Functioning of discrete dynamic circulant-type system with threshold functions

The functioning of discrete dynamic circulant-type systems with threshold functions is studied. The general properties of the functional graph of a system are described. In binary case, all states of the system are classified according to the length of $0$-series and $1$-series. As a result, some properties of cycles in the functional graph and a lower estimate for the number of connected components are given. For an arbitrary value $p$, a criterion for the existence of stable states in the system is given, the forms and the number of these states are determined.