Fig. 1: The accuracy of our theory increases with the ratio of the interaction strengths of the first layer σ₁ to that of the second layer σ₂.

The relative error in the estimate of the largest Lyapunov exponent \({{{\Lambda }}}_{\max }\) of the perturbed system decreases from a maximum of approximately 40% in region 6 (\({}^{{\sigma }_{1}}\,{/}_{{\sigma }_{2}}\approx 0.0917\)) to a minimum of approximately 0.7% in region 3 (\({}^{{\sigma }_{1}}\,{/}_{{\sigma }_{2}}\approx 3.5\)). Within each region, the error increases sublinearly with the normalized dynamical distance. Each point is averaged over 1000 realizations; error bars are smaller than the symbol size. Inset: schematic illustration of the six regions (adapted from del Genio et al.²⁷). Layer 1 is individually stable only when σ₁ is greater than a critical value (red striped regions); layer 2 is individually stable only when σ₂ lies between two critical values (blue striped regions); region 1 is the only zone of the phase diagram where both layers are already individually stable.