For a nonlinear system $\dot{x} =f(x,\alpha)$ where $\alpha$ is a parameter, with fixed points $x^*$ such that $f(x^*,\alpha) = 0$, what methods are there for computationally determining the stability of the fixed points?
Specifically in the case where the eigenvalues of the jacobian are all real and negative except one which is zero, for all values of the parameter $\alpha$.
Reducing the system down to its center manifold and examining the stability in that way is analytically difficult, which is why I am seeking computational methods.