Scenarios for the Creation of Hyperchaotic Attractors with Three Positive Lyapunov Exponents

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 three-dimensional Kaneko endomorphism and a four-dimensional Hénon map.