In this article, using orbits of the dynamical system generated by the function F, operator representations of commutation relations XX*=F(X*X) and AB=BF(A) are studied and used to investigate commuting operators expressed using polynomials in A and B. Various conditions on the function F, defining the commutation relations, are derived for monomials and polynomials in operators A and B to commute. These conditions are further studied for dynamical systems generated by affine and q-deformed power functions, and for the -shift dynamical system.