The benthic δ18O stack record (LR04) is used throughout this study as empirical ground truth. It should be noted that the frequency and the strength of each spectral peak can in principle be affected by the orbital tuning of the record; a conservative tuning strategy is taken for the LR04 δ18O record. Investigating the chronology of the record is however beyond the scope of this work. We assume that the frequency structure of the LR04 record is basically correct including the observed dominance of 95 kyr periodicity over the last 1 Myr (Fig. 1i). The 95 kyr spectral peak is indeed observed in both orbitally tuned and untuned δ18O records, although it is slightly subdued in untuned records . The conclusions are derived from model simulations that are not subject to possible circular reasoning due to the orbital tuning.

Our study is based on an Earth system model of intermediate complexity (EMIC) called CLIMBER-2 . It couples atmosphere, ocean, vegetation, global carbon and dust models and a three-dimensional thermomechanical ice sheet model . This is the most comprehensive EMIC that, thanks to its exceptional simulation speed, still allows the analysis required here to be performed, with a large number of million-year-scale simulations. The glaciogenic dust is one of the key feedback mechanisms in CLIMBER-2 . It is assumed to be sourced from the terrestrial sediments exported from the margins of ice sheets. Its deposition reduces ice albedo and enhances ablation. In CLIMBER-2, some glacial terminations are only possible with the glaciogenic dust feedback . The importance of the dust loading for complete terminations has also been proposed earlier by . The CLIMBER-2 version used in this study is the same as used by , which is slightly different from the earlier version used by with respect to the dust cycle scheme, the permafrost model and the regolith mask. Although the differences between these versions appear to be small, the underlying dynamics in the absence of forcing are different. We come back to this point in the discussion.

simulated the glacial cycles over the last 3 Myr, assuming scenarios about a long-term reduction in the volcanic-CO2 outgassing rate and gradual changes in ice sheet substratum from regolith to hard-friction crystalline bedrock by glacial erosion (Fig. 1e), which we call the background condition (BC). Older BCs are characterized by a higher volcanic-CO2 outgassing rate and wider area of regolith (the continents are assumed to be fully covered by regolith at 3 Myr BP). The present-day BC consists of the volcanic outgassing rate of 5.3 Tmol C yr-1 and the distribution of regolith cover based on present-day observations, in which large parts of North America and Scandinavia are characterized by exposed crystalline bedrock. These temporal changes in BC underlie the simulated MPT accompanying the dynamical change from 41 to ∼ 100 kyr glacial cycles . The simulated δ18O (Fig. 1d, magenta) under the optimal scenario (Fig. 1e) is shown over the last 2 Myr.

While we basically follow the previous model settings , we perform here 1 Myr scale simulations with temporarily fixed BCs, which are values taken from the optimal scenario at a specific time (Fig. 1e). Fixed BCs are not optimal for simulating observations faithfully but may make the interpretation of results easier. All simulations have been initialized using the same initial state, corresponding to an interglacial state obtained from a transient simulation of the last four glacial cycles, but with an ice-free Greenland . However, the model runs were started from different points in time between 1.1 and 1.2 Myr BP for simulations over 1–0 Myr BP and between 2.1 and 2.2 Myr BP for simulations over 2–1 Myr BP, and thus from different initial astronomical configurations. The initial 100–200 kyr data are removed from power spectral analyses (Appendix C).

In order to understand the effects of different astronomical parameters, we conduct a series of sensitivity experiments. In each, CLIMBER-2 runs for a fixed astronomical configuration or under a hypothetical astronomical forcing. In the latter case, the amplitudes of eccentricity or obliquity variations are scaled up or down.

We first simulate glacial cycles under the true astronomical forcing , which serves as a reference simulation for further experiments.

Under fixed background conditions (BCs) of volcanic-CO2 outgassing rate and regolith cover at their 1.6 Myr BP values, which we assume representative of the BC over the period from 2 to 1 Myr BP, CLIMBER-2 simulates δ18O series similar to the observed record (Fig. b). The dominant spectral power at the 41 kyr obliquity band is reproduced (Fig. b). Simulated powers at precession bands are stronger than in the record but still minor.

With BC fixed at their present-day values, the model exhibits glacial cycles with strong ∼ 100 kyr periodicity over the period from 1 Myr BP to present (Fig. d). Simulated glacial cycles depend on the initial times at which the simulations are started; there are time epochs in which trajectories starting from different initial conditions get close to each other, while the trajectories diverge in some other epochs. That is, the simulated glacial cycles are partially synchronized by the astronomical forcing. This type of temporal instability can appear generically in dynamical systems driven by quasiperiodic forcing like the astronomical forcing . Accordingly the PSD also depends on initial times. Nevertheless, a large fraction of spectral power is attracted by the periodicities of the eccentricity at 95, 124, and 405 kyr (Fig. d).

