We look at the periodic behaviour of the Earth's glacial cycles and the transitions between different periodic states when either external parameters (such as $\omega $) or internal parameters (such as $d$) are varied. We model this using the PP04 model of climate change. This is a forced discontinuous Filippov (non-smooth) dynamical system. When periodically forced this has coexisting periodic orbits. We find that the transitions in this system are mainly due to grazing events, leading to grazing bifurcations. An analysis of the grazing bifurcations is given and the impact of these on the domains of attraction and regions of existence of the periodic orbits is determined under various changes in the parameters of the system. Grazing transitions arise for general variations in the parameters (both internal and external) of the PP04 model. We find that the grazing transitions between the period orbits resemble those of the Mid-Pleistocene-Transition.

Original languageEnglish
Pages (from-to)462-491
Number of pages30
JournalIMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications)
Issue number3
Publication statusPublished - 1 Jun 2022

Bibliographical note

This research was funded in part by an award from the Botswana International University of Science and Technology (BIUST).


  • Climate model
  • Glacial cycle
  • Grazing bifurcation
  • Non-smooth dynamics
  • Transitions

ASJC Scopus subject areas

  • Applied Mathematics


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