In this paper, we consider tipping instabilities in nonlinear open systems . By open, we mean systems that are influenced by changing external conditions which we refer to as external forcings. In a mathematical dynamic model of an open system, such external forcings are represented by time-varying input parameters.

Large and abrupt changes in the state of an open system may occur when the external forcing exceeds some critical level . The points in time, or in the level of forcing, at which such changes occur are commonly referred to as bifurcation-induced tipping points . They have been identified in many domains, including ecosystems and the human brain , and are of particular concern under anthropogenic climate change . Furthermore, it has recently been recognised that critical levels can be exceeded temporarily without causing tipping . This occurs when the time of exceedance is short compared to the inherent timescale of the system .

However, there is another, less obvious potential consequence of changes in external forcing. When an external forcing changes faster than some critical rate rather than necessarily by a large amount, this can lead to rate-induced tipping points (; ; ). In contrast to bifurcation-induced tipping, rate-induced tipping occurs due to fast-enough changes in external forcing and usually does not exceed any critical levels as a result of external forcing. Such tipping points are much less widely known and yet are arguably even more relevant to contemporary issues such as climate change , ecosystem collapse , and the resilience of human systems .

This paper combines a review writing style with new results to make the concept of rate-induced tipping points accessible to a wide scientific audience. Even though the phenomenon is rarely discussed by scientists and policy makers, we argue that it is ubiquitous and likely to be prevalent in many open systems. The rationale is that the current human-dominated era of Earth history, which has been called the Anthropocene , is characterised by systems (e.g. climate, ecosystems, infrastructures, and economy) that are subject to fast-changing external conditions, and are thus kept far from the changing equilibrium as a result of human activity. These circumstances of rapidly changing external forcing are precisely the conditions that can lead to rate-induced tipping. Rate-induced tipping is therefore especially relevant to the contemporary period, even though the phenomenon is not widely known or understood. By contrast, bifurcation-induced tipping in a system that is forced slowly towards a threshold and that thus stays close to or tracks the changing equilibrium is much more widely understood but is less relevant in a rapidly forced system . We demonstrate this through an analysis of the following three distinct dynamic models of natural and human systems of differing complexity: a predator–prey ecosystem, the large-scale ocean circulation, and an electrical power grid network.

Rate-induced tipping occurs when the system deviates too much from the changing stable equilibrium and crosses some threshold. Here, we focus on examples of what call a regular threshold. Easily verifiable criteria for the occurrence of rate-induced tipping, such as threshold or basin instability, are identified in all of these examples. Furthermore, we uncover universal features of rate-induced tipping. These include multiple critical rates of change due to the interaction of different timescales of the external forcing with the inherent timescales of the system. Finally, we highlight important phenomena, such as return tipping, that are non-obvious and can be easily overlooked. For further reading on different threshold types, see for an example of an elusive quasithreshold and for an example of what appears to be a fractal-like irregular threshold.

A system is known to be susceptible to rate-induced tipping if the state the system currently resides in is threshold unstable . Suppose that, for a given initial level of external forcing, the system resides in a stable equilibrium (although, in general, it can be a stable limit cycle or an even more complicated attractor). This equilibrium will be referred to as the base state. One way of depicting threshold instability is with a moving-stability landscape, as illustrated by Fig. , where the base state is represented by the blue ball in the well in Fig. a

We note that this example is for illustrative purposes. In general, the base state can be non-stationary, the system may reside near rather than in the base state, and not all dynamical systems can be characterised by a stability landscape; see for example .

Let us denote the time-varying external forcing with λ. The level of the forcing at a time t is simply the value of λ at this time t. However, defining critical rates of change in external forcing is more subtle. On the one hand, different external forcings will have different physical units and be different, often nonlinear, functions of time. On the other hand, we would like to quantify critical rates of change in a uniform way that is independent of the physical units and the temporal profile of the forcing. Therefore, we introduce a rate parameter r in units per inverse second (or day, year, etc.); we write the external forcing as λ(rt), where u=rt is dimensionless; and we work with r as the main input parameter. Most importantly, we define a critical rate as a special value of r at which rate-induced tipping occurs, while the shift magnitude of λ(rt) remains fixed.

To avoid confusion between the rate parameter r and the rate of change of external forcing dλ/dt, we note that dλdt=dλdududt=rdλ(u)du has units of λ per second, depends on r and on the profile of λ(u), and may itself be a function of time. In other words, the rate parameter r quantifies the rate of change of external forcing with a given profile. Furthermore, if the forcing λ itself is a physical rate of some sort (e.g. freshwater flux into the North Atlantic, measured in Sverdrups (millions of m3s-1) or population growth rate, measured in individuals per unit area per year, as per the examples in Sect. ), λ(rt) will be the level of this rate at time t, referred to as the level of the forcing, and r will quantify the rate of change of this rate, referred to as the rate of change of the forcing.

