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LIT-929: Generalized Stability Theory. Part II: Nonautonomous Operators

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Properties6
StatusActive — read 2026-10-14 (NOTE-842) in full; worth reading for its main result: a time-dependent linear operator that is stable, or neutral, at every instant can be asymptotically unstable, only if it is non-normal, and random variation of the operator does this generically, with less variation needed the more non-normal and less damped the mean operator. Read it knowing that genericity is cited to stochastic stability theory rather than proved, that its examples are low-dimensional, and that in its deterministic examples variation much faster than the transient growth destabilizes nothing. Filed from it: THEORY-325.
Version2
Tagscomplex-systems, mathematics, natural-sciences
SupportsCLAIM-186 — How fast an argument grows sets how it grows: drift tracks how fast new material arrives relative to how fast it is anchored, so a fast, prolific exchange branches, as a crystal grown far from equilibrium does, and a project memory stabilizes an argument by keeping anchoring up with arrival, CLAIM-376 — A linear system that is stable at every instant can have a positive Lyapunov exponent: shifting any operator with exponent λ > 0 by −δI lowers its exponent by exactly δ and makes every instantaneous operator stable, so for 0 < δ < λ the shifted system is stable at every instant and unstable
Cited as “paper” byNOTE-842 — Farrell & Ioannou — Generalized Stability Theory II
Cited as “source” byTHEORY-325 — Random variation of a linear operator destabilizes it generically, needing less variation the more non-normal and less damped the mean operator, THEORY-724 — For a linear system du/dt = A(t)u, non-normality is necessary for time dependence to destabilize an operator that is uniformly stable at every instant, and a non-normal operator neutral at every instant can be asymptotically unstable under switching

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Brian F. Farrell and Petros J. Ioannou (1996), Journal of the Atmospheric Sciences 53(14), 2041–2053 — doi:10.1175/1520-0469(1996)053<2041:gstpin>2.0.co;2

Key takeaways

From the reading, NOTE-842.

  • No modes, but the theory carries over. For du/dt = A(t)u the temporal eigenmodes “are not defined”. The first Lyapunov exponent λ = lim sup ln‖Φ(t)‖/t replaces the spectral abscissa. The first Lyapunov vector depends only on the current time, and its optimal excitation only on the initial time.
  • Non-normality is necessary for destabilization by time variation. λ lies between the long-run averages of the extreme eigenvalues of (A + A†)/2 (eq. 12), so an operator normal and stable at every instant cannot be destabilized.
  • And with it, time variation destabilizes. A frequency-switched oscillator neutral at every instant, and random rotations of a neutral 2 × 2 matrix, have positive exponents. “Transient growth instigated at the starting time is continued with further growth when the switch to the second operator takes place.” In both, the exponent vanishes as the switching interval goes to zero.
  • Multiplicative noise destabilizes generically. Cited to Has’minskii and Colonius & Kliemann: for dimension ≥ 2 and noise matrices neither skew-symmetric nor commuting with the mean operator, noise of sufficient size gives a positive exponent. More non-normality lowers the threshold; more damping raises it (Figs. 8–9).
  • Lorenz and the atmosphere. Along the Lorenz trajectory the mean numerical abscissa is about 4.2 against a first exponent of 0.88. A three-layer baroclinic model gains a positive exponent at wind fluctuations of about 10%, which the authors take to suggest that long-range forecast error comes from the flow’s time dependence.

Standing in the record

Read on 2026-10-14 in NOTE-842. THEORY-325 is filed from it: time dependence can destabilize a non-normal system stable at every instant; non-normality is necessary; random variation destabilizes generically, needing less the more non-normal and less damped the mean operator.

CLAIM-186 cites it for the remark that an argument under development is the nonautonomous case, and this entry, when filed, asked what carries over to an operator that changes as fast as the state does. The reading answers more than the remark asked. The autonomous comfort of Part I (LIT-931), that a stable non-normal system settles after its burst, does not carry over: a structure stable at every snapshot can fail to settle if it keeps changing. The answer to the question is conditional. In the deterministic examples change much faster than the transient growth destabilizes nothing, and change on the growth timescale is what destabilizes. Under white noise the fast limit stays unstable. Damping stabilizes in all cases. Which of these anchoring corresponds to is the reading across, and the paper says nothing about arguments.

Not held in the Anthology of the SOTA; it carries no instruction for machine-learning practice.

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