论文标题
快速慢系统中的双曲线脐带奇异性
The hyperbolic umbilic singularity in fast-slow systems
论文作者
论文摘要
在快速变量中具有三个缓慢变量和梯度结构的快速慢系统，通常具有双曲线脐带，椭圆形的脐带或燕尾奇异性。在本文中，我们对双曲线脐带奇异性附近的快速慢系统进行了详细的本地分析。特别是，我们表明，在缓慢流动的某些适当的非分类条件下，吸引慢速的歧管跳到快速状态，并在跨越双曲线脐带奇异性时出来。该分析基于爆破技术，其中双曲线脐点被吹到5维球体。此外，还原的慢速流也被炸毁并嵌入吹入的快速配方中。此外，我们描述了我们的分析与诸如灾难理论和约束微分方程之类的古典理论有关。
Fast-slow systems with three slow variables and gradient structure in the fast variables have, generically, hyperbolic umbilic, elliptic umbilic or swallowtail singularities. In this article we provide a detailed local analysis of a fast-slow system near a hyperbolic umbilic singularity. In particular, we show that under some appropriate non-degeneracy conditions on the slow flow, the attracting slow manifolds jump onto the fast regime and fan out as they cross the hyperbolic umbilic singularity. The analysis is based on the blow-up technique, in which the hyperbolic umbilic point is blown up to a 5-dimensional sphere. Moreover, the reduced slow flow is also blown up and embedded into the blown-up fast formulation. Further, we describe how our analysis is related to classical theories such as catastrophe theory and constrained differential equations.
