Why Could Ice Ages Be Unpredictable?

Home , Van der Pol oscillator

Clim. Past, 9, 2253–2267, 2013 Open Access www.clim-past.net/9/2253/2013/ Climate doi:10.5194/cp-9-2253-2013 © Author(s) 2013. CC Attribution 3.0 License. of the Past

Why could ice ages be unpredictable?

M. Cruciﬁx Earth and Life Institute, Georges Lemaître Centre for Earth and Climate Research Université catholique de Louvain, Louvain-la-Neuve, Belgium

Correspondence to: M. Cruciﬁx (michel.cruciﬁ[email protected])

Received: 4 February 2013 – Published in Clim. Past Discuss.: 20 February 2013 Revised: 30 June 2013 – Accepted: 28 August 2013 – Published: 9 October 2013

Abstract. It is commonly accepted that the variations of 1 Introduction Earth’s orbit and obliquity control the timing of Pleistocene glacial–interglacial cycles. Evidence comes from power spectrum analysis of palaeoclimate records and from inspec- Hays et al. (1976) showed that Southern Ocean climate ben- tion of the timing of glacial and deglacial transitions. How- thic records exhibit spectral peaks around 19, 23–24, 42 and ever, we do not know how tight this control is. Is it, for ex- 100 thousand years (thousand years are henceforth denoted ample, conceivable that random climatic fluctuations could “ka”). More or less concomitantly Berger (1977) showed, cause a delay in deglaciation, bad enough to skip a full pre- based on celestial mechanics, that the power spectrum of cli- cession or obliquity cycle and subsequently modify the se- matic precession was dominated by periods of 19, 22 and quence of ice ages? 24 ka, and that of obliquity was dominated by a period of To address this question, seven previously published con- 41 ka. These authors concluded that the succession of ice ceptual models of ice ages are analysed by reference to ages is somehow controlled by the astronomical forcing. the notion of generalised synchronisation. Insight is being However, experiments with numerical models have also gained by comparing the effects of the astronomical forcing suggested that the precise timing of the glaciation or with idealised forcings composed of only one or two peri- deglaciation could sensitively depend on parameters that are odic components. In general, the richness of the astronom- not well known. Weertman (1976), for example, showed that ical forcing allows for synchronisation over a wider range the natural course of the ice volume from the present-day (ig- of parameters, compared to periodic forcing. Hence, glacial noring anthropogenic forcing) could either be glacial incep- cycles may conceivably have remained paced by the astro- tion or a long interglacial, depending on whether a certain pa- nomical forcing throughout the Pleistocene. rameter is set to 2.75 or 2.745. Paillard (2001) observed sim- However, all the models examined here show regimes of ilar phenomena using a model published in Paillard (1998). strong structural dependence on parameters. This means that Shown in Fig. 1, for instance, are two further examples of small variations in parameters or random fluctuations may ice volume history reproduced with models previously pub- cause significant shifts in the succession of ice ages. Whether lished in the palaeoclimate modelling literature (Saltzman the actual system actually resides in such a regime depends and Maasch, 1990; Tziperman et al., 2006). In both cases, on the amplitude of the effects associated with the astro- small changes in model parameters do, at some stage in the nomical forcing, which significantly differ across the differ- climate history, induce a shift in the sequence of ice ages. ent models studied here. The possibility of synchronisation The question of the stability of the ice age sequence is on eccentricity is also discussed and it is shown that a high not new. As early as 1980, Imbrie and Imbrie (1980) won- Rayleigh number on eccentricity, as recently found in obser- dered whether “nonorbitally forced high-frequency fluctua- vations, is no guarantee of reliable synchronisation. tions may have caused the system to flip or flop in an unpredictable fashion.” They also noted that “the regularity of the 100-ka cycle, and particularly its phase coherence with the 100-ka eccentricity cycle, argue for predictability”. The importance of eccentricity as a control of ice ages was recently

Published by Copernicus Publications on behalf of the European Geosciences Union. 2254 M. Cruciﬁx: Why could ice ages be unpredictable?

Forcing (mid−June insolation at 65N) This work is further developed here in three directions: the role of separate and combined inﬂuences of different com- 2 ponents of the astronomical forcing is being analysed more 0

A.U. systematically. This allows us to relate the instabilities noted

−2 in palaeoclimate models to the literature on quasi-periodic forcings with two components only, and more specifically to −800 −600 −400 −200 0 the concept of strange non-chaotic attractor. The possibility time [ka] of synchronisation on eccentricity is also briefly discussed. Second, the effects of additive fluctuations and those of pa- Saltzman and Maasch 1990 rameter changes are being related qualitatively. Finally, the

1 analysis is applied to six other palaeoclimate models, which provides a better basis to estimate the robustness of the con-

−1 clusions. Providing no conclusions as to whether or not glacial– Ice volume (A. U.) Ice volume −800 −600 −400 −200 0 interglacial cycles are indeed predictable, the present work focuses entirely on the possibility of dynamical stability of time [ka] simple ice age models. Making conclusions about the real Tziperman et al., 2006 world requires ﬁrst an elaborate account of the effects associated with model discrepancy (which is stochastic) that need 40 to be quantiﬁed through a process of statistical inference. 20 0 2 The van der Pol oscillator −800 −600 −400 −200 0 Ice volume (1E15 m3) Ice volume

time [ka] 2.1 Model deﬁnition

Fig. 1. Ice volume simulated with two models previously published: Before going further a word of justification is needed about Saltzman and Maasch (1990) and Tziperman et al. (2006), forced the use of the van der Pol oscillator as the starting point of by normalised insolation at 65◦ N. The blue lines are obtained with this study. the published parameters; the latter were slightly changed to obtain The succession of glaciations and terminations is com- the red ones: p0 = 0.262 Sv instead of 0.260 Sv in Tziperman et al. monly understood as a phenomenon of relaxation, during (2006), and w = 0.6 instead of w = 0.5 in Saltzman and Maasch which regimes of linear response to the astronomical forc- (1990). While the qualitative aspect of the curves are preserved, the ing alternate with non-linear phenomena during which the timing of ice ages is affected by the parameter changes. attracting point of the system changes. The idea, popularised by Paillard (1998), is already im- plicit in the works of Oerlemans (1980), based on experiments with an ice sheet-lithosphere model. However, the un- brought up by Lisiecki (2010) and Rial (2004); Rial et al. derstanding of the nature of the instabilities that are neces- (2013). sary to cause the succession of glacial–interglacial cycles Tziperman et al. (2006) already observed that a conceptual has considerably evolved over time, as it may involve the model fitting the ice age curve is no proof that the correct lithosphere (Oerlemans, 1980), the ocean biogeochemistry physical mechanisms have been identified, but is rather an (Saltzman et al., 1984), a combination of both (Saltzman indication of the powerful role of the astronomical forcing as and Verbitsky, 1993), the ocean circulation (Paillard and Par- a pacemaker. Our purpose is to take this approach one step renin, 2004) or sea-ice-atmospheric interactions (Gildor and further by showing that the powerful action of the astronom- Tziperman, 2000). ical forcing as a pacemaker does not guarantee the stability The van der Pol model is thus chosen here as it is one of of the ice age sequence. the simplest relaxation oscillators, without consideration of A first step was proposed in De Saedeleer et al. (2013), the exact physical mechanism that causes the ice age oscilla- where a highly idealised model of ice ages (in essence, a van tions. der Pol oscillator) displays a wide regime of synchronisation It is a dynamical system of two ordinary differential equa- when forced to the astronomical forcing. Inspection of the tions: time-averaged largest Lyapunov exponents, as well as basins of attractions, allowed us to conclude that that the forced sys- dx 1 tem goes through times of unreliable synchronisation, so that = − (F (t) + β + y). dt τ (1) the succession of ice ages may sensitively depend on fluctu- dy α = (y − y3/3 + x), ations added to the system. dt τ

