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Title: | On 1:3 Resonance Under Reversible Perturbations of Conservative Cubic Hénon Maps |
Author: | Gonchenko, Marina Kazakov, Alexey O. Samylina, Evgeniya A. Shykhmamedov, Aikan |
Keywords: | Teoria de la bifurcació Sistemes dinàmics diferenciables Teoria ergòdica Sistemes dinàmics diferenciables Bifurcation theory Differentiable dynamical systems Ergodic theory Differentiable dynamical systems |
Issue Date: | 16-Feb-2022 |
Publisher: | Pleiades Publishing |
Abstract: | We consider reversible nonconservative perturbations of the conservative cubic Hénon maps $H^{\pm}_3: \bar x=y, \bar y=−x+M_1+M_2 y \pm y^3$ and study their influence on the 1:3 resonance, i. e., bifurcations of fixed points with eigenvalues $e^{±i2π/3}$. 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 $H^\pm_3$ with $M_1=0$. . |
Note: | Versió postprint del document publicat a: https://doi.org/10.1134/S1560354722020058 |
It is part of: | Regular and Chaotic Dynamics, 2022, vol. 27, num. 2, p. 198-216 |
URI: | http://hdl.handle.net/2445/194357 |
Related resource: | https://doi.org/10.1134/S1560354722020058 |
ISSN: | 1560-3547 |
Appears in Collections: | Articles publicats en revistes (Matemàtiques i Informàtica) |
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