The relative role of external forcing and of intrinsic variability is a key question of climate variability in general and of our planet's paleoclimatic past in particular. Over the last 100 years since Milankovic's contributions, the importance of orbital forcing has been established for the period covering the last 2.6 Myr and the Quaternary glaciation cycles that took place during that time. A convincing case has also been made for the role of several internal mechanisms that are active on timescales both shorter and longer than the orbital ones. Such mechanisms clearly have a causal role in Dansgaard–Oeschger and Heinrich events, as well as in the mid-Pleistocene transition. We introduce herein a unified framework for the understanding of the orbital forcing's effects on the climate system's internal variability on timescales from thousands to millions of years. This framework relies on the fairly recent theory of non-autonomous and random dynamical systems, and it has so far been successfully applied in the climate sciences for problems like the El Niño–Southern Oscillation, the oceans' wind-driven circulation, and other problems on interannual to interdecadal timescales. Finally, we provide further examples of climate applications and present preliminary results of interest for the Quaternary glaciation cycles in general and the mid-Pleistocene transition in particular.

In the early 20th century, Milutin Milankovic presented his theory of ice ages

The orbital forcing comprises variations in (i) the eccentricity of the Earth's orbit around the sun with dominant spectral peaks around 400 and 100 kyr; (ii) the obliquity, or axial tilt, i.e., the angle between the Earth's rotational and its orbital axis, with dominant periodicity around 41 kyr; and (iii) the climatic precession, which determines the phase of the summer solstice along the Earth's orbit and has its most pronounced spectral power around 23 and 19 kyr

For 2 centuries or so of modern geology, records of our planet's physical and biological past were merely discrete sequences of strata with specific properties, like coloration and composition

The advent of marine-sediment cores after World War II led, for the first time, to the availability of records that were more or less continuous in time. Like all climate records, these cores covered limited time intervals and did so with limited resolution and with inaccuracies in absolute dating, as well as in the quantities being measured. Moreover, they posed the problem of inverting proxy records of isotopic and microbiotic counts to physical quantities like temperature and precipitation.

In spite of these limitations, the spectral analysis of deep-sea records allowed

Power spectrum of a composite

The work of

Comparison of six analyses of the annually and globally averaged surface temperature anomalies through 2018. The abscissa is time in years, and the ordinate is temperature anomalies in

In fact, interest in past climates was heightened not only by these striking observational discoveries but also by the growing concerns about humanity's impact on the climate

As a result of the twofold stimulation provided by data about past glaciations and concern about future ones, a number of researchers in the early-to-mid 1970s worked on energy balance models (EBMs) of climate with multiple stable steady states

New geochemical evidence, however, led in the early 1990s to the discovery of a “snowball” or, at least, “slushball” Earth prior to the emergence of multicellular life, sometime before 650 Myr b2k

On the other hand, it also became clear that these early models, whose only stable solutions were stationary, could not reproduce the wealth of variability that the proxy records were describing very well, not even in the presence of stochastic forcing

The theoretical quandary of modeling the Quaternary glaciation cycles, illustrated here by schematic diagrams of the composite power spectral densities (PSD) of

The stable self-sustained oscillations of these coupled models, however, were totally independent of any orbital or other time-dependent forcing, i.e., the solar input to their radiative budget was constant in time. Hence, they could not capture the wealth of spectral features, with their orbital and other peaks, of the paleo-records available by the 1980s.
The basic quandary of the Quaternary glaciation cycles – at least from the point of view of theoretical climate dynamics

In this paper, we try to show a path toward resolving the four fundamental questions listed in Box

Fundamental questions regarding the Quaternary glacial–interglacial cycles.

We follow

The basic workings of this climate oscillator can be represented by two coupled ordinary differential equations (ODEs), written symbolically as follows:

The latter feedback can be better understood by writing the following equations:

As first observed by George C. Simpson – the meteorologist of Robert F. Scott's

More generally, the presence of feedbacks of opposite sign in a system of two linear coupled ODEs,

This can be understood by considering the so-called normal form of a Hopf bifurcation, which leads from a stable steady state, called a fixed point in dynamical systems theory, to a stable oscillatory solution, called a limit cycle. The easiest way to see this transition is by writing the normal form in polar coordinates, as in

Supercritical Hopf bifurcation.