Clim. Past, 9, 2253–2267, 2013 www.clim-past.net/9/2253/2013/ M. Crucifix: Why could ice ages be unpredictable? 2255 with (x,y) as the climate state vector, τ a time constant, α Precession Obliquity a time-decoupling factor, β a bifurcation parameter and F (t) 1.0 1.0 the forcing. The parameter β does not appear in the original 0.8 0.8 van der Pol equations (van der Pol, 1926), and the present variant is sometimes referred to as the biased van der Pol 0.6 0.6 model. 0.4 0.4 The autonomous (i.e. F (t) = 0) model displays self- amplitude sustained oscillations as long as |β| < 1. For later reference 0.2 0.2 the period of the unforced oscillator is denoted Tn(τ). The 0.0 0.0 0 20 40 60 80 100 0 20 40 60 80 100 variable x then follows a saw-tooth periodic cycle, the asym- Period [ka] Period [ka] metry of which is controlled by β. We chose α = 30. This choice implies that variable y un- Fig. 2. Spectral decomposition of precession and obliquity given by dergoes rapid variations compared to x, and it may therefore Berger (1978), scaled here such that the strongest components have be referred to as the “fast” variable. In climate terms, x may amplitude 1. be interpreted as a glaciation index, which slowly accumulates the effects of the astronomical forcing F (t), while y, which shifts between approximately −1 and 1, might be in- components and it will be shown here that it is worth consid- terpreted as some representation of the ocean or carbon cycle ering the astronomical forcing with all its complexity. dynamics. Again, the van der Pol oscillator is only taken here as a 2.2 Periodic forcing starting point, its representativeness of the physics of ice ages being admittedly contentious. Models with more explicit in- Consider a sine-wave forcing (F (t) = γ sin(2π/P1t +φP 1)), ◦ terpretations of ice age physics will be discussed in Sect. 3. with period P1 = 23 716 yr and φP 1 = 32.01 . This is the In ice age models the forcing function is generally one or first component of the harmonic development of climatic pre- several insolation curves, computed for specific seasons and cession (Berger, 1978). If certain conditions are met – they latitudes. The rationale behind this choice is that whichever will soon be given – the van der Pol oscillator may become insolation is used it is, to a very good approximation, a lin- synchronised on the forcing. Synchronised means, in this par- ear combination of climatic precession and obliquity (Loutre, ticular context, that the response of the system displays p cy- 1993, see also Appendix A). The choice of one specific inso- cles within q forcing periods, where p and q are integers. It lation curve may be viewed as a modelling decision about the is said that the system is in a p : q synchronisation regime effective forcing phase of climatic precession, and the rela- (Pikovski et al., 2001, p. 66–67). The output is periodic, and tive amplitudes of the forcings due to precession and obliq- its period is equal to q × P1. uity. In turn, climatic precession and obliquity can be ex- There are several ways to identify the synchronisation in pressed as a sum of sines and cosines of various amplitudes the output of a dynamical system. One method is to plot the and frequencies (Berger, 1978), so that F (t) can be modelled state of the system at a given time t, and then superimpose on as a linear combination of about a dozen of dominant peri- that plot the state of the system at every time t +nP1, where n odic signals, plus a series of smaller amplitude components. is integer. The system is synchronised if only q distinct points They are shown in Fig. 2. More details are given in Sect. 2.4. appear on the graph, discarding transient effects associated With these hypotheses the van der Pol model may be tuned so with initial conditions. These correspond to the stable fixed that the benthic curve over the last 800 000 yr is reasonably points of the iteration bringing the system from t to t + qP1. well reproduced (Fig. 3). In the following we refer to this kind of plot as a “strobo- Abundant literature analyses the response of the van der scopic section” of period P1 (Fig. 4a). If the system is not Pol oscillator to a periodic forcing (e.g. Mettin et al., 1993; synchronised, then there are two options: the stroboscopic Guckenheimer et al., 2003, and references therein). The re- section is a closed curve (the response is quasi-periodic), or sponse of oscillators to the sum of two periodic forcings has a figure with a strange geometry (the response is a-periodic). been the focus of attention because it leads to the emergence There is another, equivalent way to identify synchronisa- of “strange non-chaotic attractors”, a concept first introduced tion. Suppose that the system is started from arbitrary ini- by Grebogi et al. (1984) and further studied in, among others, tial conditions. Then, plot the system state at a given time t, Wiggins (1987); Romeiras and Ott (1987); Kapitaniak and long enough after the initial conditions. Repeat the experi- Wojewoda (1990, 1993); Belogortsev (1992); Feudel et al. ment with another set of initial conditions, superimpose the (1997); and Glendinning et al. (2000). To our knowledge, result on the plot, and so on with a very large number of dif- however, there is no systematic study of the response of an ferent initial conditions. In doing so one constructs the sec- oscillator to a signal of the form of the astronomical forc- tion of the global pullback attractor at time t (henceforth re- ing. Le Treut and Ghil (1983), for example, represented the ferred to as the pullback section); the pullback attractor itself astronomical forcing as a sum of only two or three periodic being the continuation of this figure over all times (Fig. 4b). www.clim-past.net/9/2253/2013/ Clim. Past, 9, 2253–2267, 2013 2256 M. Crucifix: Why could ice ages be unpredictable?

