The Existence Of Limit Cycles For Several Classes Of Cubic Polynomical Systems

The domain qualitative theory of the ordinary differential equations, which was initiated by H. Poincaré, has got a very flourishing development for about a century, and limit cycle plays an important role. The theory of limit cycle has received a wide range of applications in celestial mechanics, radio technology, automatic control and so on. Conversely, the needs of engineering and technical promoted the research on limit cycle. The issue which is closely related to limit cycle is the focus-center problem and the calculation of the values of criticle points, saddle points and singular points. For the second system, the research on these issues has been made rich achievements while on the three systems relatively small.By applying the classic method of ordinary differential equations qualitative theory, three classes of cubic polynomial system are studied in this thesis . At first, we introduce the background of problems which will be investigated and the main results of this paper, at the same time some basic theory related to the topic are given. Then we mainly study the properties of equilibrium, existence and uniqueness of limit cycle for three classes of cubic polynomial system. With the formal series method, the center and focus are judged, with the Dulac function, the non-existence of closed orbit is discussed, with the Hopf bifurcation theory, some sufficient conditions for the existence of limit cycles bifurcated from the equilibrium point is analyzed, finally, after applying the Hopf bifurcation theory, some sufficient conditions for the existence and stability of limit cycles are established.