I was wondering if someone could give a bit of broad advice regarding working with Implicit Function Theorem (IFT) and, I guess, the Catastrophe theory. This is something completely new to me.
Suppose I have a functional F : $\mathbb{R} \times V \to \mathbb{R}$, where $V$ is a generic Banach space and I know that $F(0,u_0) = 0$ Following the treatment of IFT for Banach spaces here, I can verify that $F$ satisfies the conditions of IFT and in particular conclude that there exists an interval $I = (-\epsilon_1,\epsilon_1)$ for some $\epsilon_1 > 0$ such that for any parameter value $\epsilon$ in this interval we have unique solution $u_{\epsilon}$, which is in a ball around $u_0$. So far so good, but is there anything that can be said about what happens for $\epsilon \not\in I$? Also, can we even try to find $I$ explicitly? My first intuition is that on $\partial I$ we have some sort of bifurcation (a term usually devoted to the theory of dynamical systems, which has nothing to do with what I'm interested in). Can it be verified and categorised? Are there any general approaches to answer such questions? I have spent several hours today trying to find any decent book/lectures material covering something like this, but sadly to no avail. Sorry if that's just my laziness, but I would really appreciate any advice.
For $\epsilon \notin I$ little can be said with the IFT, think $F: (x,y) \in \mathbb R^2 \mapsto x^2 + y^2 - 1$. No sense in think what happens for $x>1$.
The implicit expression you ask about is as in the finite dimensional case. Since you know there exists $f$ such that $D(x,f(x)) = 0$ then: $$D_x F (x,f(x)) + D_y F (x,f(x)) f'(x) = 0$$ and the most you can do is say $$ f'(x) = \frac{ D_x F } { D_ y F }( x, f(x) )$$ which is an ODE.
If you think there might a bifurcation I suggest you look into "degree theory".
Cheers,
D