The claim
Let du/dt = A(t)u have propagator Φ(t) and first Lyapunov exponent λ = lim sup ln‖Φ(t)‖/t > 0, and suppose every A(t) has spectral abscissa at most 0. For δ in (0, λ), the system du/dt = (A(t) − δI)u:
- has exponent exactly λ − δ > 0, and
- is stable at every instant, with spectral abscissa at most −δ.
Proof
δI commutes with every A(t), so the propagator of A(t) − δI is Φ_δ(t) = e^(−δt)Φ(t). Then ln‖Φ_δ(t)‖/t = ln‖Φ(t)‖/t − δ, and the lim sup is λ − δ. The eigenvalues of A(t) − δI are those of A(t) shifted by −δ, which proves (2).
The switched oscillator
Take Farrell & Ioannou’s oscillator (LIT-929, eqs. 13–17) with γ = 0: A_ω = [[0, 1], [−ω², 0]], ω switching between 0.5 and 3 every T = 1. Each block integrates in closed form:
R_ω(T) = exp(A_ω T) = [[cos ωT, sin(ωT)/ω], [−ω sin ωT, cos ωT]].
The monodromy over one period, of length 2, is M = R_3(1)R_0.5(1). Its determinant is 1 and its trace is −2.1548. Since |tr M| > 2, the multipliers are real, with the largest equal to 1.4784, so λ = ln(1.4784)/2 = 0.1955. Every A_ω is neutral, with eigenvalues ±iω. With δ = 0.1, the instantaneous eigenvalues are −0.1 ± 3i and −0.1 ± 0.5i, and the exponent is 0.0955. The numbers were computed from the closed form, and checked against a matrix exponential of the shifted operators.
What it does not say
- That −δI is the paper’s damping. Farrell & Ioannou damp the velocity, with −2γ in the lower-right entry. The shift damps both coordinates. It is enough for the existence claim, and says nothing about the paper’s damped curves.
- Anything about random variation. That is THEORY-325’s, still Proposed.
- That such a shifted operator is normal. It inherits A(t)‘s non-normality, as THEORY-724’s bound requires.