Drawing bifurcation diagram

Question

I'm considering the ordinary differential equation (ODE) $du/dt = a + u^2 - u^5$. I know that the number of fixed points varies based on the value of $a$, and I've identified the intervals of $a$ which correspond with the existence of $1, 2,$ or $3$ fixed points.

I have also determined the stability of the fixed points (again, for each range of $a$-value): some are stable, one is unstable, and two are half-stable.

Now, I must draw a bifurcation diagram, plotting $u^*$ versus $a$ (where $u^*$ represents fixed points); then I will need to identify the hysteresis loop in the bifurcation diagram. I need clarification on how to draw the bifurcation diagram. Particularly, I am unsure how to represent the half-stable fixed points on the bifurcation diagram.

Answer 1

Here's an option, where the solid blue points are stable and the open red points unstable. This is a numerical solution, but it should give a good starting point

This is the algorithm to build it

1. Define $f(u) = a + u^2 - u^5$
2. Select a value of $a$
3. Find roots of $f(u) = 0$
4. For each root $u^*$ calculate the sign of $f'(u^*)$, this is the stability
5. Repeat steps 2 through 5