Some Branches Of The Higher Plane Vector Field And Its Applications,

In this thesis we study bifurcation of singular points, phase portraits and limit cycles in several planar vector fields with high degree. Then the bifurcation method is used to solve a partial differential equation. Using the bifurcation method of perturbed Harnilto-nian vector field and numerical exploration, firstly we show that the distributions'of limit cycles are the same in five perturbed Hamiltonian vector fields, and there are 11, 13 and 14 limit cycles when the degree of perturbed item is 3, 7 and 9 respectively. The accurate position of each limit cycle is obtained by using computer. We get the same result from two methods. Secondly we study three perturbed quintic Hamiltonian vector fields. The numbers and positions of limit cycyles are showed by using above two methods. Finally, by using the bifurcation method we show that Camassa- Holm equation has peakons when parameter k 0. Further the peakons are obtained from three ways. This result correct the conclusion "for k 0 the solitons are no longer peaked" appeared in three papers. The three ways are used to look for the peakons of generalized Camassa- Holm equation. Under some conditions the peakons and it's bifurcation values are got.