From Hopf bifurcation to limit cycle

Emphasizes the geometrical understanding of the application of qualitative and bifurcation theories in the process of Hopf bifurcation leading to limit circle.Illustrating an example of Hopf bifurcation by using the computer,we showed the proof of Liapunov method theorem in a much more comprehensible way.Based on previous work,we made a primary study on the generating of more limit cycles and on the change of singular point.Then we made an introduction of the methods to study the bifurcation and an interesting application,and made short comments on these two kinds of method at the end.