# Thread: very confused (structural stability)

1. ## very confused (structural stability)

I am quoting from Saunders: an introduction to catastrophe theory

'Consider an m-parameter family of functions. If the parameters vary continuously we may think of them as coordinates in an m-dimensional space, and then each individual function is represented by a point in this space. If $\displaystyle f_P$ is the function corresponding to the point P, and if for any Q sufficiently close to P, the corresponding function $\displaystyle f_Q$ is of the same form as $\displaystyle f_P$, then $\displaystyle f_P$ is a structurally stable function of the family ...

We can also define structural stability for a family of functions. In this case we require that any small perturbation leaves the qualitative nature as a family unchanged '

OK those are just the definitions for one function and for a family, but here is where I get confused.

We now look at polynomials, and consider functions of the same type/form to be ones with the 'same configuration of critical points'. What does the same configuration of critical points mean?

The book states $\displaystyle x^4$ as one function that is not structuraly stable, because $\displaystyle x^4 + \alpha x^5$ is close to it, but the critical points get further away as you get closer to $\displaystyle x^4$ ( since there is a critical point $\displaystyle x = \frac{-4}{5\alpha}$, it blows up as alpha gets small ). This makes intuitive sense.

But it says that $\displaystyle x^4$ and $\displaystyle x^4 + \alpha x^2$ are close AND of the same type, you just make alpha small to bring the critical points close to the origin - this suggests that you just want your critical points near the origin for 'same configuration of critical points' to be satisfied, since these 2 functions have different points:$\displaystyle x^4$ has a
degenerate minimum at the origin, while $\displaystyle x^4 + \alpha x^2$ has a maximum at the origin, and minima at x= plus or minus $\displaystyle \sqrt (-\frac{-\alpha}{2}$ ) ( if $\displaystyle \alpha < 0$ )

but it says that the family $\displaystyle x^4 + \alpha x^2$is NOT of the same type as the family $\displaystyle x^4 + \alpha x^2 + \beta x$? I do not understand why - they have different critical points again but if you make beta small enough the points just get closer to the origin?

Does anyone understand what I am going on about? I would really appreciate any help if anyone is familiar with whats going on here

2. I think ive mixed them up. Bringing points closer to the origin is bad!! you want to make them arbitrarily far away so the configurations of critical points at zero is preserved. Makes sense now! sorry