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

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

Here $\mathbf{1}$ indicates a vector of $1$'s, vector multiplication, addition, squaring are applied pointwise and $\mathbf{h}\propto (1^{-p},2^{-p},\ldots ,d^{-p})$ for some $p\in (1,2)$. 

$\mathbf{h}$ is small enough so that continuous approximation $\mathbf{x}_t\approx \exp(At)\mathbf{x}_0$ holds with $\mathbf{x_0}=\mathbf{h}$ and I need to know how trajectory of $\|\mathbf{x}_t\|_1$ depends on $p$ when $d\to\infty$.

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


[![enter image description here][1]][1]
<sub><sup>[Notebook](https://www.wolframcloud.com/obj/yaroslavvb/nn-linear/mathoverflow-rank1-recurrence-simple.nb)</sup></sub>


However, keeping the mixing term makes things much harder to handle. System now evolves as $\exp(At)$ where $A$ is a diagonal+rank1 matrix. Integration approach as before [gives formula](https://math.stackexchange.com/a/4668173/998) which is cumbersome. There's also numeric [approach](https://mathoverflow.net/a/443030/7655) 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 $\|\mathbf{x}_t\|_1$ in terms of $p$**?


Motivation: this equation models expected value of iterated Gaussian linear system [like this](https://mathoverflow.net/questions/443143/when-is-prod-i-0-infty-i-x-i-x-it-0-for-isotropic-gaussian-x-i) for non-isotropic Gaussian case. Used to model training curve of neural network training.

  [1]: https://i.sstatic.net/3a3jd.png