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

*2022, Volume 27, Number 2, pp. 198-216*

Author(s):

**Gonchenko M. S., Kazakov A. O., Samylina E. A., Shykhmamedov A.**

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 $M_1=0$.

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