一类平面三次多项式系统的平衡点分析

An Equilibrium Point Analysis of a Class of Planar Cubic Polynomial Systems

摘要: 研究了一类平面三次多项式系统ẋ=-y+αx²-αy²+βx³-3βxy², ẏ=x-2αxy+3βx²y-βy³的平衡点, 证明了当|α-1|≪0, |β-1|≪0时, 该系统共有4个无穷远平衡点且均为鞍点, 以及共有3个有限平衡点且均为焦点, 并给出了这3个焦点的位置、阶数和稳定性.

Abstract: The equilibrium points of a class of plane cubic polynomial systems ẋ=-y+αx²-αy²+βx³-3βxy², ẏ=x-2αxy+3βx²y-βy³ are discussed. It is proved that when |α-1|≪0, |β-1|≪0, there are four infinite equilibrium points and all of them are saddle points, and there are three finite equilibrium points and all of them are focal points. The position, order and stability of the three focal points are given.