Is the astronomical forcing a reliable and unique pacemaker for climate? A conceptual model study
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Abstract
There is evidence that ice age cycles are paced by astronomical forcing, suggesting some kind of synchronisation phenomenon. Here, we identify the type of such synchronisation and explore systematically its uniqueness and robustness using a simple paleoclimate model akin to the van der Pol relaxation oscillator and dynamical system theory. As the insolation is quite a complex quasiperiodic signal involving different frequencies, the traditional concepts used to define synchronisation to periodic forcing are no longer applicable. Instead, we explore a different concept of generalised synchronisation in terms of (coexisting) synchronised solutions for the forced system, their basins of attraction and instabilities. We propose a clustering technique to compute the number of synchronised solutions, each of which corresponds to a different paleoclimate history. In this way, we uncover multistable synchronisation (reminiscent of phase or frequencylocking to individual periodic components of astronomical forcing) at low forcing strength, and monostable or unique synchronisation at stronger forcing. In the multistable regime, different initial conditions may lead to different paleoclimate histories. To study their robustness, we analyse Lyapunov exponents that quantify the rate of convergence towards each synchronised solution (local stability), and basins of attraction that indicate critical levels of external perturbations (global stability). We find that even though synchronised solutions are stable on a long term, there exist short episodes of desynchronisation where nearby climate trajectories diverge temporarily (for about 50 kyr). As the attracting trajectory can sometimes lie close to the boundary of its basin of attraction, a small perturbation could quite easily make climate to jump between different histories, reducing the predictability. Our study brings new insight into paleoclimate dynamics and reveals a possibility for the climate system to wander throughout different climatic histories related to preferential synchronisation regimes on obliquity, precession or combinations of both, all over the history of the Pleistocene.
Keywords
Climate models Milankovitch Oscillator Generalised synchronisation Lyapunov exponent Multistability1 Introduction
This article is a contribution to the field of paleoclimate dynamics theory, which has experienced many developments in terms of ice age models since many years notably by Le Treut and Ghil (1983), Saltzman and Maasch (1990), and many others, and remains an active research field. Paleoclimate modelling is a complex problem, hence an uncomfortable situation for a scientist. On the one hand, the paleoclimatic records are often difficult to interpret physically (these are only proxies), and an independent dating is not always easy to achieve; hence it leads to uncertainties both in magnitude and time. On the other hand, there is not a single well established model, the problem is non autonomous, the forcing is aperiodic, and stochastic effects are present.
 1.
oscillations: the signal oscillates between higher and lower values of ice volume corresponding to the glacial and interglacial states,
 2.
asymmetry: in Fig. 1a typical transitions from a minimum to a maximum take much longer than transitions from a maximum to a minimum: deglaciations occur much more rapidly (τ_{ fast } ≈ 10 kyr) than glaciations (τ_{ slow } ≈ 80 kyr), giving a distinctive sawtooth structure in the glacial/interglacial (G/I) cycles, especially pronounced over the last 500 kyr,
 3.
100kyr dominant period: this has been identified by many authors since Broecker and van Donk (1970). Note that the G/I cycles are not periodic.
The asymmetry in the oscillations has been studied by many authors. In order to reproduce it, some authors use underlying physical principles to build phenomenological models that exhibit slowfast dynamics reasonably mimicking the climatic proxies (Saltzman 2002). Others assume this asymmetry by explicitly defining 2 different parameters such as the time intervals τ_{ up } = τ_{ slow } and τ_{ down } = τ_{ fast } (Ashkenazy 2006) or time constants [τ_{ R } and τ_{ F } in Paillard (1998) and T _{ w } and T _{ c } in Imbrie and Imbrie (1980)]. Whatever the model, it has to ultimately exhibit asymmetric oscillations under the effect of the forcing, as it is aimed to mimic the oscillations between G/I states. Relaxation oscillators are therefore very straightforward natural candidates of ice age models (Crucifix 2012). In this article, we will consider a slightly modified van der Pol oscillator model to illustrate the new contributions of our synchronisation concepts.
In this article, we chose the best possible approach for dynamical modelling available so far, i.e. we assume that the observed G/I climate variability essentially emerges from externally forced lowdimensional deterministic dynamics possibly subject to stochastic disturbances. However, it is worth mentioning that alternative approaches have been developed to explain the G/I variability, e.g. the stochastic resonancelike phenomena ^{1} (Benzi et al. 1982) in which highdimensional (or at least approximately stochastic) dynamics plays a much more important role. A short discussion about this alternative approach is given in Crucifix (2012). It is very likely that future works should combine different approaches in some appropriate way.
In this paper, we concentrate on the influence of the astronomical forcing on Earth’s climate. This forcing is induced by the slow variations in the spatial and seasonal distributions of incoming solar radiation (insolation) at the top of the atmosphere, associated with the slow variations of the Earth’s astronomical elements: eccentricity (e), true solar longitude of the perihelion measured with respect to the moving vernal equinox (\(\varpi\)), and Earth obliquity (\(\epsilon_E\)). These quantities are now accurately known over several tens of millions of years (Laskar et al. 2004), but analytical approximations of e, \(e\sin \varpi, \) and \(\epsilon_E\) valid back to one million years have been known since Berger (1978). They take the form of d’Alembert series (\(\sum A_i \sin[\omega_i t + \phi_i]\)).
