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