Source
Farrell & Ioannou, “Generalized Stability Theory. Part II: Nonautonomous Operators” (1996), LIT-929, read in full in NOTE-842: §2 (eqs. 1–7), §4 (eqs. 10–26), §5 (eqs. 27–37), Figs. 6–9.
What was actually shown
For du/dt = A(t)u, stability is the sign of the first Lyapunov exponent λ = lim sup ln‖Φ(t)‖/t (eq. 6).
- Non-normality is necessary (eq. 12). Since d‖u‖²/dt = u†(A + A†)u, λ lies between the long-run averages of the smallest and largest eigenvalues of (A(t) + A†(t))/2. For a normal operator the largest is the spectral abscissa, so an operator that is normal and stable at every instant has a negative exponent. A stable operator can be destabilized by time variation only if it is non-normal.
- It happens (§4).
- An undamped oscillator whose frequency switches between 0.5 and 3 every T units, neutral at every instant, has a positive exponent except near integer multiples of half a period (Fig. 6). The authors state that this persists with damping, when the operator is stable at each instant, without showing it.
- A 2 × 2 operator with eigenvalues ±i, neutral at every instant and randomly rotated at each step, has a positive exponent for every angle between its eigenvectors except π/2, increasing with the non-normality (Fig. 7).
- The mechanism: “transient growth instigated at the starting time is continued with further growth when the switch to the second operator takes place, forestalling the decay that would occur in the autonomous case.”
- Random variation destabilizes generically (§§4–5). For A(t) = Ā + εΣξ_i(t)B_i with Stratonovich white noise, the paper cites Has’minskii and Colonius & Kliemann for a positive exponent at sufficient ε, in dimension ≥ 2, when the B_i are neither skew-symmetric nor commute with Ā. If they commute, one basis diagonalizes both and noise cannot destabilize.
- Non-normality lowers the threshold; damping raises it (Figs. 8–9). For a damped oscillator with noisy restoring force, the exponent falls with damping and rises with noise. In a three-layer baroclinic model the noise needed falls as the shear, which sets the non-normality, rises. At typical shear a positive exponent appears for wind fluctuations of about 10%, and reaches about 1/5 day⁻¹ at 30%.
What could have come out otherwise: the examples and the numerical dependence on shear and damping. Necessity is a bound.
What this does not say
- It does not say any time variation destabilizes. In both deterministic examples the exponent vanishes as the switching interval goes to zero (T → 0; δt → 0 in eq. 26). Variation much faster than the transient growth averages out. Only under white noise scaled as δt^{−1/2} does the fast limit stay unstable. Which case a real system resembles depends on how its variation is correlated in time.
- It does not prove genericity itself. That rests on cited theorems not read here, and on low-dimensional numerical examples.
- It does not say damping is irrelevant. Damping “of sufficient magnitude is able to asymptotically stabilize the Lyapunov exponents, even with high temporal variability” (§5).
- It does not settle mean-square behaviour. The stability meant is sample stability, an almost surely negative exponent. For stochastic systems the second-moment exponent can be unstable when the sample exponent is not (μ₂ ≥ 2λ only).
- It does not settle where the atmosphere’s error growth comes from. That long-range forecast error comes from the flow’s time dependence rather than intrinsic chaos is the paper’s suggestion, from a three-layer model.
Connections
No relations are declared.
- THEORY-323. The autonomous account, where non-normality gives transient growth only. This one says time variation can turn that transient growth into asymptotic instability.
- CLAIM-186. Its note says an argument under development is the nonautonomous case and so has “no fixed modes to appeal to”. This account goes further, and against the note’s comfort: a structure that is stable at every snapshot need not settle if it keeps changing and is non-normal. Read across, it also names what would restore settling: a normal structure, enough damping, or, in the deterministic examples, change much faster than the transient growth. Which of these anchoring corresponds to is the reading’s question, not the paper’s.
Note of 2026-10-19: the deterministic half filed apart
Necessity and the switched examples are now THEORY-724, Active. The necessity bound is stated there with the uniform condition α(t) ≤ −c < 0, which it needs: with α(t) < 0 tending to 0, eq. 12 gives only λ ≤ 0. The case stable at every instant is CLAIM-376. Shifting by −δI lowers the exponent by exactly δ, and on the switched oscillator at T = 1 it gives λ = 0.0955 with every instantaneous operator stable. What keeps this THEORY Proposed is only the stochastic half: genericity under noise, which rests on cited theorems, and the threshold’s dependence on non-normality and damping, which is shown numerically.