关于螺线管Hopf-零奇点的表示和$\mathbb{Z}_2$ -等变规范形
On the representations and $\mathbb{Z}_2$-equivariant normal form for solenoidal Hopf-zero singularities
摘要：
本文研究了与Hopf-零奇异系统的经典规范形相联系的螺线管保守李代数。本文主要研究这类奇异微分方程的一些表示和$\mathbb{Z}_2$ -等变规范形。首先，我们列出这个李代数允许的一些表示。这个李代数的向量场可以用一组常微分方程来表示，其中前两个方程是一个单自由度哈密顿系统的标准形式，第三个方程依赖于前两个变量。这种表示由与此李代数的一个子族相关联的泊松代数所支配。欧拉形式、矢势和Clebsch表示是我们在这里列出的这个李代数的其他表示。我们还从这个李代数出发研究了具有Hopf-零奇性的向量场的非位势性质。最后，我们研究了在对称群$\mathbb{Z}_2$的存在下，这个族的非零立方项的唯一规范形。用修正的蔡氏振子说明规范形理论的理论结果。
英文摘要：
In this paper, we deal with the solenoidal conservative Lie algebra associated to the classical normal form of Hopf-zero singular system. We concentrate on the study of some representations and $\mathbb{Z}_2$-equivariant normal form for such singular differential equations. First, we list some of the representations that this Lie algebra admits. The vector fields from this Lie algebra could be expressed by the set of ordinary differential equations where the first two of them are in the canonical form of a one-degree of freedom Hamiltonian system and the third one depends upon the first two variables. This representation is governed by the associated Poisson algebra to one sub-family of this Lie algebra. Euler's form, vector potential, and Clebsch representation are other representations of this Lie algebra that we list here. We also study the non-potential property of vector fields with Hopf-zero singularity from this Lie algebra. Finally, we examine the unique normal form with non-zero cubic terms of this family in the presence of the symmetry group $\mathbb{Z}_2$ . The theoretical results of normal form theory are illustrated with the modified Chua's oscillator.
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