I have the following dynamical system:
$$\begin{eqnarray} \dot{\omega} & = & -\omega+J \\ \dot{J} & = & -\beta_{2} \\ \dot{\beta}_{1} & = & -\omega\beta_{2} \\ \dot{\beta}_{2} & = & J-\hat{\alpha}\omega(1-\beta_{1}) \end{eqnarray}$$
Where $\hat{\alpha}$ is a constant. Computing the equilibrium points I find that I have two: $(0,0,1,0)$ and $(J_{0},J_{0},1-\hat{\alpha}^{-1},0)$. Computing the Jacobian for the first point one obtains the following characteristic equation: $$\lambda(\lambda+1)(\lambda^{2}+1)=0$$ The second point has the characteristic equation: $$\lambda(\lambda^{3}+\lambda^{2}+(1-\hat{\alpha}J_{0}^{2})\lambda-\hat{\alpha}J_{0}^{2}+2)=0$$
So the first question I have is: Does the $J_{0}$ in the second equilibrium mean that I don't have an isolated equilibrium point?
What do the different eigenvalues mean? I get that I'm going to have complicated structure, and the phase plane will be a phase 4-space.
Any suggestions?