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On 1:3 Resonance Under Reversible Perturbations of Conservative Cubic Hénon Maps
—We consider reversible nonconservative perturbations of the conservative cubic H´enon maps H± 3 : ¯x = y, y¯ = −x + M1 + M2y ± y3 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 M1 = 0, M2 = −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 M1 = 0.