Structural Causal Models with cycles I this context I am referring to Structural Causal Models (SCM) such as presented here https://fabiandablander.com/r/Causal-Inference.html or here https://medium.data4sci.com/causal-inference-part-iv-structural-causal-models-df10a83be580.
From the way, these Structural Causal Models are defined, I do not see any rules that forbid Structural Causal Models to have cycles. I could easily defined two structural equations $f_i$ and $f_j$ which contain each other as inputs/arguments to their function: $X_i := f_i(X_j, \ldots, U_i)$ and $X_j := f_j(X_i, \ldots, U_j)$.
Are Structural Causal Models allowed to by cyclic or do they need to be acyclic similar to Directed Acyclic Graphs (DAGs)?
If Structural Causal Models can have cycles, then how do Structural Causal Models deal with the infinite feedback loops of redefining/updating each other's values for $X_i$ and $X_j$?
I would appreciate theoretical and applied answers or any insights to this question.

Well, the fundamental rule of causality is that causes must precede effects - that is a strict inequality in time. So it is not permissible to have $X_i(t)=f_i(X_j(t),\dots,U_i(t)),$ but then turn around and have $X_j(t)=f_j(X_i(t),\dots,U_j(t)).$ But you can have feedback show up in subsequent moments in time: $X(t)=f(Y(t)),$ and then $Y(t+1)=g(X(t)).$ This is non-trivial to deal with in the DAG setting, but one way to begin to handle it is to treat $Y(t)$ and $Y(t+1)$ as separate nodes in the causal graph.
