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

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 1:3 resonance, i. e., bifurcations fixed points with eigenvalues $$e^{\pm i2\pi/3}$$ . It follows from [1] that this resonance is degenerate for $$M_{1}=0,M_{2}=-1$$ when corresponding point elliptic. show under give rise to four 3-periodic orbits, two them are symmetric (saddles in case map $$H_{3}^{+}$$ elliptic orbits $$H_{3}^{-}$$ ), other nonsymmetric they compose couples dissipative (attracting repelling saddles Jacobians less than 1 greater ). these local symmetry-breaking can lead mixed dynamics due accompanying global nontransversal homo- heteroclinic cycles. also generalize results $$p:q$$ resonances odd $$q$$ all $$H_{3}^{\pm}$$ $$M_{1}=0$$