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

Gonchenko, Marina ; Kazakov, Alexey O ; Samylina, Evgeniya A ; Shykhmamedov, Aikan

Pleiades Publishing 2022-02

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  • Título:
    On 1:3 Resonance Under Reversible Perturbations of Conservative Cubic Hénon Maps
  • Autor: Gonchenko, Marina ; Kazakov, Alexey O ; Samylina, Evgeniya A ; Shykhmamedov, Aikan
  • Assuntos: Bifurcation theory ; Differentiable dynamical systems ; Ergodic theory ; Sistemes dinàmics diferenciables ; Teoria de la bifurcació ; Teoria ergòdica
  • Notas: Versió postprint del document publicat a: https://doi.org/10.1134/S1560354722020058
    1560-3547
    https://doi.org/10.1134/S1560354722020058
    730958
    http://hdl.handle.net/2445/194357
    Articles publicats en revistes (Matemàtiques i Informàtica)
    Regular and Chaotic Dynamics, 2022, vol. 27, num. 2, p. 198-216
  • Descrição: 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 $M_1=0$. .
  • Editor: Pleiades Publishing
  • Data de criação/publicação: 2022-02
  • Idioma: Inglês
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  • Realização: ABCD-USP USP   Logos de Redes Sociais: Freepik

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