We introduce a framework of cascading tipping, i.e. a sequence of abrupt transitions occurring because a transition in one subsystem changes the background conditions for another subsystem. A mathematical framework of elementary deterministic cascading tipping points in autonomous dynamical systems is presented containing the double-fold, fold–Hopf, Hopf–fold and double-Hopf as the most generic cases. Statistical indicators which can be used as early warning indicators of cascading tipping events in stochastic, non-stationary systems are suggested. The concept of cascading tipping is illustrated through a conceptual model of the coupled North Atlantic Ocean – El Niño–Southern Oscillation (ENSO) system, demonstrating the possibility of such cascading events in the climate system.

Earth's climate system consists of several subsystems, e.g. the ocean, the atmosphere, ice and land, which are coupled through fluxes of momentum, mass and heat. Each of these subsystems is characterized by specific processes, on very different timescales, determining the evolution of its observables. For example, processes in the atmosphere occur on much smaller timescales than in the ocean; hence, in weather prediction the upper ocean sets the background state for the evolution of the atmosphere. Similarly, in equatorial ocean–atmosphere dynamics associated with the El Niño–Southern Oscillation (ENSO) phenomenon, the global meridional overturning circulation can be considered a background state, as it evolves on a much larger timescale.

This notion that one subsystem provides a background state for the evolution of another subsystem
is important when critical transitions are considered. In the climate system, a number of tipping elements
have been identified (

Many tipping points have been analysed in separate subsystems, both for
phenomena of the present-day climate

An example in past climates is the coupling between the ocean's overturning circulation and
land ice. The rapid glaciation of the Antarctic continent around the Eocene–Oligocene boundary (34 Ma) is often explained
in terms of a

In the last few years, much work has been carried out to formulate statistical indicators
and early warning signals of tipping points. A system close to a critical transition shows features
of a “critical slowing down”

When considering cascading tipping points, the auto-correlation of two time series and their
interaction need to be analysed simultaneously.

In this paper, we provide a quantitative approach to cascading tipping events. We start with a
mathematical framework to formulate elementary cascading tipping points (Sect.

In the climate system, tipping points are usually related to rapid transitions, where an observable
in the climate system may change abruptly in a relatively short time compared with changes in the forcing of
the observable. Such rapid changes usually involve transitions from one equilibrium state
to another, which can often be explained using classical bifurcation theory for autonomous dynamical
systems. To a certain extent, these concepts can also be applied to non-autonomous systems (so-called
slow–fast systems) when the time variation of parameters can be viewed as a slow external
forcing

In this section, we present a mathematical framework for simple cascading
transitions, which acts as a first step towards analysing the more complex
real-world transitions. We focus on bifurcation-induced tipping
points, and consider two types of bifurcations that are thought to be
relevant to mechanisms of abrupt changes in the climate system; the
back-to-back saddle-node bifurcation is often used to explain transitions
between two coexisting equilibria (multi-stable systems), while the Hopf
bifurcation can explain the appearance of oscillatory behaviour

A back-to-back saddle-node bifurcation (two saddle-nodes connected by a common unstable branch) generically occurs in physical systems (that have bounded states) when one parameter varies and the simplest dynamical system has a bifurcation that is described by

Parameter values and coupling for the four types of cascading tipping as
shown in Figs.

A Hopf bifurcation also generically occurs in physical systems, and the
simplest dynamical system in which it occurs when one parameter is varied is
described by

There are two other bifurcations when one parameter is varied (the transcritical and pitchfork bifurcations); however, these bifurcations are non-generic because special conditions must hold (e.g. symmetry) and so they are not considered here. Using the saddle-node and Hopf bifurcations, cascading tipping can be viewed as a combination of two coupled subsystems, where each subsystem undergoes one of these two types of bifurcations. Combining these bifurcations leads to four types of cascading tipping, which are discussed in the following.

Stable (solid),
unstable (black dotted) and oscillatory (amplitude red dotted) regimes of
various cascading tipping types (as defined on top of the figure). Red,
orange and black dots indicate fold, Hopf and torus bifurcations, respectively. Leading system versus forcing

Example simulations for each cascading
event type: the double-fold cascade

Coupling two systems introduces a direction to the cascade and we take
account of this by defining a “leading” system, which during its
transition changes a parameter (that is, the coupling term) in the “following” system.
The changing parameter in the following system can then
bring the following system closer to a bifurcation point, potentially even
resulting in a second transition. In this section, we only look at
deterministic cases, which do not allow for noise to play a role in the
tipping. Therefore, transitions in the leading system result in a changed
coupling term that can lead to transitions in the following system. In
bifurcation diagrams versus forcing, the bifurcation points (for
deterministic systems) can overlap. However, the transitions are
distinguishable in transients, because the following system always tips after
the completion of the first transition. This is why we show the bifurcation
diagrams of both systems versus forcing (Fig.

