Limited Nature Of The Limit Cycle Of The Secondary System

In this paper we will prove that quadratic system has at most finitely limit cycles. Bamon claimed that he had finished the demonstration of the finiteness of limit cycles, but he used Il'yashenko Theorem, which involves some knowledge of complex domain, so it is the main idea to give Il'yashenko Theorem an elementary proof in this paper. What we have done in this paper is to prove that unbounded polycycle of quadratic system is finite, which involves the case of unbounded hyperbolic polycycles and the case of unbounded non-hyperbolic polycycles. The main part is the first case, that is to say, it is the Il'yashenko Theorem. To get ourresult we adopt two equalities and , where and are the eigenvalues of the singular points to the linear parts. It has been proven that the system has finite limit cycles when the equalities do not exist, so we need to study the other case. Under this condition we must study the possible existence of separatrix cycles and the accumulation of limit cycles according to different classifications of k and g. The most key case happens when k and g belong to(-oo,0) .Here are only hyperbolic saddles in the separatrix cycle. Whether theequalities exist decides whether the cycle can consist simple graph or not. Then we consider the case when those separatrix cycles consist simple graphs, that is to say, they can make the equalities exist. During the illustration, we use some theorems about the non-existence and the uniqueness of limit cycles. At the end of this paper we talk about the question of the maximal number of limit cycles in quadratic system and some applied uses in ecological circumstances.