A very natural transformation of variables,

In this subsection, we present an argument for the role of intrinsic oscillations in the mid-Pleistocene transition (MPT). The first point to be made is that while orbital forcing clearly plays a major role in the power spectrum of the Quaternary's climatic variability, it cannot be, in and of itself, the cause of the MPT. Indeed, changes in the solar system's orbital periodicities only occur on much longer timescales than the entire Quaternary's duration

Physically speaking, the presence or absence of the regular, purely periodic oscillations obtained by KCG and illustrated in

To clarify the simple physical concepts that underlie subcritical and supercritical Hop bifurcations, let us consider a purely mechanical oscillator with mass

The supercritical Hopf bifurcation in the absence of forcing is analogous to the nonlinear response of a soft, sublinear spring to periodic forcing in which the oscillations in the position

There is a clear-cut analogy with the mid-Pleistocene transition, occurring at roughly 0.8 Ma b2k, at which small-amplitude climate variability with a dominant periodicity near 40 kyr becomes larger, dominated by a periodicity that is close to 100 kyr, as well as being more irregular

In contrast, the subcritical Hopf bifurcation of the KCG and

We start this section by describing some fairly simple ways in which the orbital forcing might have modified intrinsic climate variability, thus helping to solve the mismatch between Fig.

Depending on parameter values, the periodicity

Power spectra of simulated

The results of the

Aside from the spectral features noted in the figure caption and discussed in greater detail by

In returning to the “fundamental question no. 2” in Box 1, one must recall that on the paleoclimatic timescales of interest – apart from deterministic chaos à la

The highly preliminary results summarized in Sect.

On the road to including deterministically time-dependent and random effects, one needs to realize first that the climate system – as well as any of its subsystems on any timescale – is not closed: it exchanges energy, mass, and momentum with its surroundings, whether this pertains to other subsystems or the interplanetary space and the solid earth. The typical applications of dynamical systems theory to climate variability until not so long ago have only taken into account exchanges that are constant in time, thus keeping the model – whether governed by ordinary, partial, or other differential equations – autonomous; i.e., the models had coefficients and forcings that were constant in time.

Alternatively, the external forcing or the parameters were assumed to change either much more slowly than a model's internal variability, meaning that the changes could be assumed to be quasi-adiabatic, or much faster, meaning that they could be approximated by stochastic processes. Some of these issues are covered in much greater detail by

The presentation of the key NDS and RDS concepts and tools in this subsection is aimed at as large a readership as possible and follows

Succinctly, one can write an autonomous system as

For instance, two distinct trajectories,

We know only too well, however, that the seasonal cycle plays a key role in climate variability on interannual timescales, while orbital forcing is crucial on the Quaternary timescales of many millennia. In addition, more recently it has become obvious that anthropogenic forcing is of utmost importance on the interdecadal timescales between these two extremes.

How can one take into account these types of time-dependent forcings and analyze the non-autonomous systems that they lead us to formulate? One succinctly writes such a system as follows:

To illustrate the fundamental distinction between an autonomous system like Eq. (

In the case of Eq. (

Formally, the indexed family

each snapshot

the pullback attraction occurs for all times,

Since the dynamics of the phase

The PBA with respect to the coordinate

Trajectories and PBA of the system defined by Eqs. (

Figure

Figure

If

Finally, if

Let us return now to the more general, nonlinear case of Eq. (

The noise processes may include “weather” and volcanic eruptions when

Schematic diagram of a random attractor

A key feature of the pullback point of view on noise-perturbed dynamical systems that characterizes RDS theory is the use of a single noise realization, as opposed to the traditional, forward viewpoint of the Fokker–Planck equation and associated concepts, in which multiple noise realizations play a role. For a precise definition of a random attractor – as well as the commonalities and differences between the deterministic and random cases of time-dependent forcing – please see

Heat maps of the time-dependent invariant measure

Before discussing conceptual glacial cycle models, we take a little detour and introduce a simpler – yet interesting and at the same time highly instructive – application of NDS theory to another important climate phenomenon. During past glacial periods, Greenland experienced a series of sudden decadal-scale warming events that left a clear trace in ice core records