Many attempts have been made since the eighties in order to identify any relationship between the frequencies observed in the paleoclimatic records, reproduced by a given mathematical model, and those present in the insolation, mainly by performing spectral analysis. For example, (Le Treut and Ghil 1983) consider a nonlinear climatic oscillator based on physical climatic mechanisms, and found frequency locking for some specific runs of their model. They proposed to explain the ice age cycle in terms of a beat period (or combination tone) between the 19 and 23 kyr periods. Hyde and Peltier (1985) also propose a physical ice age model, and study several individual harmonic forcing periods (ibid., Fig. 23) and also astronomical forcing (Hyde and Peltier 1987) but reject the “combination tone” hypothesis. Paillard (1998) suggested two simple threshold models with multiple states in order to reproduce the nonlinearity between the 100 kyr periodicity in the records and the insolation forcing. Gildor and Tziperman (2000) presented a sea ice climatic switch mechanism by which the insolation changes act as a pacemaker, setting the phase of the oscillations. Their box model is able to reproduce the asymmetric sawtooth structure, and results show phaselocking to the orbital variations through a nonlinear mechanism. Most of the time, however, the conclusions rely on a few particular realisations of the models, without providing a global analysis of the synchronisation phenomenon. Such an analysis is the subject of this study.
Synchronisation
There is ample evidence that the astronomical forcing influences the climate system. The phrase ’pacemaker of ice ages’ was coined in a seminal paper (Hays et al. 1976) to express the idea that the timing of ice ages is controlled by the astronomical forcing, while the ice age cycle itself is shaped by internal system dynamics. The paradigm has prevailed since then and it is still supported by the most recent analyses of palaeoclimate records (Lisiecki and Raymo 2007; Huybers 2007). The notion of ’pacemaker’ naturally evokes some sort of synchronisation. However, despite some attempts, the actual type of synchronisation has not been clearly identified or demonstrated to date. For example, (Ashkenazy 2006; Tziperman et al. 2006) speak of “nonlinear phaselocking” although they do not define suitable “phase variables” that can be used to demonstrate a fixedintime relationship between phases of the forcing and the oscillator response.
In this paper, we use a simple van der Pol oscillator model to identify and illustrate for the first time the phenomenon of generalised synchronisation between ice age cycles and astronomical forcing. The dynamical systems approach outlined in the next section (1) allows for stability analysis of such synchronisation, (2) uncovers interesting effects related to the robustness of the synchronisation with respect to external perturbations, and (3) uncovers the phenomenon of multistable synchronisation that has been overlooked by previous studies. We show that, in contrast to claims in Tziperman et al. (2006), synchronisation needs not be unique.
The article is structured as follows. Section 2 introduces a slightly modified version of the van der Pol oscillator as a suitable model for studying synchronisation of ice ages to astronomical forcing. In Sect. 3, we analyse synchronisation to periodic forcing and quasiperiodic astronomical forcing in terms of largest Lyapunov exponents. Section 4 is dedicated to the study of multistable synchronisation in terms of attracting trajectories in the phase space of the forced system, and the associated basins of attraction. In Sect. 5, we investigate effects of the symmetrybreaking parameter β for the van der Pol oscillator model. Section 6 is concerned with the robustness of the synchronisation and focuses on two aspects relating to predictability. Firstly, it shows that the local stability can be lost temporarily causing divergence of nearby climatic trajectories. Secondly, it demonstrates that in the multistable regime external perturbations (such as noise) may cause jumps between coexisting synchronised solutions when these solutions come close to their basin boundary. To be clear, all the treatment below is deterministic, except for Figs. 14 and 16.
This article requires some basics of Dynamical Systems theory (dynamical systems, nonlinear oscillations, limit cycles, bifurcations of vector fields, etc.), for which we refer the reader to Guckenheimer and Holmes (1983), Arnold (1983) and Strogatz (1994). We also refer to Saltzman (2002) for dynamical paleoclimatology, and to Savi (2005) for a review of many useful concepts such as attractors and Lyapunov exponents. For details about the van der Pol oscillator, we refer the reader mainly to van der Pol (1926), Strogatz (1994), Barnes and Grimshaw (1997), Hilborn (2000) and Balanov et al. (2009).
2 Generic ice age model: a modified van der Pol relaxation oscillator
The physical interpretation of the model (4a, 4b) is as follows. Ice volume x integrates the external forcing F(t) over time but with a drift y + β. Assuming α ≫ 1, y is the faster variable whose dynamics is controlled by a twowell potential \(\Upphi(y). \) For example, there are arguments that the dynamics of the Atlantic ocean circulation may be approximated by an equation similar to Eq. 4b (Rahmstorf et al. 2005; Dijkstra et al. 2003). Further interpretation and discussion of the fast variable can be found in Saltzman et al. (1984), Tziperman and Gildor (2003), Paillard and Parrenin (2004), Tziperman et al. (2006) and Crucifix (2012). It is however not the goal of this paper to design the best suited paleoclimatic model; on the contrary, we propose a methodology for diagnosing synchronisation which could be applied to any paleoclimatic model, and which is illustrated herein using a simple conceptual paleoclimatic model sufficiently plausible. Also, we deliberately chose a deterministic approach in order to present the concepts, while it is clear that a more appropriate paleoclimatic model should also include stochastic components.
