I know what the definition of a Poincaré map is but I am struggling a little bit with the exercises and with computing it. I am trying to solve the following problem:
Given the system (in polar coordinates):
$$ \dot{r} = r + \alpha r^3 $$ $$ \dot{\theta} = \nu $$
with $\alpha < 0$. Find the solutions of the system and write the corresponding Poincaré map for a section $\theta =$ constant. Study the dynamics of the Poincaré map (one-dimensional map). Any help with how to proceed is welcome. Thank you very much!