In natural and human systems, tipping points are often associated with crossing a critical level of the forcing, defined by a dangerous (e.g. fold) bifurcation for the frozen system with fixed-in-time forcing, causing a catastrophic, abrupt, and irreversible change to the state of the system . This type of tipping is commonly referred to as bifurcation-induced tipping (or B-tipping) and is illustrated by Fig. a and d . Suppose the system starts from the base state on the upper branch of stable equilibria for the frozen system, as indicated by the grey dot in Fig. d. Initially, as external forcing changes slowly (Fig. a), the state of the system (the red trajectory in Fig. d) tracks the moving base state (the branch of stable equilibria). However, once external forcing reaches the fold that defines the critical level of the forcing, the base state disappears (the branch of stable equilibria terminates), and the system subsequently undergoes a catastrophic transition to the alternative stable state. Crucially, the tipping occurs for any rate of change of the forcing past the fold. The alternative state is often a less desired state, such as an extinction state in an ecosystem , collapse of an ocean circulation , or a blackout on a power grid network . However, it could also be a more desired state, such as a well-being state for developing countries .

Figure  introduces a subtle but crucial difference to previous examples that have considered B-tipping , and that is to apply a tilt

Many conceptual models of bifurcation-induced tipping use the one-dimensional normal form of a fold (saddle-node) bifurcation to illustrate and study the phenomenon. While all systems that exhibit a fold bifurcation are topologically equivalent to its normal form sufficiently close to the bifurcation point, the behaviour of the branches of equilibria will typically be different away from the bifurcation point. Our tilted branches incorporate simple deviations from the normal-form behaviour expected in higher-dimensional systems away from the bifurcation point.

We now consider an example from ecology, namely that of a predator–prey system, which models the time evolution of plant and herbivore biomass densities . The model has been proposed to conceptually study tipping points in bistable ecosystems with a non-monotone functional response. Examples of such systems can be the dominance shift between submerged macrophytes and phytoplankton and between coral reefs and macro-algae or the transition of kelp forests into sea urchin barrens that are dominated by crustose coralline algae .

In this example, changes in environmental conditions affect the plant growth rate and herbivore mortality rate simultaneously and play the role of external forcing. Such a scenario can be considered to be possible under climate change, where plants benefit from the fertilisation effect due to increasing levels of CO2 but where the resulting increased temperatures are detrimental to herbivores . The plant growth rate and herbivore mortality rate vary within a range where the ecosystem has two stable equilibria. The stable coexistence equilibrium is the base state. The stable plant-only equilibrium with no herbivores is the alternative stable state.

We consider three sample time profiles of a nonlinear shift ramp external forcing, shown in Fig. a, with the same change in the level of environmental conditions but at different rates. The resulting impacts on the herbivore biomass are shown in Fig. b. For the slowest (green) change in environmental conditions, the ecosystem tracks the moving base state. Herbivores increase slightly, which represents the continued presence of coexisting plants and herbivores. Contrast this to the fastest (red) change in environmental conditions, which causes R-tipping to the alternative stable state. In this scenario, the herbivore population declines to zero, and the ecosystem becomes entirely dominated by plants.

Figure c illustrates the underlying dynamics in the phase plane of the plant and herbivore biomass. For the initial level of environmental conditions, the base state is indicated by the grey dot. The shift in the level of environmental conditions changes the position of the base state, and this change is indicated by the dotted black line. The black dot at the other end of the dotted line indicates the base state for the final level of environmental conditions. Further ramifications of the shift in environmental conditions include changes in the basin of attraction of the base state. The basin of attraction shifts (from the dotted curve to the solid grey curve) such that the initial base state (the grey dot) is not contained in the basin of attraction of the final base state (the region above the solid grey curve). A consequence of basin instability is that the behaviour of solutions starting near the initial base state depends on the rate of change of environmental conditions. For slow rates, the system is able to continually adapt and remain within the changing basin of attraction of the base state so that solutions converge to the final base state (the green trajectory). However, for a sufficiently fast rate, the ecosystem fails to track the fast moving base state and R-tips to the final alternative stable state indicated by the second black dot (the red trajectory). This rate-sensitive behaviour is expected due to basin instability. What may be surprising is that, at a critical rate of change in environmental conditions, the corresponding solution converges to an unstable edge state (black circle) on the basin boundary of the final base state (the blue trajectory).