The most intuitive system that has the potential to undergo a cascading
tipping event is a system where both the leading and the following system
have saddle-node bifurcations. Analogous to Eq. (

Implementing this system with the values shown in Table

The second type of cascading tipping event involves a saddle-node bifurcation in the
leading system and a subsequent Hopf bifurcation in the following system. Using analogous
notation as in Eq. (

The bifurcation structure of the leading system of Eq. (

A third type of cascading event involves a Hopf bifurcation in the leading system and a
subsequent saddle-node bifurcation in the following system. Using similar notation
to the previous subsection, the simplest system with a Hopf–fold cascade (see Table

Figure

When

A fourth type of cascading tipping event discussed here involves a Hopf bifurcation in the leading system and a subsequent Hopf bifurcation in the following system. Using analogous notation to the previous subsections, this double Hopf cascade is captured by the following dynamical system:

Figure

In the previous section we formulated elementary deterministic dynamical systems that can exhibit cascading tipping. In order to detect tipping events from, for example, observed time series in real systems, we need to detect whether a system is close to a critical transition. In general, systems close to critical transition recover more slowly from perturbations, which in turn increases memory in the time series. This leads to the phenomenon of “critical slowing down” prior to bifurcation points. In this section, we simulate cascading tipping events to (a) study the statistical character of such events, (b) diagnose (post-tipping) whether tipping events are causally related and (c) take a first step towards statistical indicators for the prediction of cascading tipping events.

Several methods have been suggested for the analysis of time series to detect
the approach of a single tipping point. For saddle-node bifurcations, the key
features of a time series such as this is a critical slowing down. This can be
investigated using standard quantities such as increasing auto-correlation (e.g.
the lag-1 auto-correlation), increasing variance and increasing skewness

As critical slowing down implies an increasing auto-regressive behaviour in
the time series prior to a transition, the memory component is increased. In
general, the first step in DF is the projection of the data fields onto the
leading EOF, which gives a time series

In DFA, one first chooses an integer window size

For

Cascading tipping involves two systems with their own bifurcation structure and their proximity towards bifurcation points. Although the leading system may be close to tipping, the following system might still be far from its bifurcation point and needs the critical transition of the leading system to even come close to this point; this makes the behaviour of the following system more prone to noise, less dependent on the leading system and less auto-correlated prior to the first tipping. This is why the general measures for single tipping events cannot be used, nor can regular cross (Pearson) correlation. The reason for this is that the following and leading systems do not have a one-to-one relationship (that is, weakly Pearson correlated), but are rather coupled through specific parameters, which is only visible in long-range correlations.

When approaching a cascading tipping point, the long-range cross-correlation
between the leading state

A variation on this method was proposed by

There are several a priori limitations of using the power-law scaling
coefficient and

In this section we discuss the previously described metrics applied in ensemble simulations of cascading transition events. This provides insight into the statistical characteristics of these events, the causal relation between tipping of both systems and the potential prediction of these events. We focus on the double-fold and the fold–Hopf cascading tipping cases for multiple reasons. First, these cases are most illustrative in terms of relation to physical systems. Second, in these cases the leading system starts with a fold bifurcation, which creates a clear threshold for the start of the event (which is convenient for analysis purposes).

Ensemble (100-member) simulations of a
dynamical system undergoing a double-fold cascade (Eq.

Parameter values, coupling and initial conditions for the ensemble simulations of the
double Fold and fold–Hopf systems as shown in Figs.

As in Fig.

To simulate these events and use statistical indicators, noise has to be
included. The system of equations used here is

In the LTP, we clearly see the gradual increasing leading system's variance, the AR(1) coefficient and DFA scaling coefficient. These are all evidence of the leading system slowly approaching a bifurcation point, according to the change in the parameter. There is not much evidence of long-range auto-correlations in the time series of the following system, as its variance is low and the DFA scaling exponent remains below 0.5, pointing towards the fact that the detrended fluctuations are statistically white noise. The AR(1) coefficient of the following system does increase just prior to the first tipping, but it also stays small (compared to unity).

