Based on Poincaré Algebra and strict proofs, this paper discusses the construction of the spin of the motion, which is combined with Pauli Lubankis spin of the strain and thus a complete theoretical system is formed.

By using the method of successor function to determine the behaviors of bifurcation points.

It is very important for successor function to judge the number,the stability and the relative place of the limit cycles of the perturbated system.

In this paper, by using the successive function and implicit function theorem, the number of limit cycles bifurcated from the origin of Bogdanov-Takens system under quadratic perturbations is given combined with the calculation of Mel nikov functions.

The type and stability of the bifurcation points in the subsequent reduced two dimensional system are analyzed by the method of succeeding function.

Then the type and stability of the bifurcation points are analyzed by utilizing the method of succeeding function.

In this dissertation, by using the method of the Poincare maps and the displacement functions, we firstly study the existence, stability and its criterion and bifurcation of periodic solutions of scaler periodic impulsive differential equations.