Next, we consider an example from climate, specifically the possible collapse of the Atlantic meridional overturning circulation (AMOC) under global warming. The AMOC forms part of the global thermohaline circulation, which is a large-scale ocean circulation current driven by temperature and salinity gradients. The AMOC contributes to the relatively mild climate in western Europe by transporting heat from the tropics to the North Atlantic. Once the warm salty waters reach the North Atlantic, they cool down and become denser. The higher density of these waters causes sinking (or overturning), followed by a return to the tropics along the bottom of the ocean. However, the AMOC can be easily disturbed by contemporary climate change. The amount of freshwater added to the North Atlantic (referred to as freshwater hosing) may increase under climate change, for example due to the melting of the Greenland Ice Sheet, changes in precipitation patterns, or both. Specifically, and show in coupled climate models that the AMOC can collapse under sufficiently fast rates of change in either CO2 emissions or freshwater hosing. Additionally, R-tipping of the AMOC has been observed in a global oceanic box model .

Here, we work with the global oceanic box model for the AMOC , in which changing freshwater hosing plays the role of external forcing. The forcing varies within a range where the AMOC model has two stable equilibria. The stable AMOC-On equilibrium is the base state. The stable AMOC-Off equilibrium is the alternative stable state.

Figure d shows three sample time profiles of ramp forcing that increase the freshwater hosing from zero to the same non-zero level but each at a different rate. The response of the AMOC to these freshwater-forcing scenarios is shown in Fig. e. For the slowest (green) change in freshwater hosing, there is tracking of the moving base state. The AMOC strength suffers a slight drop but ultimately remains in the AMOC-On state. However, for the fast (red) change in freshwater hosing, there is R-tipping to the alternative stable state. The AMOC strength declines to a point of complete collapse.

The phase portrait in the plane of the salinities in the North and Tropical Atlantic boxes in Fig. f illustrates the underlying dynamics. The grey dot shows the base state for the initial level of freshwater hosing. Increasing the freshwater hosing shifts the base state as indicated by the dotted black line. The black dot at the other end of the dotted line indicates the base state at the final level of freshwater hosing. Notice that the basin of attraction of the final base state (the region enclosed by the blue periodic orbit) does not contain the initial base state (the grey dot). Therefore, the initial base state is basin unstable, and R-tipping from the base state to the alternative stable state (lower black dot) will occur for sufficiently fast shifts in the level of freshwater hosing. For slow rates of increase in freshwater hosing, the system continually adapts and remains within the changing basin of attraction of the base state so that solutions converge to the final base state (the green trajectory). On the other hand, for sufficiently fast rates of increase in freshwater hosing, the system is unable to adapt and falls outside the changing basin of attraction of the base state, causing solutions to converge to the final alternative state (the red trajectory). At the critical rate of increase in freshwater hosing, surprising behaviour is again observed as the corresponding solution converges to a repelling periodic orbit that defines the basin boundary of the final base state (the blue trajectory).

To validate our choice of the simple tilted saddle-node model in Sect. , we plot in Fig.  the tipping diagrams for the plant–herbivore and AMOC models, showing regions of tracking (white), points of return (green), and points of no return (red). The tipping diagrams for the plant–herbivore model in Fig. a and the AMOC model in Fig. b are very similar to the tipping diagram for the simple model in Fig. b, although there are some differences. In both examples, there is a region of basin instability that gives rise to R-tipping for shift forcing profiles with a peak change that does not cross any critical levels. This means that the simple model with a tilted bifurcation structure indeed captures non-obvious tipping phenomena found in higher-dimensional systems. In Fig. a, the (red) region of points of no return extends to the left of the critical level (the vertical black fold line), but there are only up to two critical rates for the return forcing profile with a fixed peak change, and the (green) region of points of return vanishes for small rate parameters. In Fig. b, there are up to three critical rates for the return forcing profile with a fixed peak change, but the (red) region of points of no return does not extend to the left of the critical level (the vertical black Hopf line). Additionally the (black) boundary has small smooth wiggles that appear as non-smooth corners. A similar wiggling effect near a Hopf bifurcation has been observed in .

R-tipping instabilities are not confined to natural systems but can occur in any system, including human systems. One example is the energy sector and power grid networks . Crucially, electricity needs to be used as soon as it is produced, since it cannot be stored easily . Therefore, providing a near-constant voltage of electricity to millions of homes while power demand varies seasonally and daily and may spike during major events is a technological challenge . Some noticeable examples of power blackouts or near misses are the northeastern USA blackout in 2003, caused by a series of faults in local control systems , and the near-miss blackout in England following the conclusion of the Euro 1990 semi-final .

