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.
Keywords:
hyperchaos, Hénon-like map, Lyapunov exponents
Citation:
Karatetskaia E., Shykhmamedov A., Soldatkin K., Kazakov A. O., Scenarios for the Creation of Hyperchaotic Attractors with Three Positive Lyapunov Exponents, Regular and Chaotic Dynamics,
2025, Volume 30, Number 2,
pp. 306-324