show that the system has a limit cycle using a theorem

Question

I have the following system, and I'm trying to use a theorem to prove that it has a limit cycle. I proceeded finding the fixed points(Strogatz), $(0,0)$ in this case and then calculating the eigenvalues I got the hint that for values of $b, k, m$ such that $b^2<4km$ there might be a limit cycle. I don't really know how I can prove this using Poincaré-Bendixson given that the old trick of converting to polar coordinates doesn't seem that easy to do here. Can anyone advise me how to proceed here? Thanks. $$\text{System:}\\ \dot x_1 = x_2 \\ \dot x_2 = \frac{-k}{m}x_1-\frac{b}{m}x_2(x_1^2-1)\\ $$ This is what I've done so far using Poincaré-Bendixson: $$ \text{Having:} \quad V(x,y) = x_1^2+x_2^2 \leq c $$ $$ \frac{\partial V}{\partial x_1} \dot x_1 + \frac{\partial V}{\partial x_2} \dot x_2 = 2x_1x_2 + 2x_2[\frac{-k}{m}x_1-\frac{b}{m}x_2(x_1^2-1)]\\ =2x_1x_2 -\frac{2k}{m}x_1x_2-\frac{2b}{m}x_2^2x_1^2 + \frac{2b}{m}x_2^2\\ \text{using k,m as 1 and b as 0.1, I got:}\\ =2x_1x_2 -2x_1x_2-0.2x_1^2x_2^2 + 0.2x_2^2\\ =0.2(-x_1^2x_2^2+x_2^2) $$