I am trying to understand the following claim (I came across it while reading a paper):

Consider the map (Standard/Arnold map)

$T_{k}:(x,y)\mapsto(x+y+kf(x), y+kf(x))$, with $x\in\mathbb{R}/2\pi\mathbb{Z}$, $y\in\mathbb{R}$, $f$ is a periodic, real analytic function. $k\in\mathbb{R}^{\geq0}$ is a parameter.

We now consider a curve $(x(\theta,k),y(\theta,k))$ such that

$T_{k}(x(\theta,k),y(\theta,k))=(x(\theta+\omega,k),y(\theta+\omega,k))$

with $\omega\in(0,2\pi)$ fixed.

Assume we have a real analytic, $2\pi$-periodic (in $\theta$) function $u(\theta,k)$ such that

$y(\theta,k)=u(\theta,k)+\omega-u(\theta-\omega,k)$

Then the claim is that one can obtain the finite difference equation for $u$:

$D^{2}u-kf(\theta+u)=0$

where $D$ denotes the operator $Du(\theta)=u(\theta+\omega/2)-u(\theta-\omega/2)$.

I saw the Wikipedia article on finite differences but there only the way is discussed, how to solve a given ODE using finite differences method. Cannot construct anything "backwards" for my case.

Thank you in advance!