I'm trying to do bifurcation analysis of the system
$x'=-x((x-2)^{2} - \mu^{2} -9)((x+2)^2 - \mu^2 -9)+ \epsilon$
When $\epsilon = 0$ this is easy because we have some nice circles. But I'm stuck as to what to do when $\epsilon > 0 $. I've been advised not to find analytic expressions for equilibria. I'm not looking for an explicit solution spelled out for me, just a shove in the right direction to be able to get there myself. How can one determine location/type of bifurcation for a system like this without known equilibria? Can I use my $\epsilon = 0$ diagram and infer something from it for $\epsilon >0$? What are my options here, methodology-wise?