Crandall & Rabinowitz Theorem states what follows. We have got a Banach Space $(X,||\cdot||)$ and an equation of the following type:
$$
F(\lambda,u) = \lambda u - G(u) = 0,
$$
where $G \in C^1(X,X)$ is such that $G(0)=0$ and $G'(0)$ is compact (where ' is a Fréchet derivative).
If $\lambda\_0$ is an eigenvalue for $G'(0)$ of odd multiplicity, then one of the following occur:


* From $(\lambda\_0,0)$ there is an unbounded bifurcation curve of non-trivial solution of $F=0$ (here trivial means of the type $(\lambda,0)$),
* From $(\lambda\_0,0)$ there is a bifurcation curve which meets again the trivial branch at $(\lambda\_\*,0)$ , where $\lambda\_\*$ is another eigenvalue of $G'(0)$.


Can someone give me a (not exceedingly complex) example of when the second case occurs? I mean, it looks as though it is the case which occurs most likely, though I have not yet managed to find an example in which I can actually describe the curve.


I am sorry if the question is inadequate for this site, yet I have already tried on stackexchange (and got no response).



Crandall & Rabinowitz Theorem, bifurcation curves
Here is a simple example in $R^2$: $G(u)=(-u\_1+u\_2^3,u\_2-u\_1^3)$.


