Shearless barriers in the conservative Ikeda map

TL;DR

This paper studies shearless barriers in the conservative Ikeda map, revealing how they form, persist, and break up as system parameters change, contributing to understanding of invariant structures in area-preserving maps.

Contribution

It introduces a detailed analysis of shearless barriers in the conservative Ikeda map, highlighting their emergence and stability in a nonintegrable setting.

Findings

Shearless barriers form at extrema of the rotation number profile.

Barriers persist over a range of parameters before breaking up.

The map reduces to a uniform rotation in the integrable case.

Abstract

We investigate the dynamics of the Ikeda map in the conservative limit, where it is represented as a two-dimensional area-preserving map governed by two control parameters, $\theta$ and $\phi$. We demonstrate that the map can be interpreted as a composition of a rotation and a translation of the state vector. In the integrable case ($\phi = 0$), the map reduces to a uniform rotation by angle $\theta$ about a fixed point, independent of initial conditions. For $\phi \ne 0$, the system becomes nonintegrable, and the rotation angle acquires a coordinate dependence. The resulting rotation number profile exhibits extrema as a function of position, indicating the formation of shearless barriers. We analyze the emergence, persistence, and breakup of these barriers as the control parameters vary.