# "Damping factor" for a set of non-linear ODEs

I have a set of four non-linear ODEs representing a negative feedback. I have done parameter variation by random sampling to study the sensitivity of steady state and other dynamic properties to parameter fluctuations.

From the analytical expressions of steady state I have calculated jacobian and its eigenvalues for each parameter set and based on that I could observe the fraction of oscillating cases (complex eigenvalues). I wish to calculate something like damping factor for these cases. I am not sure what to use as a metric. I think of three choices:

1. Least eigenvalue (absolute value of the real part). This would be rate limiting.
2. Biggest eigenvalue (absolute value of the real part). This might dictate the initial damping (which is what I am basically interested in).
3. Determinant of the jacobian (?)

Please suggest me what is the right metric or let me know if there are better alternatives.

• I think sign is important here, so I would be wary about absolute values. You may also want to look at Lyapunov exponents (I am not a dynamical systems expert, but I see them used in that context). Commented Jun 10, 2015 at 13:30
• @GeoffOxberry These are all stable systems and the eigenvalues are negative. Bigger means "more negative". Commented Jun 10, 2015 at 14:54
• Yes, I understand that large magnitude negative eigenvalues mean "more negative". My comment was mainly to point out that in some applications in nonlinear systems, perturbing parameters or states can change a locally stable point into a locally unstable point, and absolute values won't be able to distinguish between the two. (Of course, if there is any component of the time derivative along the unstable subspace, you'd probably see it in a solution.) If your system is stable with respect to the perturbations you're considering, then yes, sign is unnecessary. Commented Jun 10, 2015 at 20:27