If we define a Nash equilibrium as a fixed point of the best-response mapping, i.e. a strategy $x$ s.t. $$x\in BR(x),$$ where BR denotes the (set-valued) best-response mapping, then $$\dot x \in BR(x)-x$$ is generally referred to as best-response dynamics (BR dynamics).
I have two utility functions, $u_1(c,s)$ and $u_2(c,s)$. They are derivable. $u_1(c,s)$ refers to Player 1, while $u_2(c,s)$ to Player 2. The strategic leverage of Player 1 is $s$ while the strategic leverage of Player 2 is $c$.
In order to obtain the best response functions and Nash equilibria, I have to solve the system of the following two equations: $$\frac{\partial }{\partial s}u_1(c,s)=0$$ $$\frac{\partial }{\partial c}u_2(c,s)=0$$ From this I obtain both best response functions, i.e. $$s=f_1(c)$$ $$c=f_1(s)$$ Then the best response dynamics are, \begin{align} \dot{x} &= BR(x) - x \\ \begin{pmatrix}\dot{s} \\ \dot{c}\end{pmatrix} &= \begin{pmatrix} f_1(c) \\ f_2(s)\end{pmatrix} - \begin{pmatrix} s \\ c\end{pmatrix} \end{align} I can replace the $\in$ with $=$ because the map $f_1, f_2$ are not set value maps but are single-valued.
Then I evaluated the Jacobian of this system in Nash equilibria (I have two equilibria and then two fixed points) in order to determine their stability. I found that they are both asymptotically stable because the eigenvalues of Jacobian have negative real parts.
I would like to know if this is due to a general result (e.g. BR dynamics have always asymptotically stable fixed points) or it might happen that (for another game) one equilibrium is stable and another not.
Thanks in advance!