This is a two-part question
Part 1
We have often done stability analysis by calculating the eigenvalues of the Jacobian of a nonlinear function around a stationary solution.
I accidentally repeated the process, but this time by calculating the Jacobian around a point that is not an equilibrium. Is there any meaning to the eigenvalues I obtain?
Part 2
Is there also a way to obtain a general linearized operator around a non-stationary point?
$$F(\psi_x + \Delta\psi) = F(\psi_x) + \Delta\psi F'(\psi_x)$$
If $F(\psi_x) = 0$, then $F'(\psi_x)$ is the linearized operator. However, if $F(\psi_x) \neq 0$, is it even possible to define a linearized operator around $\psi_x$ for an arbitrary $\Delta\psi?$