The 95 kyr power tends to be strongest statistically, although it is weaker than that of the observed record (Fig. S2 for enlargement). On the other hand, simulated 405 and 124 kyr powers are stronger than those in the record. These discrepancies are partly due to the fact that, in the present experimental setting, the model fails to simulate the deglaciation around ∼ 430 kyr BP. Indeed CLIMBER-2 is able to produce a stronger ∼ 100 kyr power and substantially weaker 405 kyr power if the simulation is started from an interglacial level at 410 kyr BP (Fig. S3). Also the 95 kyr spectral peak could potentially be accentuated in proxy records by the orbital tuning . Overall, we note that both the 41 kyr power in the pre-MPT experiment and the 95 kyr power in the post-MPT experiment are reproduced well, given the complexity of the model.

Previous work demonstrated that the MPT simulated in CLIMBER-2 cannot be produced by changes in the astronomical forcing alone . Indeed, CLIMBER-2 simulates 41 kyr cycles under the 1 Myr-to-present astronomical forcing if the BC at 1.6 Myr BP is used, and it simulates ∼ 100 kyr cycles under the 2 Myr-to-1 Myr astronomical forcing if the present-day BC is used (Fig. S4). This provides further evidence that changes in the internal dynamics of the Earth system are necessary to explain the MPT in CLIMBER-2.

The internal dynamics are investigated with unforced simulations for fixed orbital configurations. The configuration with zero eccentricity e=0 and mean obliquity ε=23.34∘ effectively gives an average seasonal insolation change (Fig. S5). This is reasonable since any insolation curve for a season and a latitude is well approximated by a linear combination of the obliquity ε, climatic precession esin⁡ω and co-precession ecos⁡ω . For this fixed orbital configuration, CLIMBER-2 exhibits self-sustained oscillations with timescales dependent on the BCs. Quasi-regular self-sustained oscillations with periodicity around 50 kyr arise for BC fixed at their values of 1.6 Myr BP (Fig. a). Less regular self-sustained oscillations with a timescale of a few hundred kiloyears arise for present-day BC (Fig. c). Overall, the timescale of self-sustained oscillations gradually increases when moving from ∼ 1.5 to ∼ 0.5 Myr BP BC (Fig. ). This increase in the internal timescale is accompanied by an increase in the amplitude of the oscillations (i.e., intensification of glacials). However in these unforced simulations for e=0 and ε=23.34∘, the sea level variations are limited to ∼ 80 m (Fig. ), which is smaller than that of the forced case (∼ 120 m). Similar lengthening of internal timescales also occurs for the present-day orbital configuration (Fig. S6) and for the orbital configuration at the Last Glacial Maximum (21 kyr BP) (Fig. S7). In those cases, much larger (∼ 250 m) and much longer (half-million-year scale) oscillations are observed for the post-MPT BCs, mainly due to the carbon cycle feedback (see below).

The self-sustained oscillations of CLIMBER-2 found here are generated by various feedback processes described in previous studies . The glaciogenic dust feedback and the carbon cycle feedback are especially crucial. Indeed, if the atmospheric CO2 concentration is fixed to a constant value and if the glaciogenic dust feedback is switched off, the spontaneous oscillations cease, or, even if some fluctuations remain, their amplitudes are reduced (Fig. S8).

In the unforced simulation for the BC at their 1.6 Myr BP values, e=0 and ε=23.34∘ (Fig. ), ice sheets nucleate and then grow facilitated by the ice albedo feedback. However, once the sea level reaches around -30 m, the ice sheets rapidly shrink due to an abrupt increase in glaciogenic dust deposition over the Northern Hemisphere ice sheets (Fig. b), which reduces the ice albedo and enhances ablation . While the glaciogenic dust emissions are low during glaciation periods, high glaciogenic dust emissions continue throughout the deglaciation since the terrestrial sediments, which are eroded and transported to the margins of the ice sheets by basal ice sliding if the ice sheet base is at melting point, are exposed to the air when the ice sheets retreat. In the unforced simulation with 1.6 Myr BP BC, the period of self-sustained oscillations (∼ 50 kyr) is primarily set by ice sheet dynamics and the glaciogenic dust feedback. The carbon cycle feedback slightly modifies the oscillation amplitude but does not affect the oscillation period significantly. Indeed in the unforced simulations with fixed atmospheric CO2 concentrations over 180–300 ppm and the 1.6 Myr BP BC (Fig. , left), the period of self-sustained oscillations stably remains in a narrow range between 45 and 60 kyr.

