I would like to understand the following example based on the following definition
Definition: Alternatively, one can choose to update the states of the variables according to some fixed update order, for example, a permutation or a word on the set $\{1,\dots, n\}$. That is, let $\pi=(\pi_1,\dots,\pi_t)$ be a word using the alphabet $\{1,\dots, n\}$. The function \begin{equation}\label{seq-fun} \Phi_\pi= f_{\pi_t}\circ f_{\pi_{n-1}} \circ\cdots \circ f_{\pi_1} \;, \end{equation} is called a sequential dynamical system (SDS) and, as before, the dynamics of $\Phi_\pi$ is generated by iteration. The case when $\pi$ is a permutation on $\{1,\dots, n\}$ has been studied extensively. It is clear that using a different permutation $\gamma$ may result in a different dynamical system $\Phi_\gamma$.
Example:
Let $X=\{0;1\}$. Suppose we have four variables and the local Boolean update functions are :
\begin{align*}
f_{1}&=x_{1}+x_{2}+x_{3}+x_{4},\\
f_{2}&=x_{1}+x_{2},\\
f_{3}&=x_{1}+x_{3},\\
f_{4}&=x_{1}+x_{4},
\end{align*}
where “+” represents XOR, the exclusive OR function. and let $\pi=(2,1,3,4)$.Then The dynamics of the function
$$
\Phi_\pi = f_4 \circ f_3 \circ f_1 \circ f_2 : X^4 \longrightarrow
X^4.
$$
The phase space of $\Phi_\pi$ is the directed graph on the following Figure,
while the phase space of $\Phi_\gamma$,
where $\gamma = id$ is on the following Figure
My question:
Could someone elaborate on the calculation behind that example that let me goes :
from $0000$ to $0000$.
from $0001$ to $1010$.
from $1010$ to $1101$.
from $1010$ to $0001$.
I think at least one detail is left out here. The explanation describes each $f_i$ as a function from $X^4$ to $X^4$, but the definitions of the $f_i$ are single-valued. My best guess is that what is meant by “local update function” is that each $f_i$ modifies $x_i$ and leaves the others alone:
\begin{align*} f_{1}(x_1,x_2,x_3,x_4)&=(x_{1}+x_{2}+x_{3}+x_{4},x_2,x_3,x_4)\\ f_{2}(x_1,x_2,x_3,x_4)&=(x_1,x_{1}+x_{2},x_3,x_4)\\ f_{3}(x_1,x_2,x_3,x_4)&=(x_1,x_2,x_{1}+x_{3},x_4)\\ f_{4}(x_1,x_2,x_3,x_4)&=(x_1,x_2,x_3,x_{1}+x_{4}) \end{align*}
For $0000\to0000$: Before the updates, each $x_i=0$. For each $i$, $f_i(0,0,0,0)=(0,0,0,0)$, so the result of sequentially applying the $f_i$ in any order is $(0,0,0,0)$.
The transition $0001\to1010$ in the first figure means that $f_4 \circ f_3 \circ f_1 \circ f_2 (0,0,0,1) = (1,0,1,0)$.
Working this out, $f_4 \circ f_3 \circ f_1 \circ f_2 (0,0,0,1) = f_4 \circ f_3 \circ f_1(0,0+0=0,0,1) = f_4 \circ f_3(0+0+0+1=1,0,0,1) = f_4(1,0,1+0=1,1)=(1,0,1,1+1=0)$, which agrees with the figure.
I didn't check all the transitions, but add a comment if this doesn't let you answer your question.