van der Pol oscillator forced by astronomical forcing (a) y 1.0 ● ●● ●●● ● ●

● 5 ●● ● ● ●● ●●● ● ● ●● ●● ● ● ●●● ● ●●● ●● ● ●●● ●● ● ●● ●● ● ●●● ●● ● ●● ● ●●●● ●● ●● ● ● ● ● ●● ●●● ● t 0.5 ● ● ● ●● ● ● ● ● ● ●● ●● ●●● ●● ● ● ●● ● ● ● ● ●●● ●● ●●● ● ● ● ●● ● ● ● ●● ●●● ● ● ● ●● ●●●● ●● ● ● ● ●● ●● ● ●●● ● ●●● ● ●●●●● ●●●●● ● ● ● ● ●● ● ● ● ●●●● ● ● ●●● 4.5 ●● ● ● ● ●●● ●● ● ● ● ●●●●● ● ●●● P ● ● ● ●●● ●● ● ●●●● ●● ● ● ● ● ● ● ●● ● ● ● ●● ●● ● ●●●● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● x ● ● ●● ● ●●● ●● ● ● ● ● ●● ● ● ● ●● ● x ● ●●● ● ●● ● ● ● ● ● ●● ● ● 0.0 ●● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ●● ●● ● ●●● ● ● ●●● ● ● ● ● ● ● ●●●●● ●●● ● ● ●● ●●● ● ●●●● ● ● ● ● ●● ● ● ● ● ● 4 ●●● ● ● ●● ●●● ● ● ● ●● ●● (b) y ●● ● ● ● ●●●● ● ●● ● ●●●●●● ● t0 ● ● ●● ●● ●● ●● ●● ● ● ● ● ●● ● ● ● ● ●●● ●●● ● ●●● ●● ● ● ● ● ●●●● ● ● ● ●● ● ● ●●● ●●● ● LR04 delta−18 O ● ● ● ●● ● ● ●●● ●● ●● ● ● ● ●● ●● ●● ● ●● ● ●● ● ● ● ●●●

−0.5 ●● ●● ● ●● ● ● ● ●● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● 3.5 ● ● ● ● ● ●● ●● ● ● ● ● ●● ● ● ● ● ●● ●● ● ● ●● ● t

−1.0 ● ●

−700 −600 −500 −400 −300 −200 −100 0 tback x

time [ka] Fig. 4. (a) The stroboscopic section is obtained by superimposing Time [ka] the system state every forcing period (P ), here illustrated for a 2 : 1 Fig. 3. Simulation with the van der Pol oscillator (Eqs. 1–4), forced synchronisation. (b) The pullback section at time t0 is obtained by by astronomical forcing and compared with the benthic record superimposing the system states obtained by initialising the system of Lisiecki and Raymo (2005). Parameters are: α = 30, β = 0.75, with the ensemble of all possible initial conditions far back in time, γp = γo = 0.6, τ = 36.2 ka. here at tback. The particular example shows a global pullback attractor made of two local pullback attractors (in dashed red and blue), the sections of which are seen at t. Also shown is the convergence of initial conditions towards the pullback attractors, in grey. Each component of the global pullback attractor (two are illustrated in Fig. 4b) is a local pullback attractor. Rasmussen (2000, Chap. 2) reviews all the relevant mathematical formal- cause episodes of desynchronisation. In the particular case of ism. periodic forcing the episode of desynchronisation is called In the particular case of a periodic forcing, the strobo- a phase slip, as is well explained in Pikovski et al. (2001, scopic section and the pullback section are often identical p. 54) (Fig. 5)1. Using the pullback attractor to identify synchronisation is The number of points on the pullback section may then be not a very efficient method in the periodic forcing case. Arc- estimated for different combinations of parameters and we length continuation methods are faster and more accurate can use this as a criteria to detect synchronisation. This is (e.g. Schilder and Peckham, 2007, and references therein). It done in Fig. 6 for a range of γ and τ. It turns out that syn- is shown, however, in De Saedeleer et al. (2013) that the pull- chronisation regimes are organised in the form of triangles, back section method gives results that are acceptable enough known in the dynamical system literature as Arnol’d tongues for our purpose, and it is adopted here because it provides (Pikovski et al., 2001, p. 52). A p : q synchronisation regime a more intuitive starting point to characterise synchronisation appears when the ratio between the natural period (T ) and n with multi-periodic forcings. the forcing period is close to q/p. The tolerance, i.e. how distant this ratio can afford to be with respect to q/p, in- 2.3 Synchronisation on two periods creases with the forcing amplitude and decreases with p and q. Synchronisation is weakest (least reliable) near the edge of Consider now a forcing function that is the sum of two peri- the tongues. Unreliable synchronisation characterises a sys- odic signals. Two cases are considered here: the two forcing tem that is synchronised, but in which small fluctuations may periods differ by a factor of about 2, and the two forcing periods are close. 1This property derives from the system invariance with respect a time translation by P . There will be, however, cases where dif- 2.3.1 P = 23 716 ka and O = 41 000 ka ferent initial conditions will create different stroboscopic plots. For 1 1 example two 1 : 2 synchronisation regimes co-exist in the forced We adopt F (t) = γ [sin(2π/P t + φ ) + cos(2π/O t + van der Pol oscillator, so that there are four distinct local pullback 1 P 1 1 φO1)], with P1 = 23 716 ka, O1 = 41 000 ka and attractors, while only two points will appear on a stroboscopic plot ◦ ◦ started from a single set of initial conditions. Rigorously, the global φP 1 = 32.01 and φO1 = 251.09 . P1 is the first period pullback attractor at time t is identical to the global attractor of the in the development of precession, O1 is the first period in the iteration t +nP , and the global pullback sections at two times t and development of obliquity and φP 1, φO1 the corresponding t0 are homeomorphic. phases given by Berger (1978), so that F (t) may already be

Clim. Past, 9, 2253–2267, 2013 www.clim-past.net/9/2253/2013/ M. Cruciﬁx: Why could ice ages be unpredictable? 2257

Periodic forcing (P1 ) Periodic forcing (P1 ) 1 5 Pullback section Stroboscopic on P 1.4 1 2 6 2 1.2 3 7 4 > 7 1 1.0

