I'm trying to model convergence of $x\in \mathbb{R}^d$ which follows the following recurrence

$$x\leftarrow (\mathbf{1}-h)^2 x + h\langle x, h\rangle$$

Here $\mathbf{1}$ indicates a vector of $1$'s and $v^2$ means squaring each component of vector $v$.

In my application $x$ and $h$ are non-negative, entries of $h$ follow power-law decay with known constant $p\in (1,2)$ and $h$ is small enough so that continuous approximation $x_t\approx \exp(At)x_0$ holds. I need to know how trajectory of $\|x_t\|_1$ depends on $p$ in the case of $t<d$ and $d\to\infty$

Things are easy if we didn't have the the $h\langle x, h\rangle$ term (mixing term). $A$ is diagonal, and approximating $\|x_t\|_1$ with an integral I get formulas which match observed behavior very well.

However, keeping the mixing term I get $x_t\approx \exp(At)x_0$ where $A$ is diagonal + rank1 matrix, and this becomes much harder to handle. Applying integration as before does not produce insights.

Any advice on the approaches to follow to get a nice upper bound on $\|x_t\|_1$ in terms of $p$?

Notebook