We introduce the parameter τ to have a control over the time scale of the oscillations (it is needed, as the parameters α and β both affect the period of the unforced limit cycle). The parameter β controls the asymmetry of the glaciation/deglaciation sawtooth structure (a higher value of β leads to an enhanced asymmetry), because it controls the position of the fixedpoint on the slow manifold \(\Upphi'(y) = y^3 / 3 y = x, \) and, consequently, the ratio of times spent by the system in the two branches (’glacial’ and ’interglacial’) of the slow manifold. The coupled system Eqs. 4a, 4b has one stable equilibrium solution for \( { \beta > 1}\) and a stable periodic orbit for \( { \beta < 1}\). We use T _{ ULC } to denote the period of the stable periodic orbit and ω_{ ULC } = 2π/T _{ ULC } to denote the corresponding angular frequency.
The definition of synchronisation can be applied to our model Eqs. 4a, 4b as follows. The astronomical forcing F(t) corresponds to u(t), and the state vector whose two components are the slowlyvarying ice volume x and the faster variable y corresponds to v(t). For nonperiodic forcing, relationship (2) can be very complicated (nonfunctional or even fractallike) and hence difficult to detect. Therefore, other methods of detecting (2) had to be developed. As suggested by the auxiliary system approach (Abarbanel et al. 1996), relationships (2) and (3) are implied by an (invariant) attracting trajectory in the (x, y, t) phase space of the nonautonomous forced system (4a, 4b) (Wieczorek 2011). In the remainder of the paper, such an attracting trajectory is denoted with AT and referred to as an attracting climatic trajectory or synchronised solution. All other solutions to Eqs. 4a, 4b will be referred to as climatic trajectories.
Previous approaches to nonlinear dynamics of quasiperiodically forced oscillators focused on discretetime mappings and twofrequency forcing (Glendinning and Wiersig 1999; Osinga et al. 2000; Belogortsev 1992; Broer and Simó 1998). They uncovered interesting dynamics including Arnol’d or modelocked tongues consisting of ‘interlocking’ bubbles and open regions of multistability, nonsmooth bifurcations, and strange nonchaotic attractors (Grebogi et al. 1984; Feudel et al. 1995). Here, we consider quasiperiodic forcing with 35 frequency components and focus on the regions of mode locking. Our approach is based on instabilities of attracting trajectories in the (x, y, t) phase space of the continuoustime forced system because they relate directly to the concept of generalised synchronisation. We can provide a systematic study of generalised synchronisation to astronomical forcing by demonstrating existence of such trajectories and exploring their local and global stability properties. More specifically, we perform three types of calculations. Firstly, a clustering detection technique uncovers parameter regions with monostable (unique) and multistable (nonunique) synchronisation. Secondly, the largest Lyapunov exponent along AT quantifies its long and shortterm local (linear) stability. Thirdly, a basin of attraction of AT quantifies its global (nonlinear) stability. Finally, we remark that in the theory of nonautonomous dynamical systems, attracting trajectories in the (x, y, t) phase space are related to a modern and more general concept of a pullback attractor (Kloeden 2000; Langa et al. 2002; Kosmidis and Pakdaman 2003; Wiggins 2003).
3 Synchronisation of the paleoclimatic system to the insolation forcing
3.1 Illustration of the synchronisation phenomenon
However, if we consider now an external forcing (γ > 0) then synchronisation onto this forcing may occur under certain conditions (Ashkenazy 2006; Tziperman et al. 2006). According to our definition (2–3), synchronisation is represented by an attracting climatic trajectory in the (x, y, t) phase space.
Consider first the case of a purely periodic forcing with a period of T _{ F } = 41 kyr (main obliquity term) and strength γ = 3.33. The 70 initial conditions give rise to climatic trajectories that, after a sufficiently long integration time, converge to two attracting trajectories (see Fig. 5c). Both attracting trajectories are periodic with period of T _{ R } = 2T _{ F } = 82 kyr, and timeshifted versions of each other. This phenomenon is described in the literature as 2:1 phaselocking or frequencylocking. Generally speaking a n:m frequencylocking is defined as a fixedintime relation between the frequencies of the forcing (ω_{ F }) and the oscillator response (ω_{ R }) of the form n ω_{ R } = m ω_{ F }, where m and n are integers (Pikovsky et al. 2001, p. 52).
