I try to analyze the following vector process for some time now.
We have $n$ vectors $v_1(t), \dots, v_n(t)$ where $t$ indicates the current time step. We consider discrete time steps. Initially, all vectors have length $\leq 1$ and $> 0$. Also, the vectors are connected initially. Meaning that the endpoint of $v_1(t)$ is the starting point of $v_2(t)$ and so on. The vectors at the next time step $t+1$ can be calculated as follows:
\begin{align*} v_1(t+1) &= \frac{v_1(t)}{2\|v_1(t)\|} + \frac{1}{2}v_2(t)\\ v_i(t+1) &= \frac{1}{2}v_{i-1}(t) + \frac{1}{2}v_{i+1}(t) \\ v_n(t+1) &= \frac{1}{2}v_{n-1}(t) + \frac{v_n(t)}{2\|v_n(t)\|}\\ \end{align*}
It turns out that, if the vectors do not form a line initially, this procedure converges and in the end all vectors are the same. However, I cannot predict the vector to which the vectors converge.
Does anyone have an idea?