Consider the first order ODE \begin{equation} \dot{x} = f(x), \end{equation} with $x \in \mathbb{R}^3$ and $f: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ a smooth function. We assume that the ODE only has equilibria for $x=(0,0,x_3)$ for all $x_3 \in \mathbb{R}$. Furthermore, assume that the Jacobian of $f$ at $x=(0,0,x_3)$ has one positive, one negative and one zero eigenvalue. Does it then follow that the center manifold $W^{c}(0,0,x_3)$ is unique?

It is not hard to see that $\{ (0,0,x_3) \; : \; x_3 \in \mathbb{R} \}$ is a center manifold. In the case that you have 2 positive or 2 negative eigenvalues it is easy to see that the center manifold is unique by making use that the unstable or stable manifold which foliates the whole space. This argument does not extend to the `saddle' case so I am not sure how to proceed. It also might be that it is not true so a counter example would also be highly appreciated.


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