Then consider the case of the quasiperiodic insolation forcing described in Eq. 1 with γ = 0.75 and τ = 43.86. Figure 5e shows that the 70 climatic trajectories now converge onto three attracting trajectories, which reveals that synchronisation can be multistable (Pikovsky et al. 2001, p. 348), (Balanov et al. 2009, p. 94). This phenomenon is described in the literature as modelocking (Glass and Mackey 1988; Svensson and Coombes 2009). Note that because of the quasiperiodicity of the insolation forcing, these attracting trajectories are no longer periodic nor timeshifted versions of each other. The number of attracting trajectories depends on many factors including the dynamics of the unforced system, the nature of the forcing F(t), and the amplitude γ of the forcing. We will study this in more details in Sect. 4.
3.2 Detection of synchronisation by the way of the largest Lyapunov exponent (LLE or λ_{ max })
Local or linear stability of an attracting climatic trajectory can be quantified with the largest Lyapunov exponent (LLE) denoted here as λ_{ max } (Benettin et al. 1980). The quantity λ_{ max } is a measure of the (average) exponential rate of divergence (λ_{ max } > 0) or convergence (λ_{ max } < 0) of nearby climatic trajectories. Therefore, a negative value of λ_{ max } indicates a locally attracting climatic trajectory or generalised synchronisation (Pikovsky et al. 2001; Wieczorek 2009). A transition from λ_{ max } < 0 to λ_{ max } = 0 indicates a bifurcation where the attracting climatic trajectory disappears and generalised synchronisation is lost. Null and positive values of λ_{ max } indicate lack of synchrony (positive λ_{ max } indicates chaos but this regime is not encountered here). In the case of periodic forcing, computations of λ_{ max } can be easily validated with more precise and reliable numerical bifurcation continuation techniques (see § ’41kyr periodic forcing’ below).
3.3 Longterm λ_{ max } and shortterm λ _{ max } ^{ H } LLE’s
While λ_{ max } gives the average or longterm stability information, λ _{ max } ^{ H } can tell us about the behaviour of nearby trajectories within a short time interval H. For example, λ_{ max } < 0 does not necessarily imply λ _{ max } ^{ H } < 0 for some suitably chosen H. The definition (6) will be useful in studying the robustness of generalised synchronisation in Sect. 6. Technical details for computing λ_{ max } are given in “Appendix 2”.
3.4 Influence of the parameters γ and T _{ ULC }
3.5 41kyr periodic forcing (main obliquity term)
Figure 6a corresponds to the case of the 41kyr periodic forcing (T _{ F } = 41 kyr). The synchronisation region (λ_{ max } < 0) is composed of several Vshape regions, called Arnol’d tongues (phase or frequencylocking), originating at 1, 2, 3, etc. times the forcing period T _{ F }. These regions correspond to 1:1, 2:1, 3:1 frequencylocking zones (3:2 and 5:2 can also be guessed). Periodic solutions are found within these regions which originate generally speaking at \(T_{ULC} = (m/n) \; T_F. \) No synchronisation is possible when γ is zero but synchronisation may occur already for infinitesimally small γ. Then, for increasing γ, the synchronisation region widens and synchronisation becomes more stable up to an optimum value of the forcing. Above this optimum value, the synchronisation becomes less and less effective, because at large γ the system is too much steered away from its natural dynamics; it may even be driven into chaos at yet higher forcing amplitude (Mettin et al. 1993) but this case is beyond our focus.
In order to perform an accurate validation of the synchronisation region given by the LLE (λ_{ max } < 0) method, we computed the main Arnol’d tongues boundaries with the more accurate numerical continuation methods such as AUTO (Doedel et al. 2009). The case of periodic forcing \([F(t) = \sin(\omega t)]\) with β = 0 has been already extensively studied in the literature, analytically assuming some approximations (Guckenheimer and Holmes 1983, pp. 70–75), and using numerical algorithms for pseudoarc length continuation (Mettin et al. 1993). The usual approach extends the original nonautonomous system by additional differential equations for the forcing so that the system becomes autonomous, and then explores the (ω, γ) parameter space. In this way, we computed Arnol’d tongue boundaries as saddlenode of limit cycle bifurcations for the extended system with α = 11.11 and β = 0.25. Note that the asymmetry introduced here with the parameter β adds slightly more complexity and induces additional features to the diagrams documented in these papers.
Superposition of LLE calculations and bifurcation boundaries in Fig. 6a shows that the synchronisation regions obtained with the two different techniques match perfectly. This is a confirmation that the method based on the LLE works fine and we will be able to use it for the case of the quasiperiodic insolation forcing. Note that bifurcation boundaries are also drawn in Fig. 6c in order to stress the correspondence with yet another method of detecting synchronisation that will be discussed in Sect. 4.
3.6 Astronomical quasiperiodic forcing
For the case of the quasiperiodic insolation forcing (Fig. 6b), the region of synchronisation appears to be in one single piece with some indications of tongues (mode locking) at small γ. These tongues are in fact wellseparated, as it can be even more clearly seen on Fig. 6d which is of higher resolution.
In other words, whatever the value of the natural period T _{ ULC } of the paleoclimatic system, it has a higher probability of being synchronised onto the insolation forcing.
