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

Abstract

We consider reversible nonconservative perturbations of the conservative cubic Hénon maps \(H_{3}^{\pm}:\bar{x}=y,\bar{y}=-x+M_{1}+M_{2}y\pm y^{3}\) and study their influence on the 1:3 resonance, i. e., bifurcations of fixed points with eigenvalues \(e^{\pm i2\pi/3}\). 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 \(M_{1}=0\).