I consider a discrete dynamical system.
We define the current state of the system as follows: $$s(t) = \left(x_1(t), x_2(t), x_3(t), y_1(t), y_2(t), y_3(t)\right)$$
\begin{align*} x_1(t+1) &= f_1(s(t)) = \frac{1}{2 \cdot \sqrt{x_1(t)^2 + y_1(t)^2}}x_1(t) + \frac{1}{2} x_2(t) \\ x_2(t+1) &= f_2(s(t)) = \frac{1}{2} x_1(t) + \frac{1}{2} x_3(t) \\ x_3(t+1) &= f_3(s(t)) = \frac{1}{2} x_2(t) + \frac{1}{2 \cdot \sqrt{x_3(t)^2 + y_3(t)^2}}x_3(t) \\ y_1(t+1) &= f_4(s(t)) = \frac{1}{2 \cdot \sqrt{x_1(t)^2 + y_1(t)^2}}y_1(t) + \frac{1}{2} y_2(t) \\ y_2(t+1) &= f_5(s(t)) = \frac{1}{2} y_1(t) + \frac{1}{2} y_3(t) \\ y_3(t+1) &= f_6(s(t)) = \frac{1}{2} y_2(t) + \frac{1}{2 \cdot \sqrt{x_3(t)^2 + y_3(t)^2}}y_3(t) \\ \end{align*}
This is the Jacobi matrix of the system (I omitted the time parameter t for better readability):
\begin{align*} J &=\begin{bmatrix} \frac{y_1^2}{2(x_1^2 +y_1^2)^{3/2}} & \frac{1}{2} & 0 & - \frac{x_1 \cdot y_1 }{2(x_1^2 + y_1^2)^{3/2}} & 0 & 0 \\ \frac{1}{2} & 0 & \frac{1}{2} & 0 & 0 & 0 \\ 0 & \frac{1}{2} & \frac{y_3^2}{2(x_3^2 +y_3^2)^{3/2}} & 0 & 0 & - \frac{x_3 \cdot y_3}{2(x_3^2 + y_3^2)^{3/2}} \\ - \frac{x_1 \cdot y_1}{2(x_1^2 + y_1^2)^{3/2}}& 0 & 0 & \frac{x_1^2}{2(x_1^2 +y_1^2)^{3/2}} & \frac{1}{2} & 0 \\ 0& 0 & 0 & \frac{1}{2} & 0 & \frac{1}{2} \\ 0 & 0 & - \frac{x_3 \cdot y_4}{2(x_3^2 + y_3^2)^{3/2}} & 0 & \frac{1}{2} & \frac{x_3^2}{2(x_3^2 +y_3^2)^{3/2}} \end{bmatrix} \end{align*}
In general, $x_1 = x_2 = x_3$ and $y_1 = y_2 = y_3$is a fix point of the system in case $(x_1,y_1)$ is a unit vector, but there does not seem to be any attracting fix point. Is there any way to determine the fix point this system converges to? (it does so in simulations).
