Consider the vector field $$\dot u=f(u,\epsilon)=-u+u^2-\epsilon u,\ \ \ \ u,\ \epsilon\in\mathbb R$$ The aim of this exercise is to discuss the stability of the fixed points of the field. In the following I give my approach.
We have $f(0,0)=0$. The fixed points are $$u=0, \ \ \ u=\epsilon+1$$ For the stability we calculate: $$f_u(u,\epsilon)=-1+2u-\epsilon$$ So: $$ f_u(0,\epsilon)=-1-\epsilon \begin{cases} <0, \ \ \ \ \ \ \epsilon>-1 \\ >0, \ \ \ \ \ \ \epsilon<-1 \\ \end{cases} $$ and $$ f_u(\epsilon+1,\epsilon)=1+\epsilon \begin{cases} <0, \ \ \ \ \ \ \epsilon<-1 \\ >0, \ \ \ \ \ \ \epsilon>-1 \\ \end{cases} $$ Then $u=0$ is stable for $\epsilon>-1$, unstable for $\epsilon<-1$ and $u=\epsilon+1$ is stable for $\epsilon<-1$, unstable for $\epsilon>-1$.
