The steady states and robustness of fuzzy discrete dynamic systems

Abstract

The steady states of a fuzzy discrete dynamic system correspond to invariants (eigenvectors) of the transition matrix of the system. The structure of the eigenspace of a given fuzzy matrix is considered for various max-T algebras, where T is some triangular norm (Godel, Eukasiewicz, product, drastic). A given transition fuzzy matrix is called (strongly) robust if for every starting vector of a fuzzy discrete dynamic system a multiplication of a power matrix with the starting vector produces a (greatest) eigenvector of the transition matrix. A transition matrix is called weakly robust if the only possibility to arrive at an eigenvector is to start of a fuzzy discrete dynamic system by a vector that is itself an eigenvector. We present characterizations of the eigenspace of a given transition matrix in various max-T algebras. Further results concern the robustness (weak, strong robustness) of a matrix and an interval matrix (matrix with inexact data). Polynomial algorithms for checking the equivalent conditions for (weak, strong) robustness of interval fuzzy matrices are presented.