For a certain type of discrete--time nonlinear consensus dynamics, asymptotically stable periodic orbits are constructed. Based on a simple ordinal pattern assumption, the Frucht graph, two Petersen septets, hypercubes, a technical class of circulant graphs (containing Paley graphs of prime order), and complete graphs are considered -- they are all carrying moving average monotone dynamics admitting asymptotically stable periodic orbits with period 2. Carried by a directed graph with 594 (multiple and multiple loop) edges on 3 vertices, also the existence of asymptotically stable r-periodic orbits, r=3,4,... is shown.

The weak and strong ergodic properties of the positive solutions of nonautonomous linear ordinary differential equations will be considered. We will show that if the coefficient matrix function is bounded, essentially nonnegative and uniformly irreducible, then the normalized positive solutions are asymptotically equivalent to the Perron vectors of the strongly positive transition matrix at infinity (weak ergodicity). If, in addition, the coefficient matrix function is uniformly continuous, then the convergence of the normalized positive solutions to the same strongly positive limiting vector (strong ergodicity) is equivalent to the convergence of the Perron vectors of the coefficient matrices. 

This is a joint work with Professor Christian Pötzsche (Alpen-Adria University Klagenfurt, Austria).