  Tipping elements in the climate system are large-scale subregions of the
Earth that might possess threshold behavior under global warming with large
potential impacts on human societies. Here, we study a subset of five tipping
elements and their interactions in a conceptual and easily extendable
framework: the Greenland and West Antarctic Ice Sheets, the Atlantic Meridional
Overturning Circulation (AMOC), the El-Nino Southern Oscillation (ENSO) and the
Amazon rainforest. In this nonlinear and multistable system, we perform a basin
stability analysis to detect its stable states and their associated Earth
system resilience. Using this approach, we perform a system-wide and
comprehensive robustness analysis with more than 3.5 billion ensemble members.
Further, we investigate dynamic regimes where some of the states lose stability
and oscillations appear using a newly developed basin bifurcation analysis
methodology. Our results reveal that the state of four or five tipped elements
has the largest basin volume for large levels of global warming beyond 4
{\deg}C above pre-industrial climate conditions. For lower levels of warming,
states including disintegrated ice sheets on West Antarctica and Greenland have
higher basin volume than other state configurations. Therefore in our model, we
find that the large ice sheets are of particular importance for Earth system
resilience. We also detect the emergence of limit cycles for 0.6% of all
ensemble members at rare parameter combinations. Such limit cycle oscillations
mainly occur between the Greenland Ice Sheet and AMOC (86%), due to their
negative feedback coupling. These limit cycles point to possibly dangerous
internal modes of variability in the climate system that could have played a
role in paleoclimatic dynamics such as those unfolding during the Pleistocene
ice age cycles.
