Nonnormal Transient Growth
Expose temporary amplification in a stable nonnormal system and separate state growth from worst-case growth.
Free theory articleExpose temporary amplification in a stable nonnormal system and separate state growth from worst-case growth.
Concept & interpretation
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.
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).