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, $v^2$ means squaring each component of vector $v$ and $h\propto (1^{-p},2^{-p},\ldots ,d^{-p})$ for some $p\in (1,2)$.

I know that continuous approximation $x_t\approx \exp(At)x_0$ holds and 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.

_^Notebook

However, keeping the mixing term makes things much harder to handle. Integration approach as before gives formula which is cumbersome. There's also numeric approach which works but doesn't give insight on the role of $p$.

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

Motivation: this kind of equation gives evolution of expected value of iterated Gaussian linear system like this for non-isotropic Gaussian case. Used to model training curve of neural network training.