Konstantin Soldatkin

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.