Reversible perturbations of conservative Hénon-like maps

Marina Gonchenko; Sergey Gonchenko; Klim Safonov

doi:10.3934/dcds.2020343

Article Contents

2021, Volume 41, Issue 4: 1875-1895. Doi: 10.3934/dcds.2020343

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Reversible perturbations of conservative Hénon-like maps

1.
Universitat Politècnica de Catalunya, Barcelona, Spain
2.
Mathematical Center "Mathematics of Future Technologies", Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, Russia
3.
Laboratory of Dynamical Systems and Applications, National Research University Higher School of Economics, Nizhny Novgorod, Russia

Received: April 30, 2020

Revised: July 31, 2020

Early access: October 2020

Published: April 2021

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Abstract

For area-preserving Hénon-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, a new 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 orbits in reversible families containing quadratic conservative orientable and nonorientable Hénon maps as well as a product of two Hénon maps whose Jacobians are mutually inverse.

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Mathematics Subject Classification: Primary: 37G25, 37G40; Secondary: 37C70.

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Figure 1. Elements of the bifurcation diagram in the $ (b,M) $-parameter plane for the maps (a) $ T_2 $ and (b) $ T_{2\mu} $ with small fixed $ \mu $

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Figure 2. Main bifurcations in maps (29) and (30) when $ \mu $ is small and fixed and $ M $ changes. The points $ Q_1 $ and $ Q_2 $ are symmetric elliptic 2-periodic orbits, while the points $ S_1 $ and $ S_2 $ are nonorientable saddle fixed points that compose a symmetric couple of points. The points $ S_1 $ and $ S_2 $ are conservative with the Jacobian $ -1 $ for $ \mu = 0 $ and non-conservative with the Jacobians $ -1<J_1<0 $ and $ J_2<-1 $, respectively, for $ \mu>0 $

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Figure 3. A schematic tree for the bifurcation scenario of the appearance of a symmetric couple of 6-periodic orbits in the third power of Hénon map $ H_{1+} $

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Figure 4. Phase portraits of 3- and 6-periodic orbits for the Hénon map $ H_{+1} $: the bottom plots are magnifications of some important details of the top plots

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