Nonnormal Transient Growth

Expose temporary amplification in a stable nonnormal system and separate state growth from worst-case growth.

Nonnormal Transient Growth

In linear systems where the governing matrix is nonnormal (its eigenvectors are not orthogonal), the state norm can grow transiently even if all eigenvalues strictly indicate decay. This phenomenon occurs because the individual eigenmodes decay independently, but their non-orthogonality leads to temporary constructive interference.

Worked Example

Consider a system with Coupling K = 4, Time t = 1, and Initial vector as the Second axis (0, 1)^T. The matrix is A = [[-1, 4], [0, -1]]. The state propagator is exp(tA) = e^-t [[1, Kt], [0, 1]].

Apply the propagator to the initial state: e^-1 [[1, 4], [0, 1]] applied to (0, 1)^T.
This yields the final state vector (4e^-1, e^-1)^T.
Numerically, e^-1 ≈ 0.367879. The final state is approximately (1.4715, 0.3678).
The state norm is √((4e^-1)² + (e^-1)²) = e^-1√17 ≈ 1.5168. Even though e^-1 < 1, the norm has grown from 1 to 1.5168 because of the off-diagonal coupling term.

Experiment with the Operator

Open Nonnormal Transient Growth. Try varying the controls to observe the transient growth and eventual decay.

Set Coupling to 8, Time to 1, and Initial vector to "Second axis". Observe that the state norm increases significantly before it will eventually decay.
Change the Initial vector to "First axis". Notice that the state simply decays, because this vector corresponds to the pure decay direction without interference from the nonnormal coupling.
Observe the difference between the State norm (which depends on the initial state) and the Propagator two norm (which represents the maximum possible growth for any unit vector).

Limitations

This module uses a very specific 2D nonnormal matrix to isolate the phenomenon of transient growth. In practical fluid dynamics or control theory applications, these operators are infinite-dimensional (e.g., differential operators) or very large dense matrices. Additionally, the initial state chosen heavily influences whether transient growth is actually realized. Real systems might be continually forced, leading to sustained amplification rather than just a transient spike.

References

This fundamental model of transient growth is extensively studied in hydrodynamic stability and linear algebra. The concept of the numerical abscissa bounding the initial growth rate is central to understanding why eigenvalues alone are insufficient for nonnormal operators. See Trefethen's work on pseudospectra for a comprehensive treatment: https://people.maths.ox.ac.uk/trefethen/ttrd.pdf.

Nonnormal Transient Growth