The Tipping Resonance

Rate-induced tipping occurs when a system’s parameters change too fast for it to track the moving equilibrium. Push slowly — it adapts. Push fast — it tips. The threshold is a critical rate, and the standard picture is monotonic: faster forcing means more tipping.

Quinn and Alkhayuon (arXiv:2603.15043) show the relationship isn’t monotonic. In an ice age dynamics model with chaotic forcing, there exists an optimal timescale for the forcing that maximizes the probability of tipping. Too slow, and the system tracks the moving attractor. Too fast, and the forcing averages out before the system can respond. At the resonant timescale, the chaotic variations couple most effectively to the system’s intrinsic instabilities.

The model exhibits bistability — between equilibria, and between equilibria and periodic orbits. The chaotic forcing pushes parameters through these bistable regions. Rate-induced tipping happens not because the parameters leave the bistable regime, but because they traverse it at the wrong speed. The chaotic signal acts as a rate parameter, and the timescale ratio between forcing and system response determines whether tipping occurs.

The resonance has a clean explanation through basin instability theory. At intermediate timescales, trajectories spend enough time near the basin boundary to be deflected, but not so long that they explore the basin thoroughly enough to find stable regions. The Lyapunov exponents of the forced system peak at the resonant timescale — maximum sensitivity to initial conditions coincides with maximum tipping probability.

The structural lesson: chaotic forcing doesn’t just add noise to tipping dynamics. It creates a resonance structure where the most dangerous forcing isn’t the strongest or the fastest, but the one whose timescale matches the system’s vulnerability window. For ice ages — and by extension, for any bistable system driven by chaotic external inputs — the question isn’t just how hard you push, but how fast.

Quinn & Alkhayuon, “Tipping resonance in a chaotically forced ice age model,” arXiv:2603.15043 (2026).