A Geometric Approach to the Global Attractor Conjecture
A Geometric Approach to the Global Attractor Conjecture
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DOI：
10.1137/130928170
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发表时间：
2014-01-01
影响因子：
2.1
通讯作者：
Shiu, Anne
中科院分区：
文献类型：
--
作者：
Gopalkrishnan, Manoj;Miller, Ezra;Shiu, Anne
本文介绍了一类强内趋网络，Craciun，Nazarov和Pantea介绍的内趋网络的一个子类。主要结果表明，对于强内趋的复平衡系统，全局吸引子猜想成立：每个具有正初始条件的轨迹收敛到守恒律所允许的唯一正平衡点。这扩展了安德森最近的一个结果的系统中的反应图只有一个链接类（连接组件）。这里的结果证明使用微分包含，设置，包括幂律系统。的关键思想包括反应动力学的角度来看，组合几何的反应图，投影参数，使分析一个给定的系统中的系统具有较低的维度，和伯奇定理的扩展，一个著名的结果有关的交叉点的仿射子空间与流形参数化单项式。
This paper introduces the class of strongly endotactic networks, a subclass of the endotactic networks introduced by Craciun, Nazarov, and Pantea. The main result states that the global attractor conjecture holds for complex-balanced systems that are strongly endotactic: every trajectory with positive initial condition converges to the unique positive equilibrium allowed by conservation laws. This extends a recent result by Anderson for systems where the reaction diagram has only one linkage class (connected component). The results here are proved using differential inclusions, a setting that includes power-law systems. The key ideas include a perspective on reaction kinetics in terms of combinatorial geometry of reaction diagrams, a projection argument that enables analysis of a given system in terms of systems with lower dimension, and an extension of Birch's theorem, a well-known result about intersections of affine subspaces with manifolds parameterized by monomials.