y 0 γ 0.8 -1 0.6 -2

-1 0 1 -1 0 1 0.4 x x

0.2 Fig. 5. Pullback (at t0 = 0) and stroboscopic sections (t = 0 + nP1) obtained with the van der Pol oscillator, with parameters α = 30, β = 0.7, τ = 36 forced by F (t) = γ sin(2π/P1t + φP 1), P1 = 5 10 15 20 25 30 35 40 45 50 ◦ τ 23.7 ka, φP 1 = 32.01 and γ = 0.6. The two plots indicate a case of 4 : 1 synchronisation, and they are identical because the forcing 1 2 3 4 5 P /P is periodic. N 1 Fig. 6. Bifurcation diagram obtained by counting the number of points on the pullback section in the van der Pol oscillator (α = 30, viewed as a very rough representation of the astronomical β = 0.7) and F (t) = γ sin(2π/P1t +φP 1). The two x axes indicate forcing. τ and the ratio between the natural system period and the forcing, Let us begin with τ = 36 ka, which corresponds to a limit respectively. One observes the synchronisation regimes corresponding to 1 : 1, 1 : 2, 1 : 3, 1 : 4 and 1 : 5, respectively (gray, blue, red, cycle in the van der Pol oscillator of period Tn = 98.79 ka, and consider the stroboscopic section on P (Fig. 7, line 1). green, yellow) and, intertwined, higher order synchronisations in- 1 cluding 3 : 2, 5 : 2, 2 : 3 etc. White areas are weak or no synchro- Forcing amplitude γ is set to 0.6. Due to the presence of nisation. Graph constructed using tback = −10 Ma (see Fig. 11 and the O1 forcing, the four points of the periodic case seen in text for meaning and implications). Fig. 5 have mutated into four local attractors, which appear as closed curves (some are very flat). Every time P1 elapses, the system visits a different local attractor. They are attrac- (1984), its occurrence in the van der Pol oscillator is dis- tors in the sense that they attract solutions of the iteration, cussed in Kapitaniak and Wojewoda (1990), and its rele- bringing the system from t to t + 4 · P1. In this particular ex- vance to climate dynamics was suggested by Sonechkin and ample, the system is said to be phase- or frequency-locked on Ivachtchenko (2001). In our specific example, the system is : P1 with ratio 1 4 (Pikovski et al., 2001, p. 68), because on neither frequency-locked on P1 nor on O1, but it is synchro- average, one ice age cycle takes four precession cycles, even nised in the generalised sense: the pullback section has two though the response is no longer periodic. In this example, points. Finally, with τ = 44 ka there is frequency-locking on the curves on the P1 stroboscopic section nearly touch each O1 (regime 3 : 1) but not on P1. other. This implies that synchronisation is not reliable since Clearly, the system underwent changes in synchronisation a solution captured by one of these attractors could easily regimes as τ increased from 36 to 44 ka. Further insight may escape and fall into the basin of attraction of another local be had by considering the τ-γ plot (Fig. 8). The frequency- attractor. One can also see that it is not synchronised on O1 locking regime on P1 lies in the relic of the 1 : 4 tongue vis- since the stroboscopic section of period O1 shows one closed ible in the periodic forcing case (Fig. 6). Frequency locking curve englobing all possible phases. It may also be said that on O1 belongs to the 1 : 3 tongue associated with O1. The the system is synchronised in the generalised sense (Pikovski strange non-chaotic regime occurs where the tongues associ- et al., 2001, p. 150), because the pullback section is made of ated with these different forcing components merge. only four points: starting from arbitrary conditions, the sys- The word bifurcation has been defined for non- tem converges to only a small number of solutions at any autonomous dynamical systems (Rasmussen, 2000, Chap. 2). time t. It is also stable in the Lyapunov sense, a point that This is a complex subject and we will admit here the rather will not be further discussed here, but see De Saedeleer et al. informal notion that there is a bifurcation when a local pull- (2013). back attractor appears or ceases to exist (adapted from Def. Consider now τ = 41 ka. The four closed curves on the 2.42, in Rasmussen, 2000). With this definition, there is a bi- P1-stroboscopic section have collided and merged into one furcation at least every time color changes in Fig. 8 (assum- attractor with strange geometry. A similar figure appears ing tback is far enough in the past). on the E1-stroboscopic section. The phenomenon of strange non-chaotic attractor has been described by Grebogi et al. www.clim-past.net/9/2253/2013/ Clim. Past, 9, 2253–2267, 2013 2258 M. Crucifix: Why could ice ages be unpredictable?

Quasi-periodic forcing (P1 +E1 ) Quasi-periodic forcing (P1 and O1 )

Pullback section Stroboscopic on P1 Stroboscopic on O1 2 1.4 1

τ=36 ka 0 1.2

-1 1.0 -2

0.8 2 γ 0.6 1 τ=41 ka 0 0.4 -1

-2 0.2

0.0 2 10 15 20 25 30 35 40 45 50 τ 1 1 2 3 4 5 τ=44 ka 0 PN /P1 -1 0.5 1.0 1.5 2.0 2.5 3.0 -2 PN /O1 -1 0 1 -1 0 1 -1 0 1 Fig. 8. As Fig. 6 but with a 2-period forcing: F (t) = t = t = + nP Fig. 7. Pullback (at 0 0) and stroboscopic sections ( 0 1 γ (sin(2π/P t + φ ) + sin(2π/O t + φ )) (see text for values). t = + nO 1 P 1 1 O2 and 0 1) obtained with the van der Pol oscillator with pa- Tongues originating from frequency-locking on individual periods α = β = . F (t) = γ ( ( πt/P + rameters 30, 0 7, and forced by sin 2 1 merge and give rise to strange non-chaotic attractors. Orange dots φ ) + ( πt/O ) + φ ) P = . O = . P 1 sin 2 1 O1 , 1 23 7 ka and 1 41 0 ka and correspond to the cases shown in Fig. 7. γ = 0.6, and three different values of τ. The presence of dots on the pullback section indicates generalised synchronisation. Localised closed curves on the stroboscopic sections indicate frequency locking on the corresponding period (on P1 with τ = 36 ka and O1 pullback attractors of the system obtained with τ = 41 ka. with τ = 44 ka), and complex geometries indicate the presence of The middle panel displays one local attractor obtained with = a strange attractor (τ 41 ka). τ = 40. The two τ = 41 attractors are reproduced with thin lines for comparison. Observe that this τ = 40 attractor is qualitatively similar to the τ = 41 attractors, and most of its Another view on the bifurcation structure may be obtained time is spent on a path that is nearly undistinguishable from by plotting the x and y solutions of the system at t0 = 0, ini- those obtained with τ = 41. However, on a portion of the tiated from a grid of initial conditions at tback = −5 Ma (Mil- time interval displayed it follows a sequence of osciliations lion years), as a function of τ, still with γ = 0.6 (Fig. 9). This that is distinct from those obtained with τ = 41. In fact, there plot outlines a region of sensitive dependence on the input are four pullback attractors at τ = 40. parameter, more specifically between τ = 37 and 42 ka. Let us now consider a third scenario. Parameter τ = 41, dω 2 = −1 These observations have two important consequences for but an additive stochastic term (σ dt ,σ 0.25ka , and ω our understanding of the phenomena illustrated in Fig. 1. To symbolises a Wiener process) is added to the second system see this it is useful to refer to general considerations about equation. This is thus a slightly noisy version of the original autonomous dynamical systems. A bifurcation generally sep- system. Shown here is one realisation of this stochastic equa- arates two distinct (technically: non-homeomorphic) attraction, among the infinity of solutions that could be obtained tors, which control the asymptotic dynamics of the system. with this system. Expectedly, the system spends large frac- As the bifurcation is being approached, the convergence to tions of time near one or the other of the two pullback attrac- the attractor is slower, while the attractor that exists on the tors. However, it also spends a significant time on a distinct other side of the bifurcation may already take some tempo- path. Speculatively, this distinct path is under the influence rary control on the transient dynamics of the system. This is, of a “ghost” pullback attractor. As the bifurcation structure is namely, one possible mechanism of excitable systems. One complex and dense, as shown in Fig. 9, we expect a host of sometimes refers to “remnant” or “ghost attractors” to refer ghosts to lie around, ready to take control of the system over to these attractors that exist on the other side of the bifur- significant fractions of time, and this is what happens in this cation and may take control on the dynamics of the system particular case. over significant time intervals (e.g. Nayfeh and Balachan- To further support this hypothesis, consider a second ex- dran, 2004, p. 206). periment. Figure 11 displays the number of distinct solutions The idea may be generalised to non-autonomous sys- counted at time t0 = 0, when the system is started from 121 tems. Consider Fig. 10. The upper plot shows the two local distinct initial conditions at a time back tback, as a function of

