Provide a full phase plane analysis for the model:
$\left\{\begin{array}{l} \epsilon\dfrac{dx}{dt}=-(x^3-Tx+b)\;,\;T>0\\\dfrac{db}{dt}=x-x_0\end{array} \right.$
So I'm trying to find critical points which turned out to be $(x,b)=(x_0,Tx_0-x_0^3)$ for me, next I found the Jacobian matrix (partial derivatives) and then eigenvalues: $\lambda_1=\frac12(T-3x_0^2-\sqrt{(T-3x_0^2)^2-4}\;)$ ; $\lambda_1=\frac12(T-3x_0^2+\sqrt{(T-3x_0^2)^2-4}\;)$
Now I'm not sure what to do next, how am I supposed to sketch a phase plane, while I have $2$ constants $T>0$ and $x_0$ (of course $\epsilon$ is also a small positive constant).
Also what does it mean to provide a full analysis of the phase plane? Is it just equilibrium points?
Any help will be much appreciated.
Some helping tips.
Regarding the equilibrium point, analyzing the eigenvalues we have that
$$ (T-3x_0^2)^2-4 \ge 0 $$
define real eigenvalues. This region is shown in the attached figure. In light blue where the eigenvectors are real positive and in light red where real negative . The white region is for complex eigenvalues.
Also is shown a stream plot for $x_0 = 0, T = 1, \epsilon = 0.2$ where can be observed a limit cycle around the origin.