4 Non uniqueness: multistability and basins of attraction
The detection of synchronisation using the LLE (λ_{ max } < 0) gives only a Yes/Notype of information (i.e. synchronisation inside the tongues, no synchronisation outside), without making any distinction between different tongues as this would require information about multistability. For example, Fig. 6b indicates synchronisation for the parameter settings marked with the symbol ’×’ but gives no information about the corresponding number of attracting trajectories (we know that there are three different attracting trajectories in that case, from Fig. 5e). To explore the problem of multistable synchronisation, we propose a clustering method that not only allows us to detect synchronisation, but additionally provides information about the number of attracting trajectories denoted here with N.
4.1 Multistability analysis: numerical estimate of the number of attracting trajectories N by a clustering technique
Consider the case of the quasiperiodic insolation forcing with the three attracting trajectories, i.e., N = 3 (Fig. 5e). Although N can often be easily assessed visually, we want to automatically detect and count the number of ATs (attracting climatic trajectories). As a matter of fact, N can be easily estimated in the following way. Fix a time t that defines a twodimensional (x, y)section in the (x, y, t) phase space. Then start with a grid of initial conditions at some time t _{0} < t and take t − t _{0} sufficiently large so that all the initial conditions converge to the attracting trajectories at t. Since each AT is represented by a point on the (x, y)section, the problem of counting attracting trajectories reduces to a simple clustering problem. We designed a suitable automatic cluster detection algorithm that counts the number of clusters to obtain an estimate of N. For example, Fig. 5f shows^{6} the (x, y)section of the threedimensional (x, y, t) phase space at t = 550 kyr, given 70 initial conditions at t _{0} = 0. The 70 trajectories converge onto three (highly concentrated) clusters corresponding to the three attracting trajectories.
The idea of using cluster analysis for paleoclimatic dynamics comes from the natural fact that clustering is another way of looking at generalised synchronisation where negative LLE makes the trajectories cluster more efficiently. This provides another insightful viewpoint on the problem of identification of the number of synchronised solutions of the paleoclimatic system: the more stable the synchronisation, the more efficient the formation of clusters.

the notion of a cluster is based on the threshold distance ^{8} d _{ T } that has to be carefully chosen. If d _{ T } is chosen too large, there will be just one cluster including all points; if it is too small, no clusters will form with more than one point.

In order to have sufficiently well formed clusters, the time interval t − t _{0} must be chosen large enough so that the transient behaviour is gone; an illustration of the convergence is given in Fig. 7.
In our system, the convergence was fast and the clusters were highly concentrated and clearly separated for the great majority of the parameters. Therefore, defining a right threshold was quite easy and not critical. However, for small values of γ, the convergence is weaker, as can be seen in the bottom of the tongues on Fig. 6c.
Depending on the type and amplitude of the forcing γ, we can have potentially a whole range of possible numbers of attracting trajectories N, ranging from one (Tziperman et al. 2006), to a few (two in the 41kyr periodic forcing example in Fig. 5c, d, or three in the quasiperiodic insolation forcing example in Fig. 5e, f). When no forcing is considered (Fig. 5a, b), or there is forcing but no synchronisation occurs, we find no clusters at all. This means that there are as many points in the (x, y)section at time t as initial conditions at time t _{0}. Clearly, it is difficult to numerically distinguish between no synchronisation and a large number of attracting trajectories (N ≫ 1). Therefore, we restrict ourselves to just six different regions in Fig. 6, where we use white to indicate when there are none or more than five attracting trajectories.
Now, we apply the numerical cluster analysis in the case of the periodic forcing (Fig. 6c) and of the quasiperiodic forcing (Fig. 6d). We set t = 0 and consider a grid of 49 initial conditions covering \(x \in [2.2,2.2]\) and \(y \in [2.2,2.2]\) at the initial time t _{0} = −40 T _{ F } for the periodic forcing (T _{ F } is the period of the forcing), and t _{0} = −1,600 kyr for the astronomical forcing.
Two points in the (x, y)section are estimated to belong to a different cluster if their Euclidean distance is greater than 0.1.
4.2 41kyr periodic forcing (main obliquity term)
4.3 Astronomical quasiperiodic forcing
Figure 6d shows that synchronisation occurs for most parameter configurations. The region with one attracting trajectory (N = 1), corresponding to unique or monostable generalised synchronisation (Rulkov et al. 1995), is the largest. However, there are also parameter sets with N = 2, 3 or even more attracting trajectories. They indicate multistable generalised synchronisation where different possible stable relationships (2) between the forcing and the oscillator response coexist.
