Periodic orbits for a discontinuous vector field arising from a conceptual model of glacial cycles

James Walsh, Esther Widiasih, Jonathan Hahn, Richard McGehee

Research output: Contribution to journalArticlepeer-review

13 Scopus citations


Conceptual climate models provide an approach to understanding climate processes through a mathematical analysis of an approximation to reality. Recently, these models have also provided interesting examples of nonsmooth dynamical systems. Here we develop a new conceptual model of glacial cycles consisting of a system of three ordinary differential equations defining a discontinuous vector field. Our model provides a dynamical systems framework for a mechanism previously shown to play a crucial role in glacial cycle patterns, namely, an increased ice sheet ablation rate during deglaciations. We use ad hoc singular perturbation techniques to prove the existence of a large periodic orbit crossing the discontinuity boundary, provided the ice sheet edge moves sufficiently slowly relative to changes in the snow line and temperature. Numerical explorations reveal the periodic orbit exists when the time constant for the ice sheet edge has more moderate values.

Original languageEnglish (US)
Pages (from-to)1843-1864
Number of pages22
Issue number6
StatePublished - May 12 2016

Bibliographical note

Funding Information:
The work was partially supported by NSF Grants DMS-0940366 and DMS-0940363.

Publisher Copyright:
© 2016 IOP Publishing Ltd & London Mathematical Society.


  • glacial cycle
  • limit cycle
  • nonsmooth vector field


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