The user Dinisaur conjectured that the following vector process could maybe be analyzed via a discrete version of the heat equation. I am personally not aware of the heat equation and maybe someone here has a feeling how the result could look like.
Intuitively my process can be interpreted as follows. We have an $n$ node line graph and the temperature at time step $t$ can be interpreted for every inner node as the average over its two neighbors. At the boundaries of the graph, we increase the temperature a little bit at every time step.
More formally:
I have n vectors $v_1(t), \dots, v_n(t)$. Time is divided into discrete rounds. Initially, all vectors have length $\leq 1$. 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 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?