Riccati二次多项式微分系统的相图
Phase Portrait of the Riccati Quadratic Polynomial Differential Systems
摘要：
本文研究了具有非零参数$(x,y)\in\mathbb{R}^2$，$γ_2(x)$的Riccati二次多项式微分系统$$\dot{x}= α_2(x),\quad\dot{y} = ky^2+β_1(x) y + γ_2(x), $$的相图（否则系统是伯努利微分系统），$k\neq0$（否则该系统是Lienard微分系统），$β_ 1(x)$是次数至多为$1$的多项式，$α_ 2(x)$和次数至多为2的$γ_ 2(x)$多项式，并且$ α_2(x)$和$k y^2+β_1(x) y + γ_2(x)$的度数的最大值为2。我们给出了它们在庞加莱盘中的相图的完整描述
英文摘要：
In this paper we characterize the phase portrait of the Riccati quadratic polynomial differential systems $$\dot{x}= α_2(x),\quad\dot{y} = ky^2+β_1(x) y + γ_2(x), $$ with $(x,y)\in\mathbb{R}^2$, $γ_2(x)$ non-zero (otherwise the system is a Bernoulli differential system), $k\neq0$ (otherwise the system is a Lienard differential system), $β_ 1(x)$ a polynomial of degree at most $1$, $α_ 2(x)$ and $γ_ 2(x)$ polynomials of degree at most 2, and the maximum of the degrees of $ α_2(x)$ and $k y^2+β_1(x) y + γ_2(x)$ is 2. We give the complete description of their phase portraits in the Poincare disk
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