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<p>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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11819_2025_7244_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="98" /> </InlineMediaObject> <EquationSource Format="TEX">\((-1,e^{i\phi},e^{-i\phi})\)</EquationSource> </InlineEquation>. The proposed scenarios are illustrated by examples of the three-dimensional Kaneko endomorphism and a four-dimensional Hénon map.</p>

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

  • 期刊论文

  • 发表日期：

    2025年4月7日

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  • Efrosiniia Karatetskaia,
  • Aikan Shykhmamedov,
  • Konstantin Soldatkin,
  • Alexey Kazakov

摘要

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.