We discuss the example of the FHN model at some length in order to illustrate how external forcing can act on a system's internal variability and thereby give rise to more complex dynamics. This model's concise mathematical formulation and its widespread application in paleoclimate modeling and other fields make it ideally suited for this goal. We start with a description of the autonomous model with no time-dependent forcing. Subsequently, we introduce a simple sinusoidal forcing and numerically compute the corresponding PBA. We then extend these consideration into the realm of random dynamical systems by adding stochastic forcing and discuss the resulting random attractor.
Finally, we replace the synthetic forcings by one that corresponds to a paleoclimate proxy record of past CO

The FHN model consists of two coupled ODEs that govern behavior alternating between slow evolutions and fast transitions. Typically, the timescales of the two variables are separated by introducing the parameters

First, consider the case of large timescale separation

Now, let us investigate the coupled dynamics of the slow and fast variables

In the

In fact, in the autonomous setting, the system's qualitative behavior is
controlled by the value of the parameter

Nullclines of the autonomous FHN model governed by Eq. (

FitzHugh–Nagumo (FHN) model with parameters

This behavior can be better understood by considering the nullclines of Eqs. (

If they do not, the system first relaxes along the fast direction toward the

So far we have described the formation of the limit cycle in the FHN model under the assumption of clear timescale separation and the independence of the

The highly nonlinear, two-time behavior of the FHN model somewhat modifies the way that stable limit cycles arise in it.
While we saw the oscillation's radius grow with the square root of the bifurcation parameter in the case of the normal form given by Eqs. (

Introducing a sinusoidal time dependence

The trajectories plotted in Fig.

For Fig.

Strikingly, all trajectories converge to one another during non-oscillatory time intervals, when they are simultaneously attracted by the single existing fixed point. During oscillatory intervals, phase differences between individual trajectories may, in principle, persist. Still, convergence during a single non-oscillatory interval is so strong that after it the trajectories can no longer be discriminated visually. Numerically, however, the distance between trajectories only tends to zero but never reaches it. At the end of non-oscillatory intervals, the trajectories always re-enter the oscillatory regime from the same location in the

For Fig.

Again, the PBA of this non-autonomous system can be thought of as an infinite repetition of the common trajectory structure that can be observed in Fig.

Based on the brief introduction to RDSs in Sect.

Random attractor of the periodically and stochastically forced FHN model governed by Eq. (

In order to study the random attractor of this system, we compute trajectories with random initial conditions over a time span long enough to reveal the asymptotic behavior, as shown in Fig.

For the long periodic forcing with

For the shorter period forcing with

The investigation carried out herein assesses a very special case of an FHN model's random attractor. Random attractors of FHN-type models have been studied intensively

Readers who are familiar with the NGRIP

FHN model fit to the NGRIP

Figure

In fact,

In the present framework, the FHN model's fast variable

The interaction between the fast variable and the slow one happens here in the presence of a climate forcing represented by CO

Apparently, it was

Based on the considerable success of NDS and RDS applications to other climate problems – such as ENSO

Box

Some open questions concerning Quaternary glaciations.

Such an approach can usefully complement the more common one of merely pushing onwards to higher and higher model resolution in order to achieve ever more detailed simulations of the system's behavior for a limited set of semi-empirical parameter values.

In this section, we illustrate how the PBA concept can help shed more light upon the dynamics of ice age models. As pointed out in Sect.

Glacial–interglacial cycles simulated by the modified Daruka and Ditlevsen model of Eqs. (

Among these glacial-cycle models, the model of

Our model's variables, following DD16, are a global temperature anomaly

In the original DD16 model, MPT-like behavior was produced by a slow sigmoid variation of the parameter

Figure

PBA of the M-DD16 model governed by Eq. (

We next approximate the PBA by taking 40 random initial conditions at 10 Ma b2k and integrating the model of Eqs. (

However, when keeping the parameters

There are two interesting inferences to be drawn. First, post-MPT dynamics is much more irregular and unstable than the more stable dynamics prior to the MPT.
The robustness of the 40 kyr glacial cycles and instability of 100 kyr glacial cycles against perturbations is in line with the conclusions of previous studies

Second, the separate bundles or “ropes” of trajectories in Fig.

