Hyperchaotic attractors of three-dimensional maps and scenarios of their appearance

We study bifurcation mechanisms of the appearance of hyperchaotic attractors in three-dimensional maps. We consider, in some sense, the simplest cases when such attractors are homoclinic, i.e. they contain only one saddle fixed point and entirely its unstable manifold. We assume that this manifold is two-dimensional, which gives, formally, a possibility to obtain two positive Lyapunov exponents for typical orbits on the attractor (hyperchaos). For realization of this possibility, we propose several bifurcation scenarios of the onset of homoclinic hyperchaos that include cascades of both supercritical period-doubling bifurcations with saddle periodic orbits and supercritical Neimark-Sacker bifurcations with stable periodic orbits, as well as various combinations of these cascades. In the paper, these scenarios are illustrated by an example of three-dimensional Mir\'a map.Comment: 40 pages, 24 figure

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

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 $M_1=0$.