Consider a dynamical system described by the following coupled non-linear differential equation \begin{align} \dot{x}_1(t) &= v + a_{12}\sin(x_2(t)-x_1(t)) + a_{13}\sin(x_3(t)-x_1(t))\\ \dot{x}_2(t) &= w + a_{21}\sin(x_1(t)-x_2(t)) + a_{23}\sin(x_3(t)-x_2(t))\\ \dot{x}_3(t) &= w + a_{31}\sin(x_1(t)-x_3(t)) + a_{32}\sin(x_2(t)-x_3(t)), \end{align} $t\ge 0$, $x_i(0)\in\mathbb{R}$, $i=1,2,3$, where $v$, $w\in\mathbb{R}$, $a_{ij}\in\mathbb{R}$, and $a_{12},a_{13},a_{21},a_{31}\ne 0$.

I'm studying the properties of this dynamical system. So far, I've managed to show that if $a_{21}=a_{31}$ then the system has an invariant trajectory given by $\bar{x}(t)=(x_1(t),x_2(t),x_2(t))$, $t\ge 0$ (that is, $x_2(t)=x_3(t)$, $t\ge 0$).

Open problem:Is the invariant trajectory $\bar{x}(t)$ locally attractive?

Numerical simulations seem to suggest that the answer is in the affirmative. However the proof of this conjecture does not seem trivial to me. This could be due to the fact that I'm rather new on this kind of (local) stability problems. So I would be enormously grateful in hearing any comment/criticism/suggestion from you. Also, pointers to the literature are very welcome.