In this review and research paper, we have covered the contributions of the 1970s to the rebirth of the

Finally, in Sect.

When the parameters are gradually changed in time so as to exhibit the mid-Pleistocene transition (MPT), the PBA is simply a moving fixed point. However, when the parameters are fixed at their post-MPT values, the PBA so obtained is chaotic and exhibits clusters of trajectories that we termed ropes. This suggests (a) that the stability of the system is gradually lost while crossing the MPT and (b) that the Late Pleistocene climate, albeit chaotic, may well be subject to a kind of generalized synchronization

In a broader perspective – and leaving aside various finer points of the MPT conversation outlined in Sect.

Hence the following scenario (compare Emiliani and Geiss, 1959) suggests itself for the successive climatic transitions from Pliocene to Pleistocene and from Early to Late Pleistocene: As land masses moved towards more northerly positions, small ice caps formed on mountain chains and at high latitudes. These ice caps, due to their feedback on albedo, made climate more sensitive to insolation variations than it was in the total absence of ice. The response of the climatic system to such variations during the Early Pleistocene (2000 [kyr]–1000 [kyr] ago) was still relatively weak, of a fraction of a degree centigrade in global temperature perhaps, in agreement with the quasi-equilibrium results of Sect. 10.2.

As ice caps passed, about 1000 [kyr] ago, a certain critical size, the
unforced system jumped from its stable equilibrium to its stable limit-cycle
state (Figures 12.5 and 12.9), increasing dramatically the climate's
total variability, to a few degrees centigrade in global temperature.
Furthermore, resonant response became possible (see also

The take-home message is that slow and fast processes, both intrinsic and extrinsic, interact on all paleoclimatic timescales and that we are mastering the art of modeling such interactions.

The dynamical modeling of glacial cycles dates back to the 1970s.

Shortly after

A deeper understanding of glaciation cycles cannot be obtained without process-based models that focus on the detailed physics and biogeochemical phenomena involved

List of simple conceptual glacial-cycle models with only 1–3 variables. Note that, while extensive, this list is not exhaustive.

Continued.

Continued.

For example, coupled nonlinear oscillators frequently exhibit synchronization with simple frequency ratios, either with each other

Here we study the system of two formally decoupled ODEs

First, we define

Repeated partial integration yields

and finally

Plugging this result into the ansatz (Eq.

Since the evolution in time of the phase

All code used to generate the figures presented in this article is available from the authors upon request. The NGRIP

The video supplement to this article (

MG conceived and designed the study. KR and TM carried out the major part of the article's new research in close interaction with MG and NB. All authors interpreted and discussed the results and wrote the manuscript.

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article is part of the special issue “A century of Milankovic's theory of climate changes: achievements and challenges (NPG/CP inter-journal SI)”. It is a result of the conference “One Hundred Years of Milankovic's Theory of Climate Changes: synergy of the achievements and challenges of the next century”, 17–18 November 2020.

We thank Andreas Groth for helpful comments on an earlier version of this manuscript. It is a pleasure to thank Tamás Bódai and two anonymous referees for their thorough and constructive comments. István Daruka's public comments in part motivated the addition of Appendix A and its Table A1, in order to give a broader perspective of relevant work on the Quaternary's glacial cycles and the mid-Pleistocene transition (MPT). Takahito Mitsui and Niklas Boers acknowledge funding by the Volkswagen Foundation. The present work is TiPES contribution no. 52; the TiPES (Tipping Points in the Earth System) project has received funding from the European Union's Horizon 2020 research and innovation program under grant agreement no. 820970. Michael Ghil acknowledges support by the EIT Climate-KIC; EIT Climate-KIC is supported by the European Institute of Innovation & Technology (EIT), a body of the European Union.

This research has been supported by the Horizon 2020 research and innovation program under grant agreement no. 820970 (TiPES), the Volkswagen Foundation, and the European Institute of Innovation & Technology via the EIT Climate-KIC.The publication of this article was funded by the Open Access Fund of the Leibniz Association.

This paper was edited by Marie-France Loutre and reviewed by Tamas Bodai and two anonymous referees.