# Talk:Singular perturbation theory

The article is quite nice the way it is. I would suggest two trivial changes and one additional sentence and reference. The trivial changes are

1. The references to Murdock and Van Dyke about triple decks do not contain the date (so it is not clear that they point to the reference list at the end).

2. Saying that the nonlinear oscillator problem does not have the singular perturbation form of the previous examples suggests that it is not a singular perturbation. It would be better to say that the problem is regular when considered on a finite interval of time and singular when considered on an expanding interval [0,1/epsilon]. This illustrates that a singular problem does not have to "look" singular in the sense of having epsilon multiply the highest derivative.

The other addition that I would like (but won't insist on) is a mention of Fenichel's "geometric singular perturbation theory," which has become quite important. It is a rigorous approach to ode problems with layers, showing that the slow manifold is an approximation to an actual normally hyperbolic invariant manifold (under certain conditions). The best beginning reference is probably C.R.T. Jones's article; you can get the reference from http://www.springerlink.com/content/h28565232127qv56/.

## Reviewer C:

(Ferdinand Verhulst)This is a nice article. I have a number of small remarks and a somewhat larger one.

In eq.(3) add: as \epsilon -->

The symbol ~ is used several times for 'approximates´ without explanation. I prefer putting = and adding + ... or better + o(1)

Above eq. (7): ¨can be found¨ sounds very definitive, I would add ¨often¨

Under Differential equations, solutions need not be smooth add: in the parameter \varepsilon

Under Boundary layers etc. you mention procedures of matching discussed by Kaplun etc. I would add that matching works well for many problems but that there are still fundamental issues to be solved in the theory.

In the interesting list at the end of Boundary Layers etc. add between () (especially in fluid mechanics and combustion)

A more serious problem: For short times, ... , solutions can be ... Instead of 0 < t < o(1/ \epsilon) I would say for t = O(1) or alternatively for 0 < t < L with L a constant independent of \varepsilon If you leave it like this one gets the discussion whether the regular results are correct on the timescale 1 \sqrt{\varepsilon} or other timescales smaller than 1 / \varepsilon. I suppose you donot want this. Note that the correct (if you wish uniform) asymptotic approximation depends on t and \varepsilon t which shows already that the trouble arises from timescale 1/ \varepsilon.

A second point here is that multiple scales also refers to the boundary layer type of problems. It is a confusing term in this framework. Better would be ¨multiple timescales¨ or ¨long-time dynamics and multiple scales¨.