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Pullback sections at t=0Ma pullback attractors τ =41 ka 1.0 2-periods forcing P1 +E1 1.0 0.5 0.0 y 0.5 0.5 1.0 0.0 1.5 0 200 400 600 800 1000 1200 1400 1600 1800 x 0.5 one pullback attractor τ =40 ka 1.0 1.0 0.5 0.0

1.5 y 34 36 38 40 42 44 46 0.5 1.0 3 1.5 0 200 400 600 800 1000 1200 1400 1600 1800 2

1 one stochastic realisation τ =41 ka 1.0

y 0 0.5 0.0 1 y 0.5 2 1.0 1.5 3 0 200 400 600 800 1000 1200 1400 1600 1800 34 36 38 40 42 44 46 time [kyr] τ [ka] Fig. 10. (Top) pullback attractors of the forced van der Pol oscil- Fig. 9. Pullback solutions of the van der Pol oscillator forced by two lator (β = 0.7, α = 30, γ = 0.6 with 2-period forcing) as in Fig. 8, periods (as in Figs. 7 and 8) as a function of τ. Forcing amplitude for τ = 41 ka. They are reproduced on the graphs below (very thin is γ = 0.6 and the other parameters are as on the previous figures. lines), overlain by (middle) one pullback attractor with same pa- Vertical lines indicate the cases shown in Fig. 7. rameters but τ = 40ka, and (bottom) one stochastic realisation of the stochastic van der Pol oscillator with τ = 41 ka. tback. Surprisingly, one needs to go back to −30 Ma to identify the true pullback attractor. Obviously 30 Ma is a very considered here is whether the correspondence between the long time compared to the Pleistocene and so this solution is period of ice age cycles and eccentricity is coincidental, or in practice no more relevant than the 4 or 8 solutions that can whether a phenomenon of synchronisation of climate on ec- be identified by only starting the system back in time 1 or centricity developed. 2 Ma ago. They may be interpreted as ghost pullback attrac- To address this question we need a marker of synchro- tors, and following the preceding discussion they are likely nisation on the precession beating. The Rayleigh number to be visited by a system forced by large enough random ex- has already been used to this end in palaeoclimate appli- ternal fluctuations. cations (Huybers and Wunsch, 2004; Lisiecki, 2010). Let Pb be the beating period, and Xi the system state snap- 2.3.2 P2 = 22 427 ka and P3 = 18 976 ka shot every t = t0 + iPb, the Rayleigh number R is defined P P as | Xi − X|/ |Xi − X|, where the overbar denotes an The two periods now being combined are the second and average. R is strictly equal to 1 when the solution is synchro- third components of precession, still according to Berger nised with a periodic forcing of period Pb, assuming no other (1978). These two periods were selected for two reasons. The source of fluctuations. As a reference, Lisiecki (2010) esti- first one is that the addition of the two periodic signal pro- mated 0.94 the Rayleigh number of a stacked benthic δ18O duces an interference beating with period 123 319 yr, not too signal with respect to eccentricity over the last million years. far away from the usual 100 ka cycle that characterises Late The bifurcation diagram showing the number of pullback Pleistocene climatic cycles. Second, the period of the beating solutions is displayed in Fig. 12. The frequency-locking is not close to an integer number of the two original periods tongues on P2 and P3 are easily identified at low forc- (this occurs, accidentally, when using P1 and P3). This was ing amplitude; as forcing amplitude increases the tongues important to be able to clearly distinguish a synchronisation merge and produce generalised synchronisation regimes. Re- to the beating from a higher resonance harmonic to either gions of synchronisation on Pb, identified as R > 0.95, are forcing components. hashed. They occur when the natural system period is close It is known from astronomical theory that the periodic- to Pb but narrow bands also appear near TN = Pb/2. Observe ity of eccentricity is mechanically related to the beatings of also that these synchronisation regimes are generally not the precession signal (Berger, 1978). The scientific question unique (several pullback attractors co-exist), and additional www.clim-past.net/9/2253/2013/ Clim. Past, 9, 2253–2267, 2013 2260 M. Crucifix: Why could ice ages be unpredictable?

# of distinct solutions at t =0 2-period forcing (P2 and P3 ) 0 PN /E3 0.2 0.4 0.6 0.8 1.0

8 1.2

1.0

0.8 γ 0.6

4 0.4

0.2

2 0.0 10 15 20 25 30 35 40 45 50 1 τ 1 2 3 4 5 6 P /P -1 0 1 2 N 2 10 10 10 10 2 3 4 5 6 7 t [Myr] PN /P3 − back Fig. 12. As Fig. 8 but with: F (t) = γ (sin(2π/P t + φ ) + Fig. 11. Number of distinct solutions simulated with the van der Pol 2 P 2 sin(2π/P t + φ )). Hashes indicate regions of Rayleigh number oscillator (β = 0.7, α = 30, τ = 41 ka and γ = 0.6 with 2-period 3 P 3 > 0.95 on the beating associated with P and P , the period of forcing as in Fig. 8, as a function of the time t at which 121 2 3 back which is called E in the Berger (1978) nomenclature (third com- distinct initial conditions are considered. The actual stable pullback 3 ponent of eccentricity). attracting set(s), in the rigorous mathematical sense, is (are) found for tback → −∞.

still used in many palaeoclimate applications. Compared to sensitivity experiments show that convergence is quite slow. a state-of-the-art solution such as La04 (Laskar et al., 2004), More specifically, the synchronisation diagram was com- the error on amplitude is between 0 and 25 %, and the error = − puted here using tback 10 Ma. With shorter backward on phase is generally much less than 20◦. time horizons, the number of remaining solutions identified The bifurcation diagram representing the number of pull- in the beating-synchronisation regime often exceeds 6 and back solutions as a function of forcing amplitude and τ is could not be seen on the graph, while the Rayleigh number shown in Fig. 13. We have taken γ = γp = γo. One recog- was still beyond 0.95. Hence, a high Rayleigh number is not nises the tongues originating from the individual components necessarily a good indicator of reliable synchronisation. merging gradually into a complex pattern. The number of attractors settles to 1 as the amplitude of the forcing is further 2.4 Full astronomical forcing increased. Let us call this the 1-pullback attractor regime. We The next step is to consider the full astronomical forcing, already know that synchronisation is generally not reliable in as the sum of standardised climatic precession (5) and the the region characterised by the complex and dense bifurca- deviation of obliquity with respect to its standard value (O): tion region, where more than one attractor exist. The remaining problem is to characterise the reliability of synchronisa- F (t) = γp5(t) + γoO(t), (2) tion in the 1-pullback attractor regime. The literature says little about systematic approaches to where quantify the reliability of generalised synchronisation with quasi-periodic forcings. To develop further the ideas devel- Np X oped in Sect. 2.3.1, one can plot pullback solutions at a cer- 5(t) = a sin(ω t + φ )/a (3) i pi pi 1 tain time t as a function of one or several parameters. This i=1 is done in Fig. 14. Here γ is kept constant (= 1.0) and τ is N Xo varied between 25 and 40. There is a brief episode of a 2- O = b cos(ω t + φ )/b . (4) i oi oi 1 solution regime between 31 and 32 ka. Within the 1-solution i=1 regime there is a number of abrupt transitions (at 26, 27, 34 The various coefficients are taken from Berger (1978). We and 37 ka). take Np = No = 34, so that the signal includes in total 68 Abrupt variations such as near τ = 26, 27, 34 and 37 ka harmonic components. With this choice the BER78 solu- may represent discontinuities in the pullback attractor. In one tion (Berger, 1978) is almost perfectly reproduced. BER78 is case at least (at τ = 34.46 . . . ), the discontinuous character

