Say for a system $$\dot{x}=y$$$$\dot{y}=-x+\mu y -y^3$$ I have confirmed a hopf bifurcation occurs at the origin and that the branches are stable i.e., a stable limit cycle and the origin being stable for $\mu < 0$.
I wish to sketch the phase portraits for when $\mu >0$ and $\mu < 0$. This I know - for $\mu >0$ there is a stable limit cycle, so say for an initial condition near the origin, it will spiral outwards towards the limit cycle. similarly for the other case, it will spiral in towards the origin.
My question is how are the spirals orientated? Could I say plug in some coordinates, e.g. $(x,y)=(1,-1)$ resulting in the vector [-1,-$\mu$] and see which direction it points? Thanks!
Plug some coordinates in and look at the direction of the resulting vector.