Aikan Shykhmamedov

We study hyperchaotic attractors characterized by three positive Lyapunov exponents in numerical experiments. In order to possess this property, periodic orbits belonging to the attractor should have a three-dimensional unstable invariant manifold. Starting with a stable fixed point we describe several bifurcation scenarios that create such periodic orbits inside the attractor. These scenarios include cascades of alternating period-doubling and Neimark – Sacker bifurcations which, as we show, naturally appear near the cascade of codimension-2 period-doubling bifurcations, when periodic orbits along the cascade have multipliers $(-1, e^{i \phi}, e^{-i \phi})$. The proposed scenarios are illustrated by examples of the threedimensional Kaneko endomorphism and a four-dimensional Hénon map.

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 $e^{\pm i 2\pi/3}$. 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 $M_1=0$.