Can we express binary functions in terms of differential equations with xor as differential operator?

Differential equations can be used to describe functions. Some famous ones include $$\frac{\partial f}{\partial x} - f(x) = 0$$ $$\frac{\partial^2 f}{\partial x^2} - f(x) = 0$$

The iterated running xor working on a sequence of numbers $$\{x_0,\cdots,x_N\}$$ on $$\mathbb Z_2$$ :

$$\mathcal{X}\{x\}_{k} = x_{k} \oplus x_{k-1}$$

carries some of similar properties as the differential operator with perhaps the most obvious one: constants are taken to $$0$$. A string of $$x_{k_1}=x_{k_1+1}=\cdots=x_{k_2}$$ will give $$\mathcal{X}\{x\}_{k} = 0$$ for all $$k\in\{k_1+1,k_2\}$$.

We can obviously model or represent (families of) continuous functions with differential equations.

Now to my question : how can we do the same with iterated xor operations?

"xor" corresponds to addition over the binary field $$\mathbb F_2$$. I suppose a natural generalization of differential equations in this context might be recurrence equations for sequences in $$\mathbb F_2$$. For example, the linear equation $$x_{n+1} = x_{n} + x_{n-5}$$ with initial conditions $$x_0 = 1$$, $$x_1 = \ldots = x_5 = 0$$ produces an interesting sequence with period $$63$$.