For $C^{1}$-smooth strongly monotone discrete-time dynamical systems, it is shown that “convergence to linearly stable cycles” a prevalent asymptotic behavior in the measure-theoretic sense. The results are then applied several classes of time-periodic parabolic equations and obtain prevalence convergence periodic solutions.

While the notion of chaos is well established for dynamical systems on manifolds, it is not so for dynamical systems over discrete spaces with N variables, as binary neural networks and cellular automata. The main difficulty is the choice of a suitable topology to study the limit N → ∞. By embedding the discrete phase space into a Cantor set we provided a natural setting to define topological e...

In this paper we study phase space relations in a certain class of discrete dynamical systems over graphs. The systems we investigate are called Sequential Dynamical Systems (SDSs), which are a class of dynamical systems that provide a framework for analyzing computer simulations. Specifically, an SDS consists of (i) a finite undirected graph Y with vertex set {1, 2, . . . , n} where each verte...

Let X denote a locally compact metric space and φ : X → X be a continuous map. In the 1970s L. Zadeh presented an extension principle, helping us to fuzzify the dynamical system (X, φ), i.e., to obtain a map Φ for the space of fuzzy sets on X. We extend an idea mentioned in [P. Diamond, A. Pokrovskii, Chaos, entropy and a generalized extension principle, Fuzzy Sets and Systems 61 (1994)] and we...