The detrended cross-correlation scaling exponent (abbreviated here as DXA)
does give

The quantity

During the FTP, the variance, AR(1) and DFA of the leading system are
strongly reduced. However, the gradual increase of the following system's
variance, AR(1) coefficient and DFA scaling coefficient are definitely
visible, pointing towards the approach of a bifurcation in the following
system. Also notable is the contrast in the DFA of the following system
before and after the tipping of

Comparison of the ratios of auto-regressive variables prior to and after the
first transition, using the ensembles shown in Figs.

To assess the consequences of the cascading effect on the statistics mentioned, we
compare the results with a case where the system (Eq.

Results of Student's

To quantify this effect, Table

As in Fig.

Many statistical indicators have been specifically applied on fold bifurcations,
because these transitions show a clear sign of critical slowing
down and increased auto-correlation; this is due to their irreversibility and the process of
going from one equilibrium towards another. A state approaching a Hopf
bifurcation reacts differently to perturbations than a state approaching a
saddle-node bifurcation. Therefore, we will consider the fold–Hopf cascade in
the light of the previously described statistical indicators. For this, we use
the following stochastic dynamical system:

After the transition of the leading system, the oscillation of the following
system immediately starts due to noise. The variance and AR(1) of the
following system after the bifurcation are strongly increased with respect to
before the bifurcation, despite the removal of the running average to eliminate
the oscillation's signal. On average, the DFA scaling exponent of the following system also
increases, which relates it to the leading system's
state. The DXA sharply increases just prior to the critical transition, although
it retains relatively high values throughout the whole time series. The reason
behind this might be the low level of noise that is chosen, or other
simulation-specific parameters. It could also due to the fact the
following system has a high, weakly varying DFA scaling exponent on average,
which in turn might affect the height and variability of the
cross-correlation. The

In this section, we present an application of the concept of cascading tipping (the fold–Hopf case). This application reflects the fact that cascading transitions are not only a purely mathematical concept, but also occur in idealized physical models. Here, we consider cascading tipping in a model that couples the Atlantic meridional overturning circulation (AMOC) and the El Niño–Southern Oscillation (ENSO).

To demonstrate and quantify the coupling between the AMOC
and ENSO, we use output from global climate model simulations. In the ESSENCE
project (Ensemble SimulationS of Extreme weather events under Nonlinear
Climate changE) several simulations were performed using the ECHAM5/MPI-OM
coupled climate model, including so-called “hosing” experiments

NINO3.4 statistics (of deseasonalised data) for the different ensembles. The uncertainty stated is the standard deviation among the five runs within the ensemble. It is visible that in the case of a collapsed overturning, El Niño intensifies more than without a collapsed overturning. Values in bold represent increases in the variability of NINO3.4.

From these climate model simulations we used two ensembles; the first is
the “standard” experiment, where greenhouse gases evolve according to
observations and from the year 2000 onwards follow the SRES-A1b scenario
(experiment name SRES-A1b). The second ensemble is the same as the standard
experiment but has additional freshwater input (1 Sv

The results for the evolution of the AMOC and ENSO are shown in Fig.

It is visible from Table

Several mechanisms have been suggested in the literature regarding the coupling between the
AMOC and ENSO. The first mechanism is concerned with oceanic waves. A colder North Atlantic creates density anomalies that induce
the southward propagation of oceanic Kelvin waves (along the American coast) across the Equator. In
western Africa, this energy radiates as Rossby waves towards the north and south, which induces
Kelvin waves to move along the tip of southern Africa into the Indian Ocean, which eventually reach
the Pacific. Consequently, the eastern equatorial Pacific thermocline deepens on a decadal timescale.
This deepening has a weakening effect on the amplitude of ENSO

The second mechanism is concerned with the trade winds. Cooling in the northern
tropical Atlantic (due to AMOC weakening) induces anticyclonic atmospheric
circulation

To study the possible cascading transition through the wind-stress coupling, we use a
conceptual model. For the AMOC, the classical Stommel box model

For the El Niño–Southern Oscillation, we use the conceptual model as proposed in

The subsurface temperature

Zonal equatorial wind stress versus the Atlantic
temperature gradient. Data from the ESSENCE (Ensemble SimulationS of Extreme
weather events under Nonlinear Climate changE) project are used, where the
black dots refer to five members of the standard ensemble with SRES-A1b
forcing (1950–2100 period ), and the red dots refer to five members of the
HOSING-1 ensemble where in 2000 a freshwater perturbation is applied (i.e.
2000–2100 period). A 5-year running mean is applied, yearly averages are
shown. The zonal equatorial wind stress is defined here as the average zonal
wind stress over the 0–10