In the unforced simulation with the present-day BC, e=0 and ε=23.34∘ (Fig. ), the carbon cycle feedbacks play a crucial role in setting the timescale and the amplitude of spontaneous oscillations. Starting from the sea level of zero, the ice sheets nucleate and grow in conjunction with a decrease in the atmospheric CO2 (Fig. a and d). As the sea level decreases below -50 m, enhanced dust deposition in the Southern Ocean induces a further rapid reduction in CO2 via the iron fertilization effect, which causes a further ice volume increase (Fig. a and b). However, then a sudden increase in glaciogenic dust deposition over the Northern Hemisphere ice sheets interrupts their growths by reducing albedo and enhancing ablation (Fig. a and c). These counteracting effects keep the sea level around -80 to -50 m for 100–200 kyr. During the glacial state, the atmospheric CO2 increases on average, very gradually with a rate on the order of ∼0.1 ppm kyr-1 (Fig. d), due to the following feedbacks: the volcanic outgassing rate slightly increases on average, responding to sea level rise (Fig. e, red) (cf. ), while on the other hand, the terrestrial silicate weathering consuming atmospheric CO2 weakens during glacials (Fig. e, black). Also the carbonate (alkalinity) pump is strengthened in the direction of releasing CO2 during glacials on average as well as during terminations, as shown by the increase in CaCO3 burial (Fig. f). Finally the ice sheets retreat with the help of the glaciogenic dust deposition and the rapid atmospheric CO2 rise, which is sustained during the deglaciation. In sum, the few-hundred-kiloyear self-sustained oscillations for the present-day BC emerge from the interaction of ice sheet dynamics, glaciogenic dust and carbon cycle feedbacks. Note that self-sustained oscillations arise even without the glaciogenic feedback if the carbon cycle feedback is active, but the maximum ice volume becomes much larger and the periodicity much longer (Fig. S9).

There is a weak warming trend superimposed on the self-sustained oscillations (Fig. ) due to a long-term CO2 increase. This reflects long transient dynamics, where the carbon fluxes are still slightly imbalanced. We suppose that the system will eventually achieve a more regular limit cycle behavior without a drift. However, it takes at least more than 1 million years. Therefore, we consider that the oscillatory behavior with the subtle drift is an essential character underlying the modeled ice age cycles.

The period of self-sustained oscillations for the present-day BC (∼ 250 kyr on average) is decomposed into two parts: the periods for the buildup and collapse of ice sheets and those for glacials and interglacials. The joint period for the buildup and collapse of ice sheets is 80–90 kyr and relatively stable (Fig. ), which is roughly estimated as the maximum size of ice volume divided by the difference between accumulation and ablation rate. On the other hand, the duration of glacials ranges between 100 and 250 kyr depending on the glacial atmospheric CO2 concentration (compare the first two glacial periods with the last two in Fig. 6). A similar effect of CO2 is also confirmed in the simulations with fixed atmospheric CO2 concentration, where the glacial ice sheets are more stable (that is, the glacial duration becomes longer) if the prescribed CO2 concentration is lower than a critical level of ∼ 220 ppm (Fig. , right). In sum, the few-hundred-kiloyear periodicity for the present-day BC is determined jointly by the timescale of ice sheet evolution and the length of glacials and interglacials, which is sensitive to the atmospheric CO2 concentration.

It remains to be explained how the change in the internal timescale leads to the observed frequency change across the MPT. If we compare the spectra of forced simulations with those of corresponding unforced ones (Fig. ), we find that the spectral powers of forced simulations are entrained at one or few astronomical frequencies near the frequency of internal self-sustained oscillations. The theory of synchronization may provide a general explanation for these observations. In essence, frequency entrainment (i.e., synchronization) tends to occur near the internal frequency as long as the external forcing is moderate (Fig. S1 and Appendix A).

Consistent with synchronization theory, the oscillations before the MPT are entrained by the 41 kyr obliquity cycles, due to the proximity of the internal periodicity (around 50 kyr) to the 41 kyr obliquity periodicity (Fig. a and b). The oscillations after the MPT are entrained by the eccentricity periodicities at 95, 124 and 405 kyr because the several-hundred-kiloyear timescales of internal oscillations for post-MPT BCs are closer to those eccentricity periodicities than to the much smaller obliquity periodicity (Fig. c and d). Reflecting temporarily desynchronized epochs in the simulated glacial cycles after the MPT, the magnitudes of the 95, 124 and 405 kyr band powers depend on the realizations of simulated sequences. If the ensemble average is taken for the spectra (from 27 simulations), the 95 kyr band power is the strongest (Fig. S2b). For some realizations, a noticeable peak appears between 124 and 95 kyr (Fig. S2). It might be linked with the 107 kyr peak that arises as a higher-order combination tone of 95 and 405 kyr eccentricity periodicities (1/107≃1/95-1/(2×405)) (, and Appendix A), but note that the 107 kyr peak is still not well established since it is so close to 95 and 124 kyr peaks.

