Polynomial Differential System Limit Cycle Of The Limited Nature Of Problems

The so-called finiteness problem of polynomial differential system is to determine whether the number of limit cycles of every polynomial vector field on the real plane is finite. The existing work and material about the study of this problem is quite scarce. In the study of finiteness conjecture, there are two important conclusions, that is Revised Dulac's Theorem and Il'yashenko's Theorem. But the discusses and demonstrations of these two theorems are very brief and simple in existing documents. This paper talks about these two theorems around the finiteness conjecture and mainly gives detailed proofs for them. In addition, this paper introduces a theorem which is related to the finiteness proof of quadratic system, that is H'yashenko's two-side cycle theorem.The paper has four parts. Introduction gives the origin of the problem and some definitions. The second chapter discusses and proves the Revised Dulac's Theorem, that is, if a polycycle of an analytic vector field has a monodromy map, then there is a semi-transversal for which the monodromy map is either flat, semi-regular, or the inverse of a flat. The third chapter discusses and proves Il'yashenko's Theorem, that is, if every vertex on the polycycle of an analytic vector field in the real plane is hyperbolic, then limit cycles cannot accumulate on this polycycle. The last chapter talks about the H'yashenko's two-side cycle theorem, that is, every two-side cycle of an analytic vector field in the real plane is the finite polycycle. The quadratic finiteness theorem is a direct corollary of this theorem.