Siegel disk for complexified Henon map

TL;DR

This paper demonstrates that critical phenomena related to Siegel disks in 1D complex maps also occur in 2D complex Hénon maps, introducing a numerical method for estimating the Siegel disk's scaling center in multi-dimensional invertible maps.

Contribution

It shows the persistence of Siegel disk phenomena in 2D maps and develops a new numerical method for locating the Siegel disk's scaling center in multi-dimensional invertible maps.

Findings

Siegel disk phenomena survive in 2D complex Hénon maps.

A specialized numerical method for estimating the Siegel disk center is introduced.

The method generalizes the 1D extremum estimation to multi-dimensional maps.

Abstract

It is shown that critical phenomena associated with Siegel disk, intrinsic to 1D complex analytical maps, survives in 2D complex invertible dissipative H\'{e}non map. Special numerical method of estimation of the Siegel disk scaling center position (for 1D maps it corresponds to extremum) for multi-dimensional invertible maps are developed.