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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).

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Here is a simple example in $R^2$: $G(u)=(-u_1+u_2^3,u_2-u_1^3)$.

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