关于平面全纯系统
On Planar Holomorphic Systems
摘要：
平面全纯系统$\dot{x}=u(x,y)$，$\dot{y}=v(x,y)$是指对于某个全纯函数$f(z)$满足$u=\operatorname{Re}(f)$和$v=\operatorname{Im}(f)$的平面全纯系统.它们具有重要的动力学性质，例如，它们没有极限环，中心焦点问题是微不足道的。特别地，多项式系统是全纯的假设减少了系统的参数的数量。虽然次数为$n$的多项式系统依赖于$n^2 +3n+2$参数，但多项式全纯仅依赖于$2n + 2$参数。在这项工作中，除了使全纯系统的理论的一般概述，我们分类所有可能的全球相图，庞加莱磁盘上的系统$\dot{z}=f(z)$和$\dot{z}=1/f(z)$，其中$f(z)$是一个多项式的次数$2$，$3$和$4$在变量$z\in \mathbb{C}$。我们还对Moebius系统$\dot{z}=\frac{Az+B}{Cz+D}$的所有可能的全局相图进行了分类，其中$A,B,C,D\in\mathbb{C}, AD-BC\neq0$ .最后，我们得到了全纯系统和共轭全纯系统第一积分的显式表达式，这在流体动力学的研究中有重要的应用。
英文摘要：
Planar holomorphic systems $\dot{x}=u(x,y)$, $\dot{y}=v(x,y)$ are those that $u=\operatorname{Re}(f)$ and $v=\operatorname{Im}(f)$ for some holomorphic function $f(z)$. They have important dynamical properties, highlighting, for example, the fact that they do not have limit cycles and that center-focus problem is trivial. In particular, the hypothesis that a polynomial system is holomorphic reduces the number of parameters of the system. Although a polynomial system of degree $n$ depends on $n^2 +3n+2$ parameters, a polynomial holomorphic depends only on $2n + 2$ parameters. In this work, in addition to making a general overview of the theory of holomorphic systems, we classify all the possible global phase portraits, on the Poincaré disk, of systems $\dot{z}=f(z)$ and $\dot{z}=1/f(z)$, where $f(z)$ is a polynomial of degree $2$, $3$ and $4$ in the variable $z\in \mathbb{C}$. We also classify all the possible global phase portraits of Moebius systems $\dot{z}=\frac{Az+B}{Cz+D}$, where $A,B,C,D\in\mathbb{C}, AD-BC\neq0$. Finally, we obtain explicit expressions of first integrals of holomorphic systems and of conjugated holomorphic systems, which have important applications in the study of fluid dynamics.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
