On iterating operators and on generalized periodic orbits

TL;DR

This paper introduces a generalized framework for iterative processes, including phenomena like Pomeau-Manneville and Feigenbaum scenarios, by extending concepts such as periodic orbits to more complex structures called periodic carousels.

Contribution

It proposes a new generalized approach to iterative processes, broadening the scope of classical dynamical systems analysis to include more complex orbit structures.

Findings

Generalized iterative processes encompass known bifurcation scenarios.

Introduction of periodic carousels as a broader concept than periodic orbits.

Potential for new scaling properties in complex dynamical systems.

Abstract

We try to define the more general form of iterative processes in which the Pomeau-Manneville and the Feigenbaum scenario may occur along with their specific scaling properties. Doing this we need to generalize other basic concepts. Thus, what we call a periodic carousel is a generalization of what is usually called a periodic orbit.