It is crucial to appreciate that synchronised solutions are not periodic and that, unlike in the periodic forcing case, different synchronised solutions for a given set of parameters are not timeshifted versions of each other. The idea that different synchronised solutions coexist is of practical relevance for paleoclimate theory. Namely, the set of parameters used to obtain the fit to the paleoclimatic records shown in Fig. 3 give two distinct solutions at t = 0 when started from a grid of initial conditions at t _{0} = −700 kyr. Sensitivity studies show that the choice of t − t _{0} is sometimes important for estimating correctly N. However, tests with t − t _{0} as large as 200 Myr of astronomical time suggest that several attracting trajectories may coexist at the asymptotic limit of \(t_0\rightarrow \infty. \)
4.4 Evolving geometry of the basins of attraction
Each \(AT_i\, (i=1 \ldots N)\) has its own basin of attraction ^{10} (Barnes and Grimshaw 1997), that is defined as the set of all initial conditions in the (x, y, t) phase space that converge to that AT _{ i } as time tends to infinity. For our nonautonomous system Eqs. 4a, 4b, we can study basins of attraction in the (x, y)section for different but fixed values of initial time t _{0}, and observe how they vary with t _{0}. A given initial condition at time t _{0} lies in the basin of attraction of AT _{ i } if it approaches AT _{ i } as time tends to infinity. The whole phase space can then be ’painted’ with several colours, each colour representing a specific basin. Technical details about the computation of the basins of attraction by use of the specific classification algorithm developed (see Fig. 18) are given in “Appendix 3”. Basins of attraction are of major importance because they provide the information about global or nonlinear stability of synchronisation. If we care about predictability, basin boundaries indicate when a change in the attracting climatic history is likely.
In the case of a periodic forcing (two basins), the pattern repeats itself periodically (compare the t _{0} = 0 kyr to the t _{0} = 40 kyr, and to the t _{0} = 80 kyr subfigures in Fig. 11). However, in the case of the quasiperiodic forcing (three basins), the pattern is much more intricate and seems not to repeat itself for the time horizon considered here, cf. Fig. 12.
The ratio between the area of a basin of attraction and the considered area of the phase space can be interpreted as a probability to converge to the corresponding attracting trajectory when starting from a randomly chosen initial condition. In the case of the periodic forcing, the two ATs are roughly equally likely for all t _{0} as could be guessed from Fig. 5c. However, this is not the case for the quasiperiodic forcing where the probability to reach the same attracting trajectory may vary significantly in time. For example, the yellow basin is rather small at t _{0} = 0 kyr but becomes much larger at a later time t _{0} = 90 kyr.
In the multistable regime, if an AT _{ i } happens to lie sufficiently close to its basin boundary, then small perturbations could make the climate jump to another (coexisting) AT _{ j≠i }, reducing predictability. This phenomenon is illustrated in Sect. 6.
5 Influence of the symmetrybreaking parameter β
First consider the 41kyr periodic forcing (Fig. 13a). To understand this figure, recall that the unforced oscillator (i.e., γ = 0) has a stable fixed point for \( { \beta > 1}\) and a stable limit cycle for \( { \beta < 1}\).
The system responds almost linearly to the forcing when \( { \beta}\) is sufficiently large. This explains regions of unique synchronisation (N = 1) where only one climate response is possible. The system becomes excitable when \( { \beta}\) is just slightly greater than one. If the forcing is large enough it will excite oscillations. In this case, N is equal to the number of initial conditions if synchronisation is lost, or to a smaller number if synchronisation occurs.
Consider now the interval −1 < β < 1. For this, keep in mind (1) that the period of the unforced oscillation T _{ ULC } varies by almost a factor of two within the range \( {0 <  \beta < 1}\), and (2) that synchronisation requires some relation between the period of the unforced oscillations and the forcing period. Consequently, synchronisation on the periodic forcing occurs only for fairly narrow ranges of β that are symmetric around zero. The figure reminds us of Arnol’d tongues. The main synchronisation regimes detected here correspond to 4:1, 3:1 and 5:2 frequencylocking. Outside these synchronisation regimes, the system fails to converge to a sufficiently small set of attracting trajectories, meaning that the forcing is not an efficient pacemaker.
Even if not obvious, Fig. 13a can be partly explained by considering Fig. 6c. The multistability plot can indeed be considered here as being 3dimensional in the parameter space {T _{ ULC }, γ, β}, but the intricate aspect is that Figs. 13a and 6c are not straight cuts into this 3dimensional space, as the relation between β and T _{ ULC } is not linear. For example, if you take the parameters of Fig. 13a, i.e. τ = 35.09, it corresponds to T _{ ULC } = 100 kyr; so, when performing the cut in Fig. 6c the yellow region will be reached in the upper part (pay attention that the yscale is different), what is consistent with what is found in Fig. 13a for low \( { \beta}\) values. Then, when increasing \( { \beta}\), T _{ ULC } increases, i.e. one moves a bit on the right on Fig. 6c, reaching then the red region, which is again consistent with what is found in Fig. 13a for higher \( { \beta}\) values. For \( {  \beta > 1}\), Fig. 6c is no longer relevant. To have a deeper understanding, much more views in the 3dimensional space would be required (e.g. Fig. 6c should be done for several values of β), but it is not the goal of this paper.
Finally, compare the 41kyr periodic forcing situation with that obtained with the astronomical forcing (Fig. 13b). Synchronisation now occurs in a larger area of the parameter space. Whereas the structure of the periodic forcing is preserved as long as the forcing amplitude is low enough, there is a much richer and more complex pattern of different N for larger γ. This pattern emerges from the interaction with different harmonics and their beatnotes.
