论文标题
平衡光谱类型的代数几何分类
An algebro-geometric classification of spectral types of equilibria
论文作者
论文摘要
我们给出了三个代数方程式，允许给定$ m $维的所有光谱类型的平衡类型的几何分类，并在维度3和4中彻底分析它们。这些方程定义的基因座对应于确定的分叉类型。这种基因座的补体给出了在空的域中不变空间的几何分解，在该域中，平衡具有给定的光谱类型。这种方法的有用性是一个事实，即在处理参数依赖性动力学系统时，位点的回扣从不变空间到参数空间可为手动动力学系统的参数空间的分叉分解。我们还提供有效的方法来明确计算光谱指数。
We give three algebraic equations which allow a geometric classification of all spectral types of equilibria of a given $m$-dimensional dynamical system, and we analyse them thoroughly in dimension 3 and 4. The loci defined by these equations correspond to definite types of bifurcations. The complement of such loci give a geometric decomposition of the space of invariants in open domains in which the equilibrium has a given spectral types. The usefulness of this approach lays in the fact that, when dealing with a parameter-dependent dynamical system, the pull-back of the loci from the space of invariants to the parameter space gives the bifurcation-decomposition of parameter space for the dynamical system at hand. We also give effective methods to explicitly compute the spectral indices.
