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

Abstract

We consider reversible nonconservative perturbations of the conservative cubic Hénon maps $H^{\pm}_3: \bar x=y, \bar y=−x+M_1+M_2 y \pm y^3$ and study their influence on the 1:3 resonance, i. e., bifurcations of fixed points with eigenvalues $e^{±i2π/3}$. 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 $H^\pm_3$ with $M_1=0$. .