论文标题
我的ODES系统是$ K $合作的吗？
Is my system of ODEs $k$-cooperative?
论文作者
论文摘要
如果线性动力学系统的流动映射到自身的非负轨道，则称为阳性。更确切地说，它将零符号变化的向量集映射到自身。线性动力学系统称为$ k $阳性，如果其流量映射到自身最多$ k-1 $符号变化的向量。 非线性动力学系统称为$ k $合作，如果其变分系统（这是一个时变的线性动力系统）为$ k $ prositive。这些系统具有特殊的渐近特性。例如，最近显示，强烈的2美元合作系统满足了强大的Poincaré-Bendixson物业。阳性和〜$ k $ - 积极性很容易根据动力学中矩阵的标志图案验证。但是，在坐标转换下，这些标志条件并不是不变的。一个自然的问题是确定给定的〜$ n $二维系统是否为$ k $阳性，直至坐标转换。我们研究了两个特殊类型的转换的问题：通过签名矩阵进行排列和缩放。对于任何$ n \ geq 4 $和〜$ k \ in \ {2，\ dots，n-2 \} $，我们为$ k $ - 稳定性提供了图形理论的必要条件和充分的条件，以实现此类坐标转换。我们描述了我们的结果应用于特定类别的Lotka-Volterra系统。
A linear dynamical system is called positive if its flow maps the non-negative orthant to itself. More precisely, it maps the set of vectors with zero sign variations to itself. A linear dynamical system is called $k$-positive if its flow maps the set of vectors with up to $k-1$ sign variations to itself. A nonlinear dynamical system is called $k$-cooperative if its variational system, which is a time-varying linear dynamical system, is $k$-positive. These systems have special asymptotic properties. For example, it was recently shown that strongly $2$-cooperative systems satisfy a strong Poincaré-Bendixson property. Positivity and~$k$-positivity are easy to verify in terms of the sign-pattern of the matrix in the dynamics. However, these sign conditions are not invariant under a coordinate transformation. A natural question is to determine if a given~$n$-dimensional system is $k$-positive up to a coordinate transformation. We study this problem for two special kinds of transformations: permutations and scaling by a signature matrix. For any $n\geq 4$ and~$k\in\{2,\dots, n-2\}$, we provide a graph-theoretical necessary and sufficient condition for $k$-positivity up to such coordinate transformations. We describe an application of our results to a specific class of Lotka-Volterra systems.