We couple the AMOC and ENSO models through the relation between the Atlantic
meridional temperature gradient (in the Stommel model) and the Pacific zonal
wind stress (in the Timmermann model); the Pacific zonal wind stress mechanism,
which is the more important of the two abovementioned mechanisms, is
found in the literature, and described in the previous section. Even in a simplified
model, the relation between wind stress and meridional temperature gradient
is physically justified: thermal wind balance indicates a direct connection
between the adjustment of wind stress to changes in the meridional
temperature gradient. In the Timmermann model, the zonal wind stress

In the coupling (Eq.

Bifurcation diagrams and forward runs of the Stommel

The AMOC model's bifurcation diagrams are shown in Fig.

Simulation run of the coupled Stommel–Timmermann model for different
model configurations, where the collapse of the overturning flow function
(black) leads to the crossing of a Hopf bifurcation in the eastern
equatorial Pacific SST (orange). Parameter values as in

The bifurcation diagram of the ENSO model with

Using

Using the coupling of Eq. (

Despite the parameter sensitivity, this is a typical illustration of the fold–Hopf cascading behaviour discussed in earlier sections. This re-enforces the possibility that cascading transitions are possible in real physical systems.

In this paper, we introduced the concept of cascading tipping, which can
occur when a transition in a leading system alters background conditions for
a following system that then undergoes a transition. We presented a
mathematical framework around this concept, using generic
bifurcations (saddle-node and Hopf) in both leading and following systems.
Four types of deterministic dynamical systems with the possibility for
cascading events were formulated, including the double-fold cascade, the
fold–Hopf cascade, the Hopf–fold cascade and the double-Hopf cascade. In all
cases we assumed a linear coupling between the following and leading system.
The double-fold coupled system has previously been considered in another context

We discussed statistical indicators and analysis tools for cascading
tipping points. Indicators for cascading tipping points are found in
detrended cross-correlation analysis (DCCA) and a special case of
extrapolation using the DFA of the following system. These tools were applied
in simulations involving both the double-fold and fold–Hopf cascades. The
increased variance, AR(1) and DFA scaling exponent are clearly found in each
case of single tipping. The cross-correlation indicators (DCCA and

The concept of cascading tipping was applied to study the behaviour of a
model describing a link between the Atlantic meridional overturning
circulation (MOC) and ENSO. We modelled this using a coupling between the

A potential example of a double-fold cascade, which was not further treated here,
could be the impact of a bistable MOC on the (bistable) land ice formation on the Antarctic continent.
In this case the coupling exists through the atmospheric

These two applications reflect the relevance of this paper. There are likely many cases in which these cascading events occur in climate and therefore highlight the importance of the topic. Future research will point out whether these events are likely to happen in the future climate and whether these effects also occur in fields other than climate science. Of course, this paper covers the very basics of deterministic cascading events. However, one can imagine a wide range of phenomena if more complicated transitions between attractors are considered and when noise is included. For example, when a leading chaotic system is coupled to a deterministic following system with a saddle-node bifurcation structure, a slight change in the chaotic attractor may change the background conditions for the following system such that it undergoes a transition. An application here may be the effect of a mid-latitude atmospheric jet on the Atlantic MOC. We hope that this paper will stimulate more research on the various types of cascading tipping and also on the development of well-suited indicators and early warnings of such events.

Models, analysis scripts and outputs are available on request from the corresponding author.

All authors developed the research idea, MMD carried out the computations and performed the analysis. All authors discussed the results and contributed to writing the paper.

The authors declare that they have no conflict of interest.

Anna S. von der Heydt and Henk A. Dijkstra acknowledge support from the Netherlands Earth System Science Centre (NESSC), financially supported by the Ministry of Education, Culture and Science (OCW), grant no. 024.002.001. Anna S. von der Heydt thanks Peter Ashwin for discussions related to this work. She is also grateful to the University of Exeter and the EPSRC funded Past Earth Network (grant no. EP/M008363/1) for funding an extended visit to the University of Exeter during the Summer of 2017. Edited by: Valerio Lucarini Reviewed by: Alexis Tantet and one anonymous referee