Stable difference equations associated with elementary cellular automata (Q5944158)

It is considered a cellular automaton described by \[ X^{t+ 1}_i= F(X^t_{i-1}, X^t_i, X^t_{i+1}),\quad t\geq 0,\quad i\in\mathbb{Z}, \] where \(F\) is a Boolean function called update rule. There are \(2^{2^3}= 256\) update rules and here Rule 90 is considered, defined by a certain combination of the values of \(F\). Starting from the problem of associating a difference equation corresponding to the Rule 90 cellular automaton i.e. of extending the Boolean function to a real function it is obtained a piecewise linear difference equation that is proper and stable.