Note that Fig. 13a, b have a very high level of symmetry with respect to β, which was expected as the system Eqs. 4a, 4b is invariant under the transformation \(\{x,y,\beta,F\} \rightarrow \{ x,y,\beta,F\}. \)
6 Robustness of synchronisation
Robustness or reliability of synchronisation can be studied in terms of two properties of an attracting climatic trajectory. Local stability analysis based on the shortterm LLE (λ _{ max } ^{ H } ) provides information about the shortterm local convergence towards the AT. For example, a temporary loss of local stability indicated by λ _{ max } ^{ H } > 0 will cause a temporary loss of synchrony and divergence from the AT even though the trajectory is stable on average (λ_{ max } < 0). Global stability analysis based on the geometry of the basins of attraction for different ATs provides information about the system’s response to external perturbations such as random fluctuations. For example, an external perturbation may push a climatic trajectory outside of its basin of attraction. Robustness and uniqueness of synchronisation become closely linked when there are coexisting attracting trajectories. Robustness is compromised most when a temporary loss of local stability coalesces with a weakening of the global stability. We will now briefly discuss these two effects that could restrict the prediction horizon for the evolution of climatic trajectories.
6.1 Temporary desynchronisation via loss of local stability
These results remain unchanged with respect to the most important parameters of the model. For example, our main conclusion about the stability remain qualitatively valid, even for different values of α (like α = 100), or with a different type of potential (\(\Upphi'_5(y) = (y+1.7)(y+1.58)(y+0.8)(y)(y0.5)\)), even if the shape and size of the limit cycle and the boundaries of the basins of attraction are of course different. The effect of the insolation function F(t) has also been checked: we compared the attracting trajectories for the insolation given by Eq. 1 to those for the insolation given by Laskar et al. (2004). As these insolation functions are very similar, the results are also very similar, and no difference was noticed.
At first glance, it may appear that these episodes of temporary divergence are not relevant to the robustness of synchronisation because climatic trajectories converge back to the attracting trajectory on a long term. However, other effects may be present that could strongly amplify such temporary divergence. They are identified below.
6.2 Sensitivity to perturbations: preliminary results
Consider again Fig. 12 showing (x, y)sections with coexisting attracting trajectories in the case of the quasiperiodic insolation, and their basins of attraction for different values of t _{0}. Suppose now that the system is subject to additive fluctuations (for example, these may represent volcanic eruptions). Under certain conditions, such external perturbations may cause a displacement of the trajectory to a different basin of attraction, causing a jump^{12} to another attracting trajectory.
Figure 15 indicates the exact location where the “synchronisation jumps” are likely to occur. At first glance, it seems that the most critical times are those corresponding to large positive values of x (i.e. to large ice volumes according to Fig. 3). More precisely, these critical times are actually ’just before’ the maximum ice volume, and the jump occurs then at the beginning of the deglaciation. This is consistent with the stochastic realisation shown in Fig. 16.
We conjecture that externally triggered jumps between coexisting attracting climatic trajectories are most likely when the temporary desynchronisation due to the loss of local stability coalesces with the weakening of the global stability due to the proximity to the basin boundary.
Note finally that a related result has been indicated by Paillard (2001, on Fig. 14), where the model has only one attracting trajectory, but this AT is very sensitive to changes in the model parameters. The end result is that very different trajectories for the ice volume evolution emerge from small parameter changes (in Paillard 2001, two possible trajectories depending on a change in the parameter i _{0} representing an insolation threshold).
7 Conclusions
Previous studies have shown that locking mechanisms could be found in the ice ages problem (Le Treut and Ghil 1983; Hyde and Peltier 1985; Paillard 1998; Gildor and Tziperman 2000), but most of the time, the conclusions rely on a few particular realisations of the models, without providing a global analysis of the synchronisation phenomenon, like the one provided in this study.
Also, despite some attempts, the actual type of synchronisation has not been clearly identified or demonstrated to date. For example, (Ashkenazy 2006; Tziperman et al. 2006) speak of “nonlinear phaselocking” although they do not define suitable ”phase variables” that can be used to demonstrate a fixedintime relationship between phases of the forcing and the oscillator response.
In this paper, we have for the first time identified, illustrated, and provided a systematic study of the phenomenon of generalised and multistable synchronisation between a simple conceptual model of the climatic glacial/interglacial oscillations and the astronomical forcing. A van der Poltype relaxation oscillator, designed to reproduce the slowfast dynamics of the paleoclimatic records, has been used for illustration purposes, but the methodology proposed may of course be applied to other paleoclimatic models.
The dynamical systems approach proposed herein (1) allows for stability analysis of such synchronisation, (2) uncovers interesting effects related to the robustness of the synchronisation with respect to external perturbations, and (3) uncovers the phenomenon of multistable synchronisation that has been overlooked by previous studies. We have shown that, in contrast to claims in Tziperman et al. (2006), synchronisation needs not be unique.
