial conditions and to determine how soon drastic climate events might take place.