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^±:x̅=y,y̅=-x+M_1+M_2y± 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_3^± with M_1=0 .