THE VALUES OF SINGULAR POINT OF E(N) AND SOME KINDS OF PROBLEMS OF BIFURCATION

In this paper, by discussing 2pi-periodical solutions under the polar coordinates for the complex system E(n)(epsilon, lambda), the researc h for the real planar Hopf bifurcation, center point bifurcation a.d a class of homoclinic bifurcation are all reduced to calculating the fo rmulas of the values of the singular point and to extracting the roots of an algebraic equation. Therefore, the general criteria for the exi stence of several limit cycles appearing in these bifurcations are obt ained. The examples of quadratic systems with three limit cycles appea ring in both a homoclinic loop bifurcation and a center point bifurcat ion are obtained. For the cubic system, the functions on the right-han d side have no quadratic terms, the examples about 10 and 5 limit cycl es appearing in a double homoclinic loop bifurcation and a center poin t bifurcation are also obtained.