I have a 4 dimensional dynamical system and I want to know the stability of one of its fixed points, the problem is that all the eigenvalues of the Jacobian there are purely imaginary: $$ 0 - 5.64799 i, 0 + 5.64799 i, 0 - 5.71808 i, 0 + 5.71808 i $$
The eigenvectors of the Jacobian at the fix point are linearly independent and are the following:
$$ \left(-5 \sqrt{\frac{5}{192 \sqrt{5}-125}},1,0,0\right) \\ \left(5 \sqrt{\frac{5}{192 \sqrt{5}-125}},1,0,0\right) \\ \left(0,0,-5 i \sqrt{\frac{5}{125+64 \sqrt{5}}},1\right) \\ \left(0,0,5 i \sqrt{\frac{5}{125+64 \sqrt{5}}},1\right) $$
What is the stability of this fix point and how can one figure it out?
Extra info: My system is a continuous system of ODEs describing a physical system.
(I found a similar question here but I did not fully understand how it would apply to my case since he is talking about an eigenvalue with multiplicity 2.)