论文标题
在平面全态系统上
On Planar Holomorphic Systems
论文作者
论文摘要
Planar Holomorphic Systems $ \ dot {x} = u（x，y）$，$ \ dot {y} = v（x，y）$是$ = \ u = \ operatorName {re}（re}（f）$ and $ v = \ operatatorName {im}（im}（im}（im}（f）for Some holomorphic Function它们具有重要的动力学特性，例如强调它们没有限制周期，而中心对焦问题是微不足道的。特别是，多项式系统是全态的假设减少了系统的参数数量。尽管$ n $的多项式系统取决于$ n^2 +3n +2 $参数，但多项式全态仅取决于$ 2N +2 $参数。在这项工作中，除了对全态系统理论进行一般概述之外，我们还对庞加莱磁盘上的所有可能的全球阶段肖像，系统$ \ dot {z} = f（z）$和$ \ dot {z）$ \ dot {z）= 1/f（z）= 1/f（z）$ and $ f（z）$，$ f（z）$，$ 2 $ 2 $ 2 $ 2 $ 2 $ 2 $ 2 $ 2 $ 2， \ Mathbb {C} $。我们还对Moebius Systems $ \ dot {z} = \ frac {az+b} {cz+d} $进行分类，其中$ a，b，c，c，in \ in \ in \ mathbb {c}，ad bc \ neq0 $。最后，我们获得了全体形态系统和共轭全体形态系统的第一积分的明确表达，它们在流体动力学研究中具有重要的应用。
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.
