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Say you have a system of ODE's where the Jacobian has one zero eigenvalue; what does that tell you about the stiffness of the system?

This case doesn't seem to be discussed in the cases I have been able to find (Stiffness ratio or stiffness index).

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The zero eigenvalue is not particularly harmful. Think of the linear system $$ \dot x = A x, x(0) = x_0 $$ then the components of $x(t)$ decay in proportion to the (magnitude of the) eigenvalues of $A$ (assuming the eigenvalues are all non-negative), whereas the components of $x(t)$ in direction of eigenvectors that belong to zero eigenvalues simply stay the same. In other words, these components simply don't factor at all in the determination whether a system is stiff or not.

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Like conditioning of linear systems, it is hard to boil down the difficulty of time-stepping a differential equation to a single number. In general a big gap in the spectrum is often held as the accepted definition of "stiffness" because it represents a separation of scales, often a nonphysical separation coming from a discretization technique (spectral methods come to mind).

The numbers you give are reasonable measures of stiffness in certain circumstances, given this understanding. The situation you present however show that it breaks down in some important cases. Discretizing hyperbolic problems one often obtains for example a nontrivial nullspace, and there furthermore is not an obvious separation of scales if one looks at the spectrum (even if using a very high order method), nevertheless the problem is still "stiff."

There are some excellent answers to a similar question which may be of use to you:

The definition of stiff ODE system

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