Im working through the problems of Strogtz`s Nonlinear Dynamics and Chaos. On problem 4.3.5, it does not ask, but is it possible to find a normal form for it? The equation is the following:
$$ \dot{\theta} = \mu+\cos(\theta)-\cos(2\theta)$$ which for the fix point $\theta^*=\pi$, $\mu_c = 2$ a bifurcation occurs. My attempt to finding the normal form (i ploted and its seems to be a saddle node. $$ \left.\dot{\theta}\right|_{\theta=\pi}\cong\mu+(-1+\frac{1}{2}(\theta-\pi)^2 )-(1-(\theta-\pi)^2)\\ = \mu -1 + \frac{1}{2}(\theta+\pi)^2-1 + (\theta-\pi)^2\\ \dot{\phi}= \mu + \frac{3}{2}\phi^2 $$ which is not quite the normal form $\dot{x}=r+x^2$