Is the astronomical forcing a reliable and unique pacemaker for climate? A conceptual model study

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signal involving different frequencies, the traditional concepts used to define synchronisation to periodic forcing are no longer applicable.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 frequency-locking 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). (…)

💡 Research Summary

The paper investigates how astronomical forcing—specifically, variations in summer insolation at 65° N—synchronizes the glacial‑interglacial cycles of the Pleistocene. Using a highly simplified climate model that is a modified van der Pol relaxation oscillator, the authors explore the nature of this synchronization when the forcing is a quasiperiodic signal composed of many incommensurate frequencies (precession, obliquity, and longer terms). Traditional concepts of phase‑ or frequency‑locking, which apply to strictly periodic drivers, are inadequate for such a complex driver. Instead, the authors adopt the framework of generalized synchronization, in which a functional relationship exists between the state of the forcing and the state of the oscillator, possibly non‑unique.

The model consists of two coupled differential equations for a slow variable x (ice volume) and a fast variable y (representing a rapid component such as Atlantic overturning). The equations contain three key parameters: α (time‑scale ratio), β (asymmetry controlling the relative duration of glaciation versus deglaciation), and γ (strength of the astronomical forcing). When |β| < 1 the system possesses a stable limit cycle; when |β| > 1 it settles to a fixed point. The forcing F(t) is built from 35 sinusoidal terms derived from Berger’s insolation series, reproducing the dominant 19‑, 23‑, and 41‑kyr components.

The authors first verify that for a simple periodic driver the system exhibits classic phase locking and a single stable synchronized solution, as confirmed by a negative largest Lyapunov exponent (LLE). When the full quasiperiodic astronomical forcing is applied, the behavior depends critically on the forcing amplitude γ. For low γ the system displays multistable generalized synchronization: several distinct attracting trajectories coexist, each corresponding to a different paleoclimate history. The basin of attraction for each trajectory is mapped by sampling thousands of initial conditions and clustering the long‑term outcomes. The number of clusters provides a quantitative measure of how many synchronized solutions exist.

As γ increases, the basins merge and the system converges to a monostable regime: only one synchronized solution remains, and the LLE stays negative for all times, indicating robust local stability. However, even in this regime the authors detect short intervals (≈ 50 kyr) where the LLE becomes temporarily positive, signalling temporary desynchronization. During these episodes nearby trajectories diverge before re‑converging, a phenomenon the authors attribute to the trajectory approaching the boundary of its basin of attraction.

Global stability is further examined by tracking how the basins evolve over time. Near basin boundaries, even modest external perturbations (e.g., stochastic noise, volcanic forcing) can push the system from one synchronized attractor to another, producing abrupt shifts in the timing of glacial terminations. This mechanism offers a dynamical explanation for the irregularities observed in marine δ¹⁸O stacks and ice‑core CO₂ records.

The role of the asymmetry parameter β is also explored. Larger |β| enhances the asymmetry between slow glaciation and rapid deglaciation, lengthening the desynchronization intervals and making the basins more irregular. Conversely, β ≈ 0 yields a more symmetric oscillator with tighter basins and fewer desynchronization events.

In summary, the study demonstrates that astronomical forcing does not act as a unique, immutable pacemaker. Instead, depending on forcing strength and internal model parameters, the climate system can settle into multiple coexisting synchronized states, each with its own basin of attraction. The presence of temporary desynchronization and sensitivity to small perturbations implies that the predictability of glacial cycles is limited, especially when the system operates in the multistable regime. The methodology—combining Lyapunov analysis, basin mapping, and clustering—provides a powerful toolkit for assessing synchronization and stability in other non‑autonomous climate models.