SAME EIGENVALUES
Same eventual stability. Different excursions.
Every start is the same. Every system eventually returns toward zero. Change the coupling and watch how far the disturbance travels first.
Stable can still mean a detour.
One start · two fixed scales · exact trajectories
Start paused near the large excursion. Restart to release the same disturbance from t = 0, or scrub time.
The eigenvalues stay put.
The second coordinate decays while feeding the first. Stronger coupling can carry the state farther from zero before both coordinates fade.
λ₁ = −1 λ₂ = −2
x(0) = (0, 1) ‖x(0)‖₂ = 1
- Norm now
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- Peak / initial norm
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- Time of peak
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Peak over all t ≥ 0 for this fixed starting vector. It is not the largest possible amplification over other starting directions.
Read the study
What stays the same
These continuous linear systems satisfy ẋ = Ax, with A = [[−1, k], [0, −2]]. The triangular matrix always has eigenvalues −1 and −2. The initial vector is always (0, 1). Both coordinates use the same fixed dimensionless units, and distance always means the Euclidean norm √(x₁² + x₂²).
State-space units are equal horizontally and vertically. Neither plot rescales with k or time. The time unit is fixed. The faint complete curves extend to t = 6; the formulas continue to zero as t → ∞.
What changes
Only the off-diagonal coupling k varies, from 0 to 12. The matrix norm and eigenvector geometry are not fixed. This is a different system in the same coordinates, not a change of units applied to one system.
For k ≠ 0 the matrix is nonnormal: AAᵀ ≠ AᵀA. Its stable eigenvalues establish asymptotic decay, but do not determine the norm at every earlier time. Nonnormality alone does not guarantee a transient rise for every k or initial vector.
Exact motion, no integrator
x₁(t) = k(e⁻ᵗ − e⁻²ᵗ)
x₂(t) = e⁻²ᵗ
Each displayed state is evaluated directly from this solution. Curves are sampled for drawing; time scrubbing does not accumulate integration error. At t = 0 the norm initially decreases for every k. The stronger cases then grow before their eventual decay.
The norm is a geometric distance. This exhibit does not assert a mechanical energy interpretation or demonstrate an unstable system.
Which peak?
The readout maximizes ‖x(t)‖₂ / ‖x(0)‖₂ for the single stated initial vector over all t ≥ 0. It checks t = 0 and every positive stationary time analytically; the limit at infinity is zero. The maximum can remain 1 at t = 0.
This differs from optimizing over all initial directions, which would require the operator norm of eᴬᵗ. Context: Oxford’s introduction to pseudospectra and nonnormal behavior.