For a given rational map f : Ĉ→ Ĉ, the Julia set consists of those points in Ĉ around which the dynamics of the map is chaotic (a notion that can be defined rigorously), while the Fatou set is defined as the complement. The Fatou set, where the dynamics is well-behaved, is an open set, and one can classify its periodic connected components into five well-understood categories. This classificati...

The topological recurrence algorithm provides a fast and robust method for detecting the presence of unstable periodic orbits (UPO's) in short, noisy experimental data files. We present here a technique for improving this method by using a matrix fitting algorithm to extract dynamical information about the system from these UPO's. This method greatly increases the sensitivity of the algorithm, ...

This is an introduction to the algebraic theory of periodic arithmetical maps. The topic is connected with number theory, combinatorics, algebra and analysis. 1. Various Backgrounds I. Combinatorial Background. For n ∈ Z = {1, 2, 3, · · · } and a ∈ R(n) = {0, 1, · · · , n− 1}, let a(n) = a+ nZ = {x ∈ Z : x ≡ a (mod n)} = {· · · , a− n, a, a+ n, · · · } and call it a residue class with modulus n...

Abstract— In this paper, the generalized analysis and novel application of the multiple periodic (MP) structure are proposed. Both transmission and radiation performances of one dimensional MP structures are studied. The dispersion relations are analyzed from both layered media (distributive) and lumped circuits aspects. Regarding each aspect, both non-dispersive (conventional) and dispersive (...

We report on the existence of stable, completely delocalized response regimes in a nonlinear defective periodic structure. In this state of complete delocalization, despite the presence of the defect, the system exhibits in-phase oscillation of all units with the same amplitude. This elimination of defect-borne localization may occur in both the free and forced responses of the system. In the a...

This study concerns the properties of physical systems that can spontaneously form non-periodic structures. It focuses on the Socolar-Taylor tiling model in which a single space-filling prototile forces a limit-periodic pattern–a state made up of the union of infinite levels of periodic structures with ever-increasing sizes–through local interaction rules. A two-dimensional lattice model, which...

Heat transfer by electromagnetic radiation is one of the common methods of energy transfer between objects. Using the fluctuation-dissipation theorem, we have studied the effect of particle arrangement in the transmission of radiative heat in many-body systems. In order to show the effect of the structure morphology on the collective properties, the radiative heat transfer is studied and the re...

Recent work has identified nonlinear deterministic structure in neuronal dynamics using periodic orbit theory. Troublesome in this work were the significant periods of time where no periodic orbits were extracted — “dynamically dark” regions. Tests for periodic orbit structure typically require that the underlying dynamics are differentiable. Since continuity of a mathematical function is a nec...