Suppose, I have a system of 4 coupled non-linear ode in 4 variables(2 variables are from a compact space, two from non-compact space). Let $\mathbf{x}_0$ be a fixed point of the system. Now, I want to determine the stability of the system at the fixed point $\mathbf{x}_0$.
I used linearization to find the local Jacobian of the stability matrix and determined its eigenvalues and eigenvectors. I have two cases:
- All four eigenvalues are zero and the eigenspace is of dimension 2
- Two eigenvalues are zero and the rest two are purely imaginary conjuagte pairs. The geometric multiplicity of the zero eigenvalue is 1
Thus in both cases I have geometric multiplicity< algebraic multiplicity for the zero eigenvalue.
Can I then conclude that my fixed point is unstable?