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Open AccessFeature PaperArticle

A Discontinuous ODE Model of the Glacial Cycles with Diffusive Heat Transport

Department of Mathematics, Oberlin College, Oberlin, OH 44074, USA
Department of Mathematics, University of Hawaii–West Oahu, Kapolei, HI 96707, USA
Author to whom correspondence should be addressed.
Mathematics 2020, 8(3), 316;
Received: 31 January 2020 / Revised: 20 February 2020 / Accepted: 21 February 2020 / Published: 1 March 2020
(This article belongs to the Special Issue Applied Analysis of Ordinary Differential Equations 2020)
We present a new discontinuous ordinary differential equation (ODE) model of the glacial cycles. Model trajectories flip from a glacial to an interglacial state, and vice versa, via a switching mechanism motivated by ice sheet mass balance principles. Filippov’s theory of differential inclusions is used to analyze the system, which can be viewed as a nonsmooth geometric singular perturbation problem. We prove the existence of a unique limit cycle, corresponding to the Earth’s glacial cycles. The diffusive heat transport component of the model is ideally suited for investigating the competing temperature gradient and transport efficiency feedbacks, each associated with ice-albedo feedback. It is the interplay of these feedbacks that determines the maximal extent of the ice sheet. In the nonautonomous setting, model glacial cycles persist when subjected to external forcing brought on by changes in Earth’s orbital parameters over geologic time. The system also exhibits various bifurcation scenarios as key parameters vary. View Full-Text
Keywords: differential equation; invariant manifold; limit cycle; differential inclusion differential equation; invariant manifold; limit cycle; differential inclusion
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Walsh, J.; Widiasih, E. A Discontinuous ODE Model of the Glacial Cycles with Diffusive Heat Transport. Mathematics 2020, 8, 316.

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