  The Eocene-Oligocene transition marks a watershed point of earth's climate
history. The climate shifts from a greenhouse state to an icehouse state in
which Antarctica glaciated for the first time and periodic dynamics arise which
are still relevant for our current climate. We analyse a $CaCO_3$ concentration
time series which covers the Eocene-Oligocene transition and which is obtained
from a Pacific sediment core at site DSDP1218. Therefore, we introduce a
simplified autoregression-based variant of the dominant eigenvalue (DEV)
estimation procedure. The DEV works as leading indicator of bifurcation-induced
transitions and enables us to identify the bifurcation type. We confirm its
reliability in a methodological study and demonstrate the crucial importance of
proper detrending to obtain unbiased results. As a remark, we discuss also
possible pathways to estimate the stability of limit cycles based on the DEV
and the alternative drift slope as a proof of principle. Finally, we present
the DEV analysis results of the $CaCO_3$ concentration time series which are
reproducible in a wide parameter range. Our findings demonstrate that the onset
of Oligocene's periodic dynamics might be announced by a Neimark-Sacker/Hopf
bifurcation in course of the Eocene-Oligocene transition 34 mya. (We follow the
convention and use mya$\widehat{=}$"million years ago" and
Ma$\widehat{=}$"million years" throughout the article.)