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Full astronomical forcing

1.4

1.2

1.0

γ 0.8

0.6

0.4

0.2

10 15 20 25 30 35 40 45 50 τ

Fig. 13. As Fig. 8 but with the full astronomical forcing, made of 34 precession and 34 obliquity periods (Eq. 2). could be verified up to machine precision. The abrupt variation at τ = 37.59. . . appears numerically as a continuous Fig. 14. As Fig. 9 but with the full astronomical forcing, with pa- variation of the attractor spread over an interval of 10−7 ka2. rameters as in Fig. 13 and γ = 1.0. Further work is required to understand the origin of these discontinuities, and in particular to understand to what extent thus a mathematical transcription of the hypothesis ac- they relate to the strange nature of the attractor. The relevant cording to which the origin 100 000 yr cycle is to be aspect in the present context is that these discontinuities in- found in the biological components of Earth’s climate. duce similar phenomena as those described in Sect. 2: the presence of ghost attractors and sensitive dependence to fluc- – SM91: this model is identical to SM90 except for a dif- tuations. Hence, being in the 1-pullback attractor regime does ference in the carbon cycle equation. not guarantee a reliable synchronisation on the astronomical forcing. One needs to be deep into that zone. – PP04: the Paillard–Parrenin model (Paillard and Par- renin, 2004) is also a 3-differential-equation system, featuring Northern Hemisphere ice volume, Antarctic 3 Other models ice area and carbon dioxide concentration. The carbon dioxide equation includes one non-linear term asso- We now consider 6 previously published models. Mathemat- ciated to a switch on/off of the Southern Ocean ven- ical details are given in the Appendix and the codes are avail- tilation. Astronomical forcing is injected linearly at able online at https://github.com/mcrucifix. three places in the model: in the ice-volume equation, – SM90: this is a model with three ordinary differen- in the carbon dioxide equation, and in the ocean ven- tial equations representing the dynamics of ice volume, tilation parameterization. The autonomous version of carbon dioxide concentration and deep-ocean temper- the model also features a limit cycle. As in SM90 and ature. The astronomical forcing is linearly introduced SM91 the non-linearity introduced in the carbon cycle in the ice volume equation, under the form of inso- equation plays a key role but the bifurcation structure lation at 65◦ N on the day of summer solstice. Only of this model differs from SM90 and SM91 (Crucifix, the carbon dioxide equation is non-linear, and this non- 2012). linearity induces the existence of a limit-cycle solution – T06: the Tziperman et al. (2006) model is a mathemat- – spontaneous glaciation and deglaciation – in the cor- ical idealisation of more complex versions previously responding autonomous system. The SM90 model is published by Gildor and Tziperman (2000). T06 fea- 2Note also that at least one discontinuity up to machine precision tures the concept of sea-ice switch, according to which could also be identified in the 1-pullback attractor regime when α = sea-ice growth in the Northern Hemisphere inhibits ac- 1, which suggests that the feature is not conditioned by the slow-fast cumulation of snow over the ice sheets, and vice versa. character of the system. Mathematically, T06 is presented as a hybrid model, www.clim-past.net/9/2253/2013/ Clim. Past, 9, 2253–2267, 2013 2262 M. Crucifix: Why could ice ages be unpredictable?

which is the combination of a differential equation in 0.75 which the astronomical forcing is introduced linearly 0.50

PP04 0.25 as a summer insolation forcing term, and a discrete 0.00 variable, which may be 0 or 1 to represent the absence 120 or presence of sea ice in the Northern Hemisphere. 80

PP12 40 0 – I11: the Imbrie et al. (2011) was introduced by its au- 45 thors as a “phase-space” model. It is a 2-D model, 30 T06 15 for which the equations were designed to distinguish 0 an “ice accumulation phase” and an “abrupt deglacia- 1.5 tion” phase, which is triggered when a threshold de- I11 0.0 ﬁned in the phase space is crossed. I11 was speciﬁcally 1.5 tuned to reproduce the phase-space characteristics of 2 the benthic oxygen isotopic dynamics. A particularity 1 S91 0 of this model is that the phasing and amplitude of the 1 forcing depend on the level of glaciation. 1.5 0.0 S90 – PP12: similar to Imbrie et al. (2011), the Parrenin 1.5 and Paillard (2012) model distinguishes accumula- 800 700 600 500 400 300 200 100 0 time (kyr) tion and deglaciation phases. Accumulation is a linear accumulation of insolation, without restoring force Fig. 15. Pullback attractors obtained for 6 models over the last (hence similar to Eq. (1) of the van der Pol oscilla- 800 ka, forced by astronomical forcing with the parameters of the tor); deglaciation accumulates insolation forcing but original publications. Shown is the model component representing a negative relaxation towards deglaciation is added. ice volume. Units are arbitrary in all models, except in T06 (ice Contrary to Imbrie et al. (2011), the trigger function, volume in 1015 m3) and PP12 (sea-level equivalent, in m). which determines the regime change, is mainly a function of astronomical parameters. An ice volume term only appears in the function controlling the shift from In all generality, the equation (or its numerical approxima- “accumulation” to “deglaciation” regime. tion) may be rewritten as follows, posing τ = 1 and γ = 1:

