Correlational properties of two-dimensional solvable chaos on the unit circle

TL;DR

This paper analyzes the correlational properties of two-dimensional chaotic maps on the unit circle, providing analytical forms of covariances and characteristic functions involving Bessel functions, revealing non-positive covariances and asymmetries.

Contribution

It introduces analytical expressions for higher-order covariances and characteristic functions of 2D chaotic maps on the circle, highlighting their unique correlation structures.

Findings

Higher-order covariances are non-positive.

Characteristic functions involve three types of 2D Bessel functions.

Asymmetries between cosine and sine functions are explained.

Abstract

This article investigates correlational properties of two-dimensional chaotic maps on the unit circle. We give analytical forms of higher-order covariances. We derive the characteristic function of their simultaneous and lagged ergodic densities. We found that these characteristic functions are described by three types of two-dimensional Bessel functions. Higher-order covariances between x and y and those between y and y show non-positive values. Asymmetric features between cosine and sine functions are elucidated.