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Aikan Shykhmamedov


Gonchenko M. S., Kazakov A. O., Samylina E. A., Shykhmamedov A.
On 1:3 Resonance Under Reversible Perturbations of Conservative Cubic Hénon Maps
2022, vol. 27, no. 2, pp.  198-216
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
Citation: Gonchenko M. S., Kazakov A. O., Samylina E. A., Shykhmamedov A.,  On 1:3 Resonance Under Reversible Perturbations of Conservative Cubic Hénon Maps, Regular and Chaotic Dynamics, 2022, vol. 27, no. 2, pp. 198-216

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