On Small Perturbation Of Quasi-periodic Systems With Degenerate Equilibrium Point

Degenerate systems appear commonly in celestial mechanics. In this the-sis, we summarize research methods of small perturbation of quasi-periodic systems with degenerate equilibrium point. And they are illustrated through three examples. We also give a recent result on small perturbation of four-dimensional quasi-periodic systems with completely elliptic-type degenerate equilibrium point.In the first chapter of this paper, we give some definitions of the dynam-ic system theory and introduce the research background and progress about the small perturbation problem of quasi-periodic systems near the equilibrium point. Moreover, we summarize three methods for the perturbation problem of quasi-periodic systems with degenerate equilibrium point.In the second chapter, we elaborate respectively the application of three methods which are Herman method, stability analysis of real roots of polyno-mials and zero’s translation of vector field average in dealing with the perturba-tion problem of quasi-periodic systems with degenerate equilibrium point. The first article discussed the stability of the equilibrium about the one-dimensional quasi-periodic system, we introduce the using of Herman method in dealing with such problem by giving thought of the article’s proof; The question that studying in the second article is discussing the small perturbation of planar quasi-periodic system. We can find the stability analysis of real roots of polyno-mials play an important role in dealing with the problem of small divisor and controlling the shift of equilibrium point through summing up the proof in this paper. We analyze the application of the method of the zero’s translation of vector field average by introducing the third article in which a mapping be con-sidered. In addition, we give the latest conclusion about on small perturbation of quasi-periodic system with completely elliptic-type degenerate equilibrium point.The third chapter is research prospect in which we raise some questions to be solved.