The latter example in particular was arguably the result of R-tipping effects. The power demand on the network, following the conclusion of the football match, was expected to be high. Hence, the national grid took measures to ensure that the network would be able to cope with the high power demand. However, the national grid failed to envisage the match going to extra time and penalties. Thus, the rapid increase in power demand, following the eventual conclusion of the match, gave controllers insufficient time to react. Therefore, in this case, the limiting factor was not the peak in demand but instead the rate at which the demand on the network rose.

Here, we use a conceptual model of a power grid network , where changing power demand plays the role of external forcing. The situation with stable states in this model is more complicated than in the previous examples. The power demand changes within a range where there are infinitely many stable equilibria with the same fixed voltage magnitude and a 2π difference in the phase angle. Since each of these stable equilibria has the same voltage magnitude, they can be thought of as a single base state of the system. Furthermore, there are two alternative states. An alternative transient state is a temporary drop in the voltage magnitude accompanied by a 2π shift in the phase angle, caused by the coupling within the system. This state corresponds to a transition between two neighbouring stable equilibria within the base state. An alternative stable state is at zero voltage magnitude and corresponds to an electrical blackout

In the model, the voltage drops to negative infinity, but we restrict it to physically relevant non-negative voltage magnitudes.

We now demonstrate that a rapid increase in the level of power demand can lead to R-tipping in the form of different disruptions in power supply. First, we note that, for slow-enough increases in the level of power demand, the power grid network always tracks the moving base state (not shown here). Then, in Fig. , we show the response of the power grid network to ramp and return forcing profiles with higher rates of change.

We start with three ramp shifts that vary between the same levels of power demand without crossing any critical levels but at different rates. These are given by a concatenation of the left half of a colour curve and the dashed black curve in Fig. a. Owing to the higher rates of change, all three ramp shifts cause R-tipping to the alternative stable state, resulting in blackout; see the dashed curves in Fig. b. However, reversing the power demand (see solid colour pulses in Fig. a) can avoid blackout and restore the power grid network to the base state if the reversal is fast enough (see the corresponding solid colour responses in Fig. b). The slowest (blue) reversal is too slow to avoid irreversible R-tipping to blackout. The medium-rate (orange) reversal avoids blackout but involves reversible R-tipping to the alternative transient state – a temporary voltage drop followed by a long recovery towards the base state. The fastest (purple) reversal avoids both types of R-tipping and does not cause any noticeable disruptions to power supply.

The tipping diagram for ramp and return forcing profiles in Fig. c has, in addition to multiple critical rates already observed in the previous examples, an important new feature. Owing to the existence of two alternative states, there are two different red regions corresponding to different types of R-tipping for a return forcing profile: irreversible R-tipping to blackout (dark-red region), where the voltage magnitude drops to zero permanently, and reversible R-tipping (light-red region) involving a temporary drop in the voltage magnitude accompanied by a 2π phase slip. The three scenarios illustrated in Fig. a and b correspond to the three coloured dots in the tipping diagram in Fig. c. Note that all three dots are located past the basin instability boundary and below the critical level of power demand (the vertical black fold line), meaning that these transitions are purely rate induced.

We have shown that many natural and human systems can experience rate-induced tipping (R-tipping). Such instabilities usually occur for sufficiently fast increases in external forcing despite never crossing any critical levels of external forcing. In other words, systems are able to continually adapt to a moving base state and therefore avoid tipping when external forcing changes sufficiently slowly but fail to adapt to or track a moving base state when external forcing changes faster than some critical rate. Reversing the external forcing can prevent a system from suffering R-tipping, but the rates required for this add an additional layer of complexity in the presence of bifurcation-induced tipping (B-tipping). Previously, it has been shown that safe overshoots of critical levels for B-tipping require fast rates of change . However, faster rates of change make a system more susceptible to R-tipping. To make the concept of R-tipping accessible to a wide scientific audience, we

described an easily verifiable criterion of threshold or basin instability for R-tipping to occur

The dynamic models used in this study for representing a predator–prey ecosystem, the Atlantic overturning circulation, and an electrical power grid network are relatively simple. Since our forcing profiles are very idealised by design, we focus on the qualitative behaviour that can arise for different rates of forcing rather than on quantitative predictions. For quantitative predictions, further research on R-tipping is required in more-realistic higher-complexity models, such as state-of-the-art global circulation models, and with more-realistic forcing profiles. The base state in such models may not necessarily be a steady state (an equilibrium) but, for example, could take the form of a periodic orbit or even a chaotic state. This could lead to more complex tipping behaviour, such as phase tipping (P-tipping) .

R-tipping is likely to be prevalent in many systems given contemporary rates of change such as unprecedented anthropogenic climate change. This paper highlights the importance of considering how fast external forcing is changing as opposed to solely focusing on levels of change. Consequently, the actions taken to control the rate of change in forcing are equally as important as the actions taken to control the level at which forcing is halted.