How to linearize the following difference equation? I'm doing economic modelling where $x_t$ is the intertemporal cash-flow variable. I need to solve the following recurrence relation

$$x_{t+1}=\frac{x_{t+8}}{x_{t+1}}$$

My problem is that I don't exactly know how to linearize this function. Maybe I should introduce a new variable $y_{t} = x_{t+1}$ and get

$$y_{t}^2={x_{t+8}}$$

Could you help me out with the correct linearization technique, please? I'm looking forward to your answer.
We have the recurrence relation

$$x_{k+1} = \frac{x_{k+8}}{x_{k+1}}$$

If the denominator is nonzero, this recurrence relation can be rewritten as follows

$$x_{k+7} = x_k^2$$

Assuming positivity and taking the logarithm of both sides, we obtain a linear recurrence relation

$$\ln (x_{k+7}) = 2 \ln (x_k)$$

Shifting,

$$\ln (x_{k+1}) = 2 \ln (x_{k-6})$$

Let

$$\eta_k := \left( \ln (x_k), \ln (x_{k-1}), \dots, \ln (x_{k-6}) \right)$$

be a $7$-dimensional state vector. In matrix form,

$$\eta_{k+1} = \begin{bmatrix} 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 2\\ 1 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\ 0 &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\ 0 &amp; 0 &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\ 0 &amp; 0 &amp; 0 &amp; 1 &amp; 0 &amp; 0 &amp; 0\\ 0 &amp; 0 &amp; 0 &amp; 0 &amp; 1 &amp; 0 &amp; 0\\ 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 1 &amp; 0\end{bmatrix} \eta_k$$

Hence, we need $7$ initial conditions. We recover $x_k$ via $x_k = \exp( \mathrm e_1^{\top}\eta_k )$.