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Is the center manifold theorem applicable if say for a planar (2D) system of non-linear ODE, the stability matrix has both eigenvalues zero? Of course, there is only one eigenvector.

If not, what is the way to approach such a problem?

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    $\begingroup$ Of course the center manifold theorem is applicable to this case. It just does not reduce the system beyond what it is. $\endgroup$
    – Michael Renardy
    Commented Jul 19, 2018 at 16:31

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Yes, with a two dimensional center manifold. If you have a planar system with an equilibrium at which the Jacobian has two zero eigenvalues but only one linearly independent eigenvector, then you may have a Bogdanov-Takens (double-zero bifurcation) in your system.

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