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Regular and Chaotic Dynamics, 2022, Volume 27, Issue 2, Pages 198–216
DOI: https://doi.org/10.1134/S1560354722020058
(Mi rcd1160)
 

This article is cited in 1 scientific paper (total in 1 paper)

Alexey Borisov Memorial Volume

On 1:3 Resonance Under Reversible Perturbations of Conservative Cubic Hénon Maps

Marina S. Gonchenkoa, Alexey O. Kazakovb, Evgeniya A. Samylinabc, Aikan Shykhmamedovb

a Universitat de Barcelona, Gran Via de les Corts Catalanes, 585, 08007 Barcelona, Spain
b National Research University Higher School of Economics, ul. Bolshaya Pecherskaya 25/12, 603155 Nizhny Novgorod, Russia
c Lobachevsky State University of Nizhny Novgorod, pr. Gagarina 23, 603950 Nizhny Novgorod, Russia
Citations (1)
References:
Abstract: We consider reversible nonconservative perturbations of the conservative cubic Hénon maps $H_3^{\pm}: \bar x = y, \bar y = -x + M_1 + M_2 y \pm y^3$ and study their influence on the 1:3 resonance, i. e., bifurcations of fixed points with eigenvalues $e^{\pm i 2\pi/3}$. It follows from [1] that this resonance is degenerate for $M_1=0, M_2=-1$ when the corresponding fixed point is elliptic. We show that bifurcations of this point under reversible perturbations give rise to four 3-periodic orbits, two of them are symmetric and conservative (saddles in the case of map $H_3^+$ and elliptic orbits in the case of map $H_3^-$), the other two orbits are nonsymmetric and they compose symmetric couples of dissipative orbits (attracting and repelling orbits in the case of map $H_3^+$ and saddles with the Jacobians less than 1 and greater than 1 in the case of map $H_3^-$). We show that these local symmetry-breaking bifurcations can lead to mixed dynamics due to accompanying global reversible bifurcations of symmetric nontransversal homo- and heteroclinic cycles. We also generalize the results of [1] to the case of the $p:q$ resonances with odd $q$ and show that all of them are also degenerate for the maps $H_3^{\pm}$ with $M_1=0$.
Keywords: cubic Hénon map, reversible system, 1:3 resonance, homoclinic tangencies, mixed dynamics.
Funding agency Grant number
Russian Science Foundation 19-71-10048
19-11-00280
Ministry of Science and Higher Education of the Russian Federation 075-15-2019-1931
Federación Española de Enfermedades Raras PGC2018-098676-B-I00
Ministerio de Ciencia e Innovación de España IJCI-2016-29071
This paper was supported by the RSF grant No. 19-71-10048. Numerical experiments described in Section 7 were supported by the Laboratory of Dynamical Systems and Applications NRU HSE, of the Russian Ministry of Science and Higher Education (Grant No. 075-15-2019-1931). The work presented in Section 3 was supported by the RSF grant No. 19-11-00280. M. Gonchenko is partially supported by Juan de la Cierva-Incorporación fellowship IJCI-2016-29071 and the Spanish grant PGC2018-098676-B-I00 (AEI/FEDER/UE). A. Kazakov and E. Samylina also acknowledge the Theoretical Physics and Mathematics Advancement Foundation BASIS for financial support of scientific investigations.
Received: 22.10.2021
Accepted: 16.02.2022
Bibliographic databases:
Document Type: Article
MSC: 37G25, 37G35
Language: English
Citation: Marina S. Gonchenko, Alexey O. Kazakov, Evgeniya A. Samylina, Aikan Shykhmamedov, “On 1:3 Resonance Under Reversible Perturbations of Conservative Cubic Hénon Maps”, Regul. Chaotic Dyn., 27:2 (2022), 198–216
Citation in format AMSBIB
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\by Marina S. Gonchenko, Alexey O. Kazakov, Evgeniya A. Samylina, Aikan Shykhmamedov
\paper On 1:3 Resonance Under Reversible Perturbations
of Conservative Cubic Hénon Maps
\jour Regul. Chaotic Dyn.
\yr 2022
\vol 27
\issue 2
\pages 198--216
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