I have the following recurrence relations, modelling the movement of robots. There are $n+1$ robots, $1$ to $n+1$. Each robot $i$ with $2 \leq i \leq n$ has two neighbors, namely robots $i-1$ and $i+1$. Robot $1$ has only the neighbor $2$. Robot $n+1$ has only the neighbor $n$.
Robots move in discrete time steps (rounds). In every round, robots $2$ to $n$ move to the midpoint between their neighbors. The robots $1$ and $n+1$ move as far as possible away from their neighbors (the maximal allowed distance is $1$ and they cannot see the robot $n-1$ (3, respectively)).
The following recurrence relations describe the vector representation of the robot movement. The vector $v_i(t) = (x_i(t),y_i(t))$ is the vector pointing from robot $i$ to $i+1$ in time step $t$. The following equations describe how $x_i(t)$ and $y_i(t)$ can be computed based on the time step $t-1$.
$x_1(t) = \frac{1}{2\sqrt{x_1(t-1)^2 + y_1(t-1)^2}} x_1(t-1) + \frac{1}{2} x_2(t-1)$
$x_i(t) = \frac{1}{2} x_{i-1}(t-1) + \frac{1}{2} x_{i+1}(t-1)$
$x_n(t) = \frac{1}{2\sqrt{x_n(t-1)^2 + y_n(t-1)^2}} x_n(t-1) + \frac{1}{2} x_{n-1}(t-1)$
$y_1(t) = \frac{1}{2\sqrt{x_1(t-1)^2 + y_1(t-1)^2}} y_1(t-1) + \frac{1}{2} y_2(t-1)$
$y_i(t) = \frac{1}{2} y_{i-1}(t-1) + \frac{1}{2} y_{i+1}(t-1)$
$y_n(t) = \frac{1}{2\sqrt{x_n(t-1)^2 + y_n(t-1)^2}} y_n(t-1) + \frac{1}{2} y_{n-1}(t-1)$
Moreover, for all $i$: $x_i(0)$ and $y_i(0)$ are initial values between $0$ and $1$.
Do you think that there is any chance to find a closed formula for this?