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<p>Differentially positive systems are nonlinear systems whose linearization along trajectories preserves a cone field on a smooth Riemannian manifold. One of the embryonic forms of cone fields originates from general relativity. By utilizing the Perron–Frobenius vector field (which generalize the Perron–Frobenius theory to a differential framework) and the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3203_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Gamma \)</EquationSource> </InlineEquation>-invariance of cone field (which arises from the homogeneous structure of a manifold), we show that generic (i.e.,“almost all” in the topological sense) orbits are convergent to certain single equilibrium. This solved a reduced version of Forni-Sepulchre’s conjecture in 2016 for globally orderable manifolds.</p>

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Generic behavior of differentially positive systems on a globally orderable manifold

  • 期刊论文

  • 发表日期：

    2025年7月8日

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  • Lin Niu,
  • Yi Wang

摘要

Differentially positive systems are nonlinear systems whose linearization along trajectories preserves a cone field on a smooth Riemannian manifold. One of the embryonic forms of cone fields originates from general relativity. By utilizing the Perron–Frobenius vector field (which generalize the Perron–Frobenius theory to a differential framework) and the \(\Gamma \) -invariance of cone field (which arises from the homogeneous structure of a manifold), we show that generic (i.e.,“almost all” in the topological sense) orbits are convergent to certain single equilibrium. This solved a reduced version of Forni-Sepulchre’s conjecture in 2016 for globally orderable manifolds.