I have a function ${\dot{\varphi } = \gamma - F(\varphi )}$ (where $\varphi$ - is 2${\pi }$-periodic function) and graph of function $F(\varphi)$. So it's needed to research this graph (to find the equilibrium states and find out if they are stable, unstable, or semi-stable, build phase portraits and so on),to build a phase portrait and to calculate this integral: $\int \frac{d\varphi }{\gamma -F(\varphi ))}=\int dt$ to get $\varphi (t)$.
^
|
|1 ______
| /| \
| / | \
| / | \
__-π_______-a____|/___|________\π____>
\ | /|0 a
\ | / |
\ | / |
\ |/ |
¯¯¯¯¯¯ |-1
As I see, a segment ${[a, \pi -a]}$ is the segment of equilibrium states and any straight line $y = k$, where k ${\in (0,1)}$ would have 2 e.s. Also if $\gamma=1$ there would be only one e.s. Is it right? How can I find out what they are? And could you please help me to build the phase portrait?