Stability of the equilibrium states

Question

I have a function defined by the following differential equation $$ \frac{\mathrm{d}\varphi}{\mathrm{d}t} = \gamma - F(\varphi) $$ where

• $F(\varphi)$ is a $2\pi$-periodic function) and
• the chart of the function $F(\varphi)$ is known.

I need to find the equilibrium states (e.s.) and find out if they are stable. I've managed to find out what are they only in some cases (but this is not accurate). Can you help with others?

                 ^ F(φ)
                1|
                 |    /\  /\
                b|   /  \/  \
                 |  /        \
                 | /          \    
__-π_______-a____|/____________\____>φ
   \            /|0    a π/2   π 
    \          / |
     \        /  |
      \  /\  /   |-b
       \/  \/    |
                 |-1

If $|\gamma| > 1$ then there are no e.s.

If $|\gamma| = 1$ there are 2 (at level ±γ each) e.s. in points (a,1), (π-a,1). maybe they're semi-stable?

If $b < |\gamma| < 1$ there are $4$ at level $\pm \gamma$ each e.s.

If $|\gamma| = b$ there are $3$ at level $\pm\gamma$ each e.s.: point $(\pi/2, b)$, $\varphi = a\gamma$, $\gamma = \pi - a\gamma$.

If $0 < |\gamma| < b$ there are $2$ e.s. at level $\pm±\gamma$ each $\varphi = a\gamma$ (is it stable?) , $\varphi = \pi - a\gamma$ (and is this one unstable?).

The form of $F(\varphi)$ is the following one: $$F(\varphi)=\begin{cases} & {-\dfrac{\varphi}{a}-\dfrac{\pi}{a}}, &\text{if } {\varphi \in [-\pi, -\pi + a]},\\ \\ & \dfrac{1-b}{\dfrac{\pi }{2}-a }\varphi + \dfrac{(1-b)(\pi-a))}{\dfrac{\pi }{2}-a }-1, &\text{if } {\varphi \in [-\pi + a,-\frac{\pi }{2}}], \\ \\ & \dfrac{b-1}{\dfrac{\pi }{2}-a }\varphi + \dfrac{a(1-b)}{\dfrac{\pi }{2}-a }-1, &\text{if } {\varphi \in [-\dfrac{\pi }{2}}, -a], \\ \\ & \dfrac{\varphi}{a}, &\text{if } {\varphi \in [- a, a]}, \\ \\ & \dfrac{b-1}{\dfrac{\pi }{2}-a }\varphi - \dfrac{a(1-b)}{\dfrac{\pi }{2}-a }+1, &\text{if } {\varphi \in [a, \frac{\pi }{2}}], \\ \\ & \dfrac{1-b}{\dfrac{\pi }{2}-a }\varphi - \dfrac{(1-b)(\pi-a))}{\dfrac{\pi }{2}-a }+1, &\text{if } {\varphi \in [\frac{\pi }{2}}, \pi-a], \\ \\ & {-\dfrac{\varphi}{a}+\dfrac{\pi}{a}},&\text{if } {\varphi \in [\pi-a, \pi]}. \end{cases}$$ P.S. sorry for this chart :)