The ice volumes (or, equivalently, glaciation index or ti+1 = ti + δt sea level) simulated by each of these models are shown in 1 Fig. 15. Shown here are estimates of the pullback attractors; x + = x + δt f (x ,γ F (t )). i 1 i τ i i more specifically, the trajectories obtained with an ensemble of initial conditions at tback = −20 Ma. In some cases The parameters τ and γ introduced this way have a similar the curves actually published (in particular in SM90) are not meaning as in the van der Pol oscillator, since τ controls the pullback attractors, but ghost trajectories in the sense illus- characteristic response time of the model, while γ controls trated in Fig. 11. In some cases (PP04 and PP12) the param- the forcing amplitude. eters had to be slightly adjusted to reproduce the published Bifurcation diagrams, similar to Fig. 13 are then shown in version satisfactorily. Details are given in Appendix. Fig. 16. Remember that γ = τ = 1 corresponds to the model Some of these models include as much as 14 adjustable as originally published. parameters (e.g.: PP04) and a full dynamical investigation of A first group of four models appears (SM90, SM91, each of them is beyond the scope of the study. Rather, we T06 and PP04), on which one recognises a similar tongue- proceed as follows. Every model responds to a state equa- synchronisation structure as in the van der Pol oscillator. tion, which may be written, in general (assuming a numerical This was expected since these models are also oscillators implementation), as with additive astronomical forcing. Depending on parameter choices synchronisation may be reliable or not. Synchro- ti+1 = ti + δt nisation is clearly not reliable in the standard parameters xi+1 = xi + δt f (xi,F (ti)), used for SM90 and SM91. In T06 the standard parameters are not far away from the complex multi-pullback-attractor where ti is the discretised time, xi the climate state (a 2-D or regime and this explains why transitions such as displayed 3-D vector) at ti and F (ti) is the astronomical forcing, which in Fig. 1 could be obtained under small parameter changes is specific to each model because the different authors made (or, equivalently, with some noise, as discussed in the orig- different choices about the respective weights and phases of inal Tziperman et al., 2006 study). The standard parameter precession and obliquity. choice of PP04 is further into the stability zone and, indeed,

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Saltzman - Maasch 1990 Saltzman - Maasch 1991 Paillard - Parrenin 2004 Imbrie et al. 2011 Parrenin and Paillard, 2012 (*)

1.2 1.2 1.2 0.14 1.4 1.0 1.0 1.0 0.12 1.2 γ 0.8 γ 0.8 γ 0.8 0.6 0.6 0.6 0.10 1.0 0.4 0.4 0.4 γ 0.08 γ 0.8 0.2 0.2 0.2 0.06 0.6 0.4 0.6 0.8 1.0 1.2 1.4 0.4 0.6 0.8 1.0 1.2 1.4 0.4 0.6 0.8 1.0 1.2 1.4 τ τ τ 0.04 0.4 Tziperman et al. 2006 Parrenin and Paillard, 2012 Imbrie et al. 2011 0.02 0.2 1.2 1.2 1.2 0.00 0.0 1.0 1.0 1.0 0.6 0.8 1.0 1.2 1.4 0.6 0.8 1.0 1.2 1.4 γ 0.8 γ 0.8 γ 0.8 τ τ 0.6 0.6 0.6 0.4 0.4 0.4 Fig. 17. As Fig. 16 for (left) the I11 model, but with a zoom on the x 0.2 0.2 0.2 axis, and (right) for a slightly modiﬁed version of the PP12 model, 0.4 0.6 0.8 1.0 1.2 1.4 0.4 0.6 0.8 1.0 1.2 1.4 0.4 0.6 0.8 1.0 1.2 1.4 τ τ τ in which the transition between deglacial and glacial states is also controlled by ice volume, as opposed to the original PP12 model. Fig. 16. As Fig. 13 but for 6 models previously published. Orange dots correspond to standard (published) parameters of these models.

cyclic dynamics in the presence of a forcing. Once the forc- experimenting with this model shows that instabilities such ing is present, the two scenarios will result in qualitatively as displayed in Fig. 1 are harder to obtain. similar dynamics (see Crucifix, 2012, for more discussion on I11, with standard parameters, is also fairly deep in the this point). stable zone. One has to consider much smaller forcing val- In this study we paid attention to the dynamical aspects ues than published to recognise the synchronisation tongue that may affect the stability of the ice age sequence and its structure that characterises oscillators (Fig. 17). predictability. First, the astronomical forcing has a rich har- PP12 finally turns out to be the only case not showing monic structure. We showed that a system like the van der Pol a tongue-like structure. This may be surprising because this oscillator is more likely to be synchronised on the astronom- model also has limit-cycle dynamics (self-sustained oscilla- ical forcing (as nature provides it) than on a periodic forcing, tions in the absence of astronomical forcing). Some aspects because the fraction of the parameter space corresponding to of its design resemble the van der Pol oscillator. The role of synchronisation is larger in the former case. A synchronised the variable y in the van der Pol oscillator is here played by system is Lyapunov stable, so that at face value this would the mode, which may either be g (glaciation) or d (deglacia- imply that the sequence of ice ages is stable. However, – this tion). Also similar to the van der Pol, the direct effect of the is the second point – even if the dynamical structure of the astronomical forcing on the ice volume (x in the van der Pol; Pleistocene climate was correctly identified, there would be v in PP04) is additive. T06 has also similar characteristics. at least two sources of uncertainty : random fluctuations as- The distinctive feature of PP12 is that the transition between sociated with the chaotic atmosphere and ocean and other deglaciation and glaciation modes is controlled by the astro- statistically random forcings like volcanoes; and uncertainty nomical forcing and not by the system state, as in the other on system parameters. In theory both types of uncertainties models. To assess the importance of this element of design point to different mathematical concepts: path-wise stabil- we considered a hacked version of PP12, where the d → g ity to random fluctuations in the first case, and the structural occurs when for v < v1 (cf. Appendix B6 for model details). stability in the second. In practice, however, lack of either In this case the tongue synchronisation structure is recovered, form of stability will result in similar consequences: quan- with standard parameters marginally in the reliable synchro- tum skips of insolation cycles in the succession of ice ages. nisation regime (Fig. 17). This was the lesson of Fig. 10. It was shown here that, compared to periodic forcing, the richness of the harmonic structure of astronomical forcing 4 Conclusions favours situations of weak structural stability. The interpre- tation of the phenomenon relates directly to the theory of The present article is built around the paradigm of the “pace- quasi-periodically forced dynamical systems, and the insta- maker”, that is, the timing of ice ages arises as a combination bility develops under relatively weak forcing. It is to be dis- of climate’s internal dynamics with the variations of incom- tinguished from the chaos known to develop at high forcing ing solar radiation induced by the variations of our planet’s amplitude, and which has been implied in El-Niño theories orbit and obliquity. Mathematically, this implies that the Tziperman et al. (1994). models of ice ages tested here either present self-sustained Out of the seven models tested here, we ignore which one oscillations in absence of astronomical forcing, or that the best captures ice ages dynamics. The overwhelming com- structure of the vector field is organised such as to excite plexity of the climate system does not allow us to securely www.clim-past.net/9/2253/2013/ Clim. Past, 9, 2253–2267, 2013 2264 M. Crucifix: Why could ice ages be unpredictable? select the most plausible model on the sole basis of our B2 SM91 model knowledge of physics, biology and chemistry. Consequently, while we have understood here how and why the sequence dx of ice ages could be unstable in spite of available evidence = −x − y − v z − uF (t) dt (astronomical spectral signature; Rayleigh number), estimat- dy ing the stability of the sequence ice ages and quantifying our = −pz + r y − s y2 − y3 t ability to predict ice ages is also a problem of statistical in- d dz ference: calibrating and selecting stochastic dynamical sys- = −q(x + z) tems based on both theory and observations, which are sparse dt and characterised by chronological uncertainties. A conclu- p = 1.0, q = 2.5, r = 1.3, s = 0.6, u = 0.6 and v = 0.2, and sive demonstration of our ability to reach this objective is still F (t) is (here) insolation on the day of summer solstice, at awaited. 65◦ N, normalised (results are qualitatively insensitive to the exact choice of insolation). One time unit is 10 ka.

