Aikan Shykhmamedov
Publications:
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. 198216
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 3periodic 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 symmetrybreaking
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 $M_1=0$.
