Reversible perturbations of conservative H\'enon-like maps

TL;DR

This paper investigates smooth perturbations of reversible, area-preserving Hénon-like maps that break their conservative nature, using two methods, and studies the resulting symmetry-breaking bifurcations in various Hénon map families.

Contribution

It introduces two methods for constructing perturbations that preserve reversibility but break conservativity in Hénon-like maps and analyzes the resulting bifurcations.

Findings

Constructed perturbations that break conservativity while preserving reversibility.

Analyzed symmetry-breaking bifurcations in quadratic Hénon maps.

Studied bifurcations in both orientable and nonorientable cases.

Abstract

For area-preserving H\'enon-like maps and their compositions, we consider smooth perturbations that keep the reversibility of the initial maps but destroy their conservativity. For constructing such perturbations, we use two methods, the original method based on reversible properties of maps written in the so-called cross-form, and the classical Quispel-Roberts method based on a variation of involutions of the initial map. We study symmetry breaking bifurcations of symmetric periodic points in reversible families containing quadratic conservative orientable and nonorientable H\'enon maps as well as the product of two asymmetric H\'enon maps (with the Jacobians $b$ and $b^{-1}$).