# Do we need more [stability]?

Stability of solutions is a pretty big topic in differential equations, both in ODE and PDE. And there are many such questions in tag, but they are not easy to search for, since one has to try "stability", "stable", "unstable".

Also, the tag appears to be one of the most "monolithic" popular tags, measured by the percentage of questions within the tag that have no other tag:

So, I suggest introducing a tag for questions involving stability of solutions of differential equations, and seek input on its possible name. Claiming for this purpose might intrude on other areas. So far, the best I have is . Any other suggestions?

• Why not two tags: Stability in ODEs (this will be the most common) and Stability in PDEs. – Moishe Kohan Sep 14 '14 at 2:31
• How much of a distinction is there between 'stability' and 'perturbation theory'? I may be biased from my physics background, but when I hear 'stability' I usually think 'stability under small perturbations.' (Though the 'pertubation-theory' tag really needs a summary for it...) – Semiclassical Sep 14 '14 at 2:39
• @Semiclassical In my mind, perturbation theory deals with equations being perturbed, while stability of solutions refers to perturbation of initial data (in the context of ODE, which are the main supplier of stability questions here). I think that users asking about stability of equilibria for $y'=y-y^2$ are unlikely to know the term perturbation theory, hence will not use the tag. – user147263 Sep 14 '14 at 2:51
• @Thursday: The distinction between perturbing the ODE and perturbing the boundary data is a good point. (They may be two sides of the same coin, but they're still different sides.) As to the tag name, maybe 'stability-theory' would be specific enough to avoid overlap with other topics? 'Stability-of-equilibria' might also work, but that's pretty specific. – Semiclassical Sep 14 '14 at 3:01
• @Semiclassical I like [stability-theory], because this usage is backed up by the authority of Wikipedia: Stability theory. – user147263 Sep 14 '14 at 3:35
• When you say stability, I read [model-theory]. And besides, only horses live in the stable. – Asaf Karagila Sep 14 '14 at 3:51
• It must be managed carefully, because there is one more stability type (in numerical methods), there must be no misunderstanding – cool Sep 14 '14 at 13:27
• If you use stability-theory, people will use it for stability in GIT theory, stability in other numerical algorithms, stable reduction in elliptic curves, and other things I'm not thinking of right now. These do not belong in the same tag. I'm not saying that those topics need their own tag, since I haven't seen so much of them, but stability-in-odes will reduce the number of questions about other kinds of stability showing up in the proposed tag. – David E Speyer Sep 14 '14 at 16:53
• ... and even more folks might assume that "stability theory" is about "stable homotopy groups of spheres", "stably homeomorphic manifolds", "structural stability" in general dynamical systems, etc. Given that many questions on this site routinely assume that "General topology" deals with arbitrary topological problems, I am sure that such confusions, in addition to the ones mentioned by David, will be very common. – Moishe Kohan Sep 14 '14 at 17:09
• Good points (and welcome back to Meta, @DavidSpeyer). I think (stability-in-odes) is the way to go, and its PDE counterpart can wait until there's a genuine need for it. – user147263 Sep 14 '14 at 17:12

Based on the feedback from commenters, I created , suggested tag wiki for it, and applied the tag to a few questions. No plans for a mass retag.

Asaf pointed out that now exists. Recall what was said about it last September:

If you use stability-theory, people will use it for stability in GIT theory, stability in other numerical algorithms, stable reduction in elliptic curves, and other things I'm not thinking of right now. [...] – David Speyer Sep 14 '14 at 16:53

... and even more folks might assume that "stability theory" is about "stable homotopy groups of spheres", "stably homeomorphic manifolds", "structural stability" in general dynamical systems, etc. [...] – studiosus Sep 14 '14 at 17:09

Let's see what happened in practice. The tag was created in April, for the question #1 on the list (model-theoretic concept of stability). No tag excerpt or wiki was written. Subsequently, the tag was used 20 times — for anything but model theory. Topics include: ODE, PDE, discrete dynamical systems, difference equations, numerical methods for ODE/PDE, numerical linear algebra.

1. (stability-theoretic) ¨weakly normal groups" are closed under subgroups
2. Schur-Cohn Stability Criterion
3. Is a feedback system with an unstable component and the other component being zero internally stable?
4. Lyapunov equation for stability analysis - what's the point?
5. Changing direction of Nyquist plot with PID-controller
6. What does a 3D periodic solution of a differential equation look like?
7. Backwards Stability of systems
8. What is 'bursting' in least squares estimation, and what causes it?
9. Theorem to show trajectories of differential equations are close after small change to initial condition
10. Example of BIBO stable system that is not internally stable
11. Stability using lyapunov function
12. How to interpret complex eigenvectors of the Jacobian matrix of a (linear) dynamical system?
13. Frequency Response of unstable systems
14. does an exponential bound on a Lyapunov candidate implies asymptotic stability?
15. How do i show stability of fixed points of this two.dimensioanl system:f(x,y)=(y,y²−x²)f(x,y)=(y,y²−x²)?
16. Finding a Ljapunov function for discrete dynamical system with 3 variables.
17. Stability of Gauss elimination [closed]
18. Definition of positive definite function
19. Stability in partial differential equations
20. Stability of non-homogeneous and non-autonomous first-order difference equation
21. Bounded Input Bounded Output stability for Heat Equation (Cross-Post from Sci-comp stackexchange)

Sigh. I will retag as where appropriate. (That tag does get used, 60+ times by now). Questions without an answer:

1. Should the tag be renamed as , to capture PDE and difference equations/dynamical systems?

2. Should remain, and if so, what should it mean?

• The downvote indicates a disagreement on what exactly? – Did Jul 29 '15 at 6:52
• I'd say: yes to 1. For 2: the most common use (by far) seems to be around modified 1. So if we keep it, it'd be a synonym of that. Maybe this is not needed. Personally, I am really not DE or DS affine, but this is certainly the first association I have with stability theory. Now, somebody might uses this to tag a question about stable reduction or homotopy groups, but then some tag their high-school homework arithmetic-geometry, so this criterion is not so useful. – quid Jul 29 '15 at 14:56