On 1:3 resonance under reversible perturbations of conservative cubic Hénon maps

Abstract

We consider reversible non-conservative 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 the work by Dullin and Meiss, 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 non-transversal homo- and heteroclinic cycles. We also generalize the results of Dullin and Meiss 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 $M_1=0$.

25 pages, 10 figures

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