微分正性的算子理论方法
An operator-theoretic approach to differential positivity
摘要：
微分正系统是沿轨迹线性化为正的系统。在适当的假设下，它们的解渐近收敛到一个一维吸引子，如果极限集中没有不动点，则该吸引子是一个极限环。本文研究了微分正系统的（几何）性质与Koopman算子的（谱）性质之间的一般联系。特别是，我们得到了相反的结果，微分的积极性，例如，任何双曲极限环是微分积极的吸引区域。我们还提供了一个收缩锥场的建设。
英文摘要：
Differentially positive systems are systems whose linearization along trajectories is positive. Under mild assumptions, their solutions asymptotically converge to a one-dimensional attractor, which must be a limit cycle in the absence of fixed points in the limit set. In this paper, we investigate the general connections between the (geometric) properties of differentially positive systems and the (spectral) properties of the Koopman operator. In particular, we obtain converse results for differential positivity, showing for instance that any hyperbolic limit cycle is differentially positive in its basin of attraction. We also provide the construction of a contracting cone field.
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