Limit Distribution of Averages over Unstable Periodic Orbits Forming Chaotic Attractor

TL;DR

This paper investigates whether averages over unstable periodic orbits in chaotic attractors converge to a limit distribution, concluding that they do, with standard deviations decreasing as 1/sqrt(N), confirming the representativeness of long orbits.

Contribution

The study proves that the limit distribution of averages over unstable periodic orbits is a delta-function, refuting previous hypotheses of a non-trivial distribution.

Findings

Limit distribution is a delta-function for large N.

Standard deviations decay as 1/sqrt(N).

Supports the representativeness of long unstable periodic orbits.

Abstract

We address the question of representativeness of a single long unstable periodic orbit for properties of the chaotic attractor it is embedded in. Y. Saiki and M. Yamada [Phys. Rev. E 79, 015201(R) (2009)] have recently suggested the hypothesis that there exist a limit distribution of averages over unstable periodic orbits with given number of loops, N, which is not a Dirac delta-function for infinitely long orbits. In this paper we show that the limit distribution is actually a delta-function and standard deviations decay as 1/sqrt(N) for large enough N.