Abstract
It is well known that periodic forcing of a nonlinear system, even of a two-dimensional autonomous system, can produce chaotic responses with sensitive dependence on initial conditions if the forcing induces sufficient stretching and folding of the phase space. Quasiperiodic forcing can similarly produce chaotic responses, where the transition to chaos on changing a parameter can bring the system into regions of strange non-chaotic behaviour. Although it is generally acknowledged that the timings of Pleistocene ice ages are at least partly due to Milankovitch forcing (which may be approximated as quasiperiodic, with energy concentrated near a small number of frequencies), the precise details of what can be inferred about the timings of glaciations and deglaciations from the forcing is still unclear. In this paper, we perform a quantitative comparison of the response of several low-order nonlinear conceptual models for these ice ages to various types of quasiperiodic forcing. By computing largest Lyapunov exponents and mean periods, we demonstrate that many models can have a chaotic response to quasiperiodic forcing for a range of forcing amplitudes, even though some of the simplest conceptual models do not. These results suggest that pacing of ice ages to forcing may have only limited determinism.
Highlights
Forced nonlinear oscillators are textbook examples of nonlinear systems whose attractors can exhibit chaotic behaviour [Ott, 1993]
The observed glacial cycles of the Pleistocene transition from rather weak ice ages cycling on a 40 kyr time scale during the early Pleistocene (∼ 2.5 Myr – 1 Myr ago) to large-amplitude, asymmetric ice ages cycling on a 100 kyr time scale during the Late Pleistocene (∼ 1 Myr ago – present) [Huybers, 2007, Lisiecki and Raymo, 2005]
All have been proposed as possible models for the Pleistocene ice ages but we highlight the van der Pol model stands out as being dynamically simpler than the others, in that chaos is confined to very narrow regions
Summary
Forced nonlinear oscillators are textbook examples of nonlinear systems whose attractors can exhibit chaotic behaviour [Ott, 1993]. Feedback processes internal to the climate system, for example affecting the ice mass balance, could amplify the response to astronomical forcing on specific time scales such that the 100 kyr time scale appears in the late Pleistocene climate response [AbeOuchi et al, 2013] Another possibility is that the emergence of the long late-Pleistocene cycles is related to an internal oscillation in the climate system, with approximate period of 100 kyr or longer [Ditlevsen, 2009, Crucifix, 2012, Daruka and Ditlevsen, 2016], excited by the Milankovitch forcing starting around the Mid-Pleistocene transition.
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