The frequency entrainment at the eccentricity frequencies does not result from the eccentricity forcing itself but results from the climatic precession forcing esin⁡ω (ecos⁡ω), whose amplitude is modulated by the eccentricity cycles. As shown in Fig.  in Appendix B, the deglaciations occur near peaks of climatic precession (i.e., boreal summer insolation peaks) in rising or high phases of eccentricity .

Given that the eccentricity has the strongest power at 405 kyr over the last 1 Myr, it is unclear why – and actually rather surprising that – the internal oscillations with a timescale of a few hundred kiloyears are entrained most strongly by the 95 kyr periodicity and not by 405 kyr. In what follows we show that the 41 kyr obliquity variations play a crucial role in synchronizing the climate system to the 95 kyr rather than the 405 kyr periodicity, via a new nonlinear phenomenon that we term vibration-enhanced synchronization.

First, the scale A of eccentricity–climatic precession forcing is changed while keeping the real obliquity forcing. The time series and spectra for several values of A are shown in Fig. S10; the summary plots are in Fig. a and b. As long as the scale A of eccentricity–climatic precession forcing is moderate (A≲1.5), the 41 kyr power simply dominates; for A≳1.5 the mean precession-band power surpasses the 41 kyr band power (Fig. a and b).

Second, we change the scale B of obliquity variations while keeping the actual eccentricity-precession forcing (see Fig. S11 for the time series and spectra for several values of B). If the obliquity forcing is absent or very weak (B≲0.3), the sequences of simulated glacial cycles starting from different initial conditions are not fully synchronized to each other and have dominant powers at the precession bands (Fig. S11a). However, if the obliquity forcing is strong enough (B≳0.5), the sequences of simulated glacial cycles starting from different initial conditions are synchronized well to each other and to the 41 kyr obliquity cycles (Fig. S11). The 41 kyr band power increases rather rapidly over 0.5≲B≲1.2, exceeding the average precession-band power (Fig. d). This nonlinear increase in the 41 kyr power with B can be partly explained as a synchronization. However, the amplitude of simulated glacial-cycle oscillations moderately increases as the forcing amplitudes A and B increase (Figs. S10 and S11). This is the aspect of linear response.

First, the eccentricity–climatic precession is scaled while keeping the actual obliquity forcing. The δ18O series and spectra for several values of A are plotted in Fig. , and the summary plots are given in Fig. a and b. In the absence of eccentricity–climatic precession change (A=0), the obliquity forcing alone cannot constrain the sequence of glacial cycles, and the PSDs are not substantially different from those of internal oscillations, having the largest powers in between the 400 and 100 kyr bands (Fig. a). If the scale A of eccentricity is increased up to around A=1, the oscillations are roughly synchronized to the 95 kyr periodicity, which is strongest statistically, although also the 124 and 405 kyr bands may receive a noticeable fraction of the spectral power, depending on initial conditions (Figs. c and S2 for an enlarged version). For A>1.8, the 405 kyr band receives the maximum strength (Fig. a and b). For the extreme case A=2.5, huge ice sheets appear near every 400 kyr eccentricity minimum (Fig. e). The system achieves a synchronized state with prominent ∼ 100 kyr periodicity for a realistic scale of the eccentricity–climatic precession forcing (A≈1), and the dynamics shift toward a nonlinear resonance mode with the 405 kyr eccentricity cycles for much larger A. It is worth mentioning that the relative strength of the 41 kyr power is largest near A=1 (Fig. b).

Second, the scale B of obliquity variations is varied while we keep the actual eccentricity–climatic precession forcing (Figs. , c and d). In the absence of obliquity forcing (B=0), glacial–interglacial cycles are likely to occur when the eccentricity is large, giving rise to a roughly 400 kyr periodicity (Fig. a); the 95 kyr mode still exists, while it is weaker than the 405 kyr mode. As B increases, the 405 kyr band power is suppressed, and the 41 kyr band power increases. Statistically, the 95 kyr power becomes strongest in the range of 0.7≲B≲1.2 (Fig. c and d). This is nontrivial because the dominance of 95 kyr periodicity is enabled by the 41 kyr obliquity forcing, which is directly related to neither 95 kyr nor 405 kyr periodicity. We call this novel nonlinear mechanism vibration-enhanced synchronization (see discussion below). While the enhancement of the ∼ 100 kyr power by the 41 kyr obliquity forcing is consistent with previous modelling studies , we have further shown that ∼ 100 kyr cycles become dominant only for a limited amplitude range of obliquity variations (Fig. b).