The following differential equation depends on a parameter $a$. Sketch the corresponding bifurcation diagram. 
The following differential equation depends on a parameter $a$. Sketch the corresponding bifurcation diagram.
$$x' = x^3 - x + a$$

Our professor did not get to finish talking about this during lecture and I am trying to figure it out but I'm stuck and was wondering if someone could point me in the right direction. From my understanding, in order to sketch a bifurcation diagram:

*

*You need to solve for $a$: $a = -x^3 + x$

*Then find the inverse of that in order to graph it

*In order to know what direction your arrows are pointing, you need to test some values using your equilibrium points

I am stuck on finding the inverse for $a$. I graphed $a$ and found the roots to be at $(0,1), (0, -1)$, and $(0,0)$ so I assumed the inverse points would be at $(-1, 0), (1, 0)$, and $(0,0)$ and sketched it using that. However, I can't figure out what inverse function graph passes through those points. Any help would be greatly appreciated.
I have also tried solving for $x$ using $x^3 - x + a = 0$ to try and find the equilibrium points and got $0, 1$, and $-1$ which I think are incorrect so any help with this would also be greatly appreciated. Thank you in advance!
 A: We want to find the bifurcation diagram for (there is a nice theory write-up at LibreTexts).
$$x' = x^3 - x + a$$
Let's get a qualitative idea of the behaviors using phase portraits for a few $a's$ and a bifurcation
$a = -1$

$a = 0$

$a = 1$

From these, we see that there a single unstable critical point at $a = -1$, three critical points, $x = -1, 0, 1$ (two unstable and one stable) at $a = 0$, and another single and unstable critical point at $a = 1$.
What happens if we make $a$ larger and smaller than one?
There is a description of these phase portraits at this Math Stack Exchange answer (this answer is different because it adds phase portraits and a bifurcation diagram).
Now, let's draw a contour plot of $x'$ as $a$ versus $x$ to see these behaviors described by the portraits and the linked description.

Notice the behaviors and compare to the phase portraits. The green area is the stable area, and the red are unstable values of $a$.
Please fill in the details so you learn and determine the type of bifurcation.
Note, we can also draw the phase portraits like a phase-line, which is equivalent to looking at the sign of $x'$ on either side of the critical point .
$a = -1$

$a = 0$

$a = 1$

Here is another example of this approach.
