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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. We investigate dynamic regimes where some of the states lose stability and oscillations appear using a newly developed basin bifurcation analysis methodology. We perform a large-scale robustness analysis using a Monte Carlo approach to propagate the large uncertainties associated with the critical temperature thresholds and the interaction strengths of the tipping elements. In our multistability analysis, we reveal that the state of four or five tipped elements is dominant for large levels of global warming beyond 4°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. 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.

Basin stability and limit cycles in a conceptual model for climate tipping cascades