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Scenarios for the creation of hyperchaotic attractors with three positive Lyapunov exponents

Regular and Chaotic Dynamics. 2025. Vol. 30. No. 2. P. 306–324.
Karatetskaia E., Aikan Shykhmamedov, Konstantin Soldatkin, Kazakov A.

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,eiϕ,e−iϕ). The proposed scenarios are illustrated by examples of the three-dimensional Kaneko endomorphism and a four-dimensional Hénon map.

Research target: Mathematics
Language: English
Full text
DOI
Keywords: показатели ЛяпуновабифуркацииLyapunov exponentsОтображение ЭноHénon mapHyperchaotic attractorгиперхаотический аттракторbifurcation analysis Kaneko endomorphismтрехмерные эндоморфизмы
Publication based on the results of:
Differential-geometric properties of dynamical systems and mathematical modeling of processes (2025)
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