Our illustrations give a snapshot of very recent research showing the current status of predictive studies. They show that tipping events, corresponding mathematically to dangerous bifurcations, pose a likely threat to the current state of the climate because they cause rapid and irreversible transitions. Also, there is evidence that tipping events have been the mechanism behind climate transitions of the past. Model studies give hope that these tipping events are predictable using time-series analysis: when applied to real geological data from past events prediction is often remarkably good but is not always reliable. With today's and tomorrow's vastly improved monitoring, giving times-series that are both longer (higher N) and much more accurate, reliable estimates can be confidently expected. However, if a system has already passed a bifurcation point one may ask whether it is in fact too late to usefully apply geo-engineering because an irreversible transition is already under way.

Techniques from non-linear dynamical systems enter the modelling side of climate prediction at two points. First, in data assimilation, which plays a role in the tuning and updating of models, the assimilated data is often Lagrangian (for example, it might come from drifting floats in the ocean). It turns out that optimal starting positions for these drifters are determined by stable and unstable manifolds of the vector field of the phase space flow (Kuznetsov et al., 2003). Second, numerical bifurcation-tracking techniques for large-scale systems have become applicable to realistic large-scale climate models (Huisman et al., 2009). More generally, numerical continuation methods have been developed (for example, LOCA by Salinger et al. (2002)) that are specifically designed for the continuation of equilibria of large physical systems. These general methods appear to be very promising for the analysis of tipping points in different types of deterministic climate models. These developments will permit efficient parameter studies where one can determine directly how the tipping event in the model varies when many system parameters are changed simultaneously. This may become particularly useful for extensive scenario studies in geo-engineering.

For example, Dijkstra et al. (2004) demonstrated how bifurcation diagrams can help to determine which perturbations enable threshold-crossing in the bi-stable THC system, and Biggs et al. (2009) studied how quickly perturbations have to be reversed to avoid jumping to coexisting attractors in a fisheries model. Furthermore, subtle microscopic non-linearities, currently beyond the reach of climate models, may have a strong influence on the large spatial scale. For example, Golden (2009) observes that the permeability of sea-ice to brine drainage changes drastically (from impermeable to permeable) when the brine volume fraction increases across the

5 per cent mark. This microscopic tipping point may have a large-scale follow-on effect on the salinity of sea water near the arctic, and thus, the THC. Incorporating microscopic non-linearities into the macroscopic picture is a challenge for future modelling efforts.

Concerning the techniques of time-series analysis, two developments in related fields are of interest. First, theoretical physicists are actively developing methods of time-series analysis that take into account unknown non-linearities, allowing for short-term predictions even if the underlying deterministic system is chaotic (Kantz

6 Schreiber, 2003). These methods permit, to a certain extent, the separation of the deterministic, chaotic, component of the time-series from the noise (see also Takens, 1981). As several of the tipping events listed in Table 3.1 involve chaos, non-linear time-series analysis is a promising complement to the classical linear analysis.

Second, much can perhaps be learned from current predictive studies in the related field of theoretical ecology, discussing how higher-order moments of the noise-induced distributions help to detect tipping points. See Section 3.4 for a brief description and Biggs et al. (2009) for a recent comparison between indicators in a fisheries model.

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