Find all bifurcation points (ODEs)

[anonymity]

I'm at a loss on this question...my troubles seem to be algebraic or that I'm simply missing something.x' = \mu - x² +4x⁴

my method for these questions has basically been to do everything required to draw bifurcation diagram bar drawing the actual diagram itself (ie, find equilibria, what values of mu create/destroy them, and the intervals of stability). Here solving for x in terms of the parameter mu has been a challenge. I've been trying to think of what it means to have mu as a function of x and what that can do for me, but so far I have nothing.

Is this the correct method, or am I making this harder than it needs to be? I'm taking the course independent as an independent study, so every once in a while I can't help but wonder.

If I am doing it right can someone give me a hint here? -.-

 

[Dick]

If you want to find the equilibrium points you want to solve mu-x^2+4x^4=0, yes? That's not too hard. Substitute u=x^2 first. Now you have a quadratic in u. Solve it for u and then find x.