I have the following system $$\frac{dx}{dt} = ax+y$$ and $$\frac{dy}{dt} = -x+ay$$ By setting the derivative equals to $0$, we get the equilibrium point $(0,0)$. After drawing the phase portrait for various parameter $a$, we conclude that $(0,0)$ is stable when $a<0$ (see 1), a center when $a=0$ (see 2), and unstable when $a>0$ (see 3). Also, the eigenvalues are $\lambda = a \pm i$.
Do we have a bifurcation here? I was guessing that maybe we have Hopf bifurcation when $a=0$ because $(0,0)$ lose its stability when $a=0$. If it's true, I was also asked to draw the bifurcation diagram. I am not sure what it means or how to draw it.