Appendix A B3 PP04 model

Insolation The three model variables are V (Ice volume), A (Antarctic Ice Area) and C (Carbon dioxide concentration): In the following models, the forcing is computed as a sum of precession (5 = e sin$/a ), co-precession (5˜ = dV 1 1 = − − + − = − ( xC yF1(t) z V) e cos$/a1) and obliquity (O (ε ε0)/b1) computed ac- dt τV cording to the Berger (1978) decomposition (Fig. 2, and dA 1 Eqs. 3–4). More precisely, we use here these quantities scaled = (V − A) dt τA ˜ (P , 5 and O) such as they have unit variance. All insolation dC 1 quantities used in climate models may be approximated as = (αF1(t) − βV + γ H + δ − C), dt τC a linear combination of 5, 5˜ and O. For example: τV = 15 ka, τC = 5 ka, τA = 12 ka, x = 1.3, y = 0.4 (was – normalised summer solstice insolation at 65◦ N = 0.5 in the original paper), z = 0.8, α = 0.15, β = 0.5, γ = 0.89495 + 0.4346O 0.5, δ = 0.4, a = 0.4, b = 0.7, c = 0.01, d = 0.27; H = 1 if ◦ aV −bA+d −c F2(t) < 0, and H = 0 otherwise. F1(t) is the – normalised insolation at 60 S on the 21 February = normalised, summer-solstice insolation at 65◦ N, and F (t) is ˜ 2 −0.49425 + 0.83995 + 0.2262O. insolation at 60◦ S on the 21 February (taken as 330◦ of true solar longitude). Other quantities (V , A, C) have arbitrary units. Appendix B B4 T06 model Model deﬁnitions The two model variables are x (ice volume) and y (sea-ice B1 SM90 model area). x is expressed in units of 1015 m3. dx dx = − − − + = −x − y − v z − uF (t) (p0 K x)(1 αsi) (s smF (t)) dt dt dy = −pz + r y + s z3 − w y z − z2 y The equation represents the net ice balance, as accumula- dt tion minus ablation, and αsi is the sea-ice albedo. αsi = dz . y y x × 6 3 = −q (x + z) 0 46 . switches from 0 to 1 when exceeds 45 10 km , dt and switches from 1 to 0 when x decreases below 3 × 106 km3. The parameters given by Tziperman et al. (2006) p = 1.0, q = 2.5, r = 0.9, s = 1.0, u = 0.6, v = 0.2 and w = are: p = 0.23 Sv, K = 0.7/(40) ka, s = 0.23 Sv and s = 0.5, and F (t) is (here) insolation on the day of summer sol- 0 m 0.03 Sv, where 1 Sv = 106 m3 s−1. F (t) is the normalised, stice, at 65◦ N, normalised (results are qualitatively insensi- summer-solstice insolation at 65◦ N. tive to the exact choice of insolation).

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B5 I11 model Ice volume v, expressed in sea-level equivalent, responds to the following equation: Define first: ( dv a − v/τ if state is d ( = − ∗ − ˜ ∗ − ˜ + d π 10 − 25y where y < 0, a55 a5˜ 5 aO O φ = · (B1) dt ag if state is g, 180 10 elsewhere. ( −1 0.135 + 0.07y where y < 0, with the following parameter values: a5 = 1.456 mka , θ = a = 0.387 mka−1, a = 1.137 mka−1, a = 0.978 mka−1, . . 5˜ O g 0 135 elsewhere −1 ad = −0.747 mka , τ = 0.834 ka, k5 = 14.635 m, = = = k5˜ 2.281 m, kO 23.5162 m, v0 122.918 m and a = 0.07 + 0.015y v1 = 3.1031 m. This parameter set is the one originally presented by Parrenin and Paillard in Climate of the Past h = ( . − . y)O O 0 05 0 005 Discussion (which differs from the final version in Climate h5 = a5sinφ of the Past), except that kO is 18.5162 m in the original paper. This modification was needed to reproduce the exact h = a5˜ cosφ 5˜ sequence of terminations shown by the authors. Subtle = + + F h5 h5˜ hO details, such as the numerical scheme or the choice of the (F + 0.28) astronomical solution might explain the difference. d = −1 + 3 0.51 All codes and scripts are available from GitHub at https: //github.com/mcrucifix. With these definitions:

 ˙2 Addendum y¨ = 0.5[0.0625(d − y) − y ] where y˙ > θ,  d−y y˙ = −0.036375 + y[0.014416 + y(0.001121 (B2) The recently published article by Mitsui and Aihara (2013)   −0.008264y)] + 0.5F elsewhere, further discusses this topic, by providing further support to the existence of strange non-chaotic attractors in van der Pol ˙ = dy ¨ = dy˙ where y dt and y dt . Note that the polynomial on the model as well as SM90, SM91 and PP04. They discuss also right-hand side of the equation for y˙ is a continuous ﬁt to the the question of sensitive dependence to ﬂuctuations and their piece-wise function used in the original Imbrie et al. (2011) connection with the occurrence of strange non-chaotic attrac- publication. Time units are here ka. tors, with references to the works of Khovanov et al. (2000) in addition to those cited here. B6 PP12 model

This is a hybrid dynamical system, with ice volume v (ex- Acknowledgements. Thanks are due to Peter Ditlevsen (Niels pressed in m of equivalent sea level) and state, which may be Bohr Institute, Copenhague), Frédéric Parrenin (Laboratoire de g (glaciation) or d (deglaciation). Glaciologie et de Géophysique, Grenoble), Bernard De Saedeleer, Define first Ilya Ermakov and Guillaume Lenoir (Université catholique de ( √ Louvain) for comments on an earlier version of this manuscript. x + 4a2 + x2 − 2a wherex > 0, Two anonymous reviewers as well as Takahito Mitsui (FIRST, f (x) := x elsewhere. Aihara Innovative Mathematical Modelling Project, and University of Tokyo) provided useful comments that improved the Climate of with a = 0.68034. Then define the following quantities, stan- the Past Discussion version. Thanks are also due to the numerous benevolent developers involved in the R, numpy and matplotlib dardised as follows: projects, without which this research would have taken far more time. MC is research associate with the Belgian National Fund of 5∗ = (f (5) − 0.148)/0.808 Scientific Research. This research is a contribution to the ITOP 5˜ ∗ = (f (5)˜ − 0.148)/0.808 project, ERC-StG grant 239604.

Edited by: J. Chappellaz = + ˜ + the threshold θ k55 k5˜ 5 kO O, and ﬁnally the following rule controlling the transition between state g and d: ( d → g if θ < v 1 . g → d if θ + v < v0

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