Convergence Time Towards Periodic Orbits in Discrete Dynamical Systems

TL;DR

This paper analyzes how quickly points in discrete dynamical systems tend to converge to periodic orbits, providing theoretical insights, algorithms, and convergence probabilities.

Contribution

It introduces a theorem characterizing phase space regions leading to periodic orbits and develops algorithms for practical implementation.

Findings

Derived bounds on convergence times to periodic orbits

Established probability estimates for convergence within fixed iterations

Provided algorithms for applying theoretical results in practice

Abstract

We investigate the convergence towards periodic orbits in discrete dynamical systems. We examine the probability that a randomly chosen point converges to a particular neighborhood of a periodic orbit in a fixed number of iterations, and we use linearized equations to examine the evolution near that neighborhood. The underlying idea is that points of stable periodic orbit are associated with intervals. We state and prove a theorem that details what regions of phase space are mapped into these intervals (once they are known) and how many iterations are required to get there. We also construct algorithms that allow our theoretical results to be implemented successfully in practice.