Combinatorial approaches to Hopf bifurcations in systems of interacting elements
Combinatorial approaches to Hopf bifurcations in systems of interacting elements
复制标题
相互作用元素系统中 Hopf 分岔的组合方法
DOI：
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
Casian Pantea
中科院分区：
文献类型：
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作者：
D. Angeli;M. Banaji;Casian Pantea
我们描述了解决实矩阵族是否允许通过虚轴的非实特征值对的问题的组合方法。当在动力系统研究中矩阵作为雅可比矩阵出现时，这些条件为此类系统的参数化族中发生 Hopf 分岔提供了必要条件。这些技术取决于加性化合物矩阵的光谱特性：特别是，我们将矩阵的乘积与一个带符号、标记的有向图（称为 DSR^[2] 图）关联起来，该有向图对有关该乘积的第二个加性化合物的信息进行编码。该有向图的循环结构的条件被证明排除了具有正实部的非实特征值的可能性。所开发的技术应用于被称为“相互作用网络”的相互作用元素系统，其中化学反应网络是一个特例。
We describe combinatorial approaches to the question of whether families of real matrices admit pairs of nonreal eigenvalues passing through the imaginary axis. When the matrices arise as Jacobian matrices in the study of dynamical systems, these conditions provide necessary conditions for Hopf bifurcations to occur in parameterised families of such systems. The techniques depend on the spectral properties of additive compound matrices: in particular, we associate with a product of matrices a signed, labelled digraph termed a DSR^[2] graph, which encodes information about the second additive compound of this product. A condition on the cycle structure of this digraph is shown to rule out the possibility of nonreal eigenvalues with positive real part. The techniques developed are applied to systems of interacting elements termed "interaction networks", of which networks of chemical reactions are a special case.