一类动力系统的大范围定性研究

本文考虑动力系统：ｄｘ／ｄｔ＝Ｐ３（ｘ，ｙ），ｄｙ／ｄｔ＝ｘ（１（．研究系统（１）具有代数曲线解：Ｘ２一ｋｙ２＝１ （ｋ＞０）（２）全局结构。 容易得到这时系统（１）等价于系统,ｄｘ／ｄｔ＝ａ１ｘ＋ａ２ｙ－ａ１ｘ３＋（ｋ－ａ２）ｘ２ｙ＋ｋａ１ｘｙ３＝ｋ（ａ２－ｋ）ｙ３，ｄｙ／ｄｔ＝ｘ（３）．由（３）我们得到所有奇点（有限远和无穷远）的类型。并用Ｄｕｌａｃ函数证明 （３）不存在极限环。进而得出（３）的所有可能的全局相图（ｌ）－（１３）．

THE GLOBAL QUALITATIVE RESEARCH OF A CLASS OF DYNAMICAL SYSTEM

In this paper, we consider the dynamical system:dx/dt=P3(x,y),dy/dt=x(1), and study the overall qualitative structure of this system , which has an algebraic curvilinear solution in the form x2-ky2-1=O (k>O) (2). It is easy to see that such a system is equivalent to the system dx/dt=a1x+ +a2y-a1x3+(k-a2)x2y+ka1xy2+k(a2-k)y3,dy/dt=x(3), from which we obtain the types of all its singular points. By means of Dulac's function, we prove that there is no limit cycle in system (3) Hence we obtain the possible global phase portrait for system (3) in Fig(1)-(13).