To study the uniqueness of synchronisation, we proposed a convenient concept of the number of attracting trajectories in the phase space of the nonautonomous forced system, each of which corresponds to a synchronised solution. We computed the number of synchronised solutions using a numerical clustering technique, and uncovered that in addition to a unique or monostable synchronisation, there are parameter settings where one finds a nonunique or multistable synchronisation. At low forcing amplitude we found regions of mode locking where the system synchronises on the individual components of the astronomical forcing in a way that is similar to frequencylocking on periodic forcing (Arnol’d tongues), giving rise to coexisting synchronised solutions. As the forcing amplitude is increased, the combined effects of precession and obliquity restrict the number of possible synchronised solutions. The emerging stability diagram consists of a large region of monostable synchronisation mixed with smaller regions of multistable synchronisation. A comparison with periodic forcing shows that the system finds it easier to synchronise to quasiperiodic insolation forcing. It is therefore conceivable that the climate system wandered throughout preferential synchronisation regimes on obliquity, precession, or combinations of both, all over the history of the Pleistocene.
The robustness of generalised synchronisation was investigated in terms of the key indicators of stability of synchronised solutions: the long and shortterm largest Lyapunov exponent (local stability), and the geometry of the basins of attraction (global stability). We found that even though the synchronised solutions are locally stable on a long term, there exist episodes where the shortterm largest Lyapunov exponent becomes positive, leading to temporary desynchronisation. As a result, climatic trajectories could diverge from the synchronised solution for some short period of time (it is shown here for 50 kyr). Moreover, we computed the evolving geometry of the basins of attraction for the coexisting synchronised solutions, and uncovered that these solutions sometimes approach the basin boundary where they become very susceptible to external perturbations. As a result, a small perturbation could make the climate jump from one synchronised solution to another, reducing predictability. We conjecture that such jumps are most likely when the temporary loss of the local stability coalesces with the proximity to the basin boundary. In this context, we briefly discussed the effect of stochastic perturbations on the timing of the iceages. We also illustrated the difference between the evolving geometry of the basins of attraction for periodic and quasiperiodic insolation forcing. In the case of the insolation forcing, we obtained an intricate pattern of basins of attraction that does not appear to repeat itself in time.
In this way, our results contribute to the emerging theory of predictability of ice ages. Future works will of course have to take into account the physical constraints like reproducing satisfactorily enough the socalled MidPleistocene Transition.
Footnotes
 1.
 2.
A quasiperiodic signal is the superposition of several periodic signals with incommensurate periods.
 3.
The van der Pol oscillator, or slightly different versions of it [a similar one is the Poincaré oscillator (Glass and Sun 1994)], has been mathematically largely studied under many aspects, most of them being related to features used in the present study: fixed points and Arnol’d tongues, basins of attraction (Barnes and Grimshaw 1997), analytical expressions for the amplitude and period of the limit cycle (D’Acunto 2006), slow manifold equation (Ginoux and Rossetto 2006), bifurcation structure (Mettin et al. 1993), chaotic dynamics (Chen and Chen 2008; Parlitz and Lauterborn 1987), additive noise (Degli Esposti Boschi et al. 2002), etc.
 4.
 5.
Even if differential versions of the LLE have sometimes been developed mainly for computational efficiency purposes, we however preferred within this article to stick on the original definition of the LLE, because it is more standard and there is no insistent need for lowering computation time in the present framework, as the number of degrees of freedom of the system is reduced.
 6.
See also Fig. 18 for a detailed view.
 7.
Cluster analysis or clustering is the assignment of a set of observations into subsets (called clusters) so that observations in the same cluster are similar in some sense. This is a common technique for statistical data analysis used in many fields for countless applications. There exists many types of clustering, along with several methods, among which: hierarchical, partitional, spectral, kernel PCA (principal component analysis), kmeans, cmeans and QT clustering algorithms.
 8.
 9.
This statement relies on the system invariance with respect to a timeshift of one forcing period (Tziperman et al. 2006 show a very nice illustration of this point).
 10.
 11.
 12.
In the periodic forcing case the phenomenon of jumping from one attracting trajectory to another in response to a perturbation is called a phase slip (Pikovsky et al. 2001, p. 238).
 13.
The proof of the existence of such a limit has been given by Oseledec (1968).
 14.
Notes
Acknowledgments
We are grateful to Guillaume Lenoir for his thorough review of several versions of the paper. The original idea of using cluster analysis for automatically identifying the number of stable locking states cropped up after presentations and discussions at the 458. WEHeraeusSeminar on ’SYNCLINE 2010: Synchronisation in Complex Networks’, held on 26–29 May 2010 at the Physikzentrum Bad Honnef (Germany), where some preliminary results of this research have been presented in a poster. The project is funded by the ERC (European Research Council) starting grant ITOP (’Integrated Theory and Observations of the Pleistocene’) under the convention ERCStG2009239604. M. Crucifix is Research Associate with the Belgian National Fund of Scientific Research, and B. De Saedeleer is Postdoctoral Research Assistant with the ITOP Project. Some figures and calculations have been made with the R language and the Intel Fortran Compiler.
Open Access
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