Estimate The Lower Bound Of Hilbert Number

David Hilbert stated 23 open problems at the International Conference of Mathematics in Paris in 1900. The 16th problem discussed the classification of algebraic curve and the number of limit cycle of polynomial system which is a very important and difficult problem for the quality theory of ordinary differential equation. It is surely that the most parts of results for quality theory are related to this problem directly or indirectly, for example, [Bull. Amer. Math. Soc. (N. S.), 2002.. 39(3): 301-354] and [The Quality Theory of Polynomial differential system, ed: Ye Y. Q., Shang Hai Sci. and Tech. Press, Shang Hai, 1993] have listed more than 160 and 600 references respectively. Moreover, it has become the 13th problem among 18 open problems which are stated by Stephen Smale for the 21th century in [Math. Intelligencer, 1998, 20(2): 7-15].In the first chapter, we introduce the limit cycles of polynomial system and the high degree of weak focus. There are many methods can be used to discuss these problems, we mainly stated the bifurcation of vector fields and Poincare mapping method and these latest results.In order to understand the Hilbert 16th problem and its latest results clearly, we summary it in the second chapter, including three Levels: individual finiteness problem, existential Hilbert problem and constructive Hilbert problem.In the third chapter, we mainly discuss the third one. In details, we give the some better lower bound of Hilbert number H(n) for the odd degree polynomial systems constructively, where H(n) denotes the maximum number of limit cycles of n-degree polynomial system. Bai J. X. and Liu Y. R. get H(n)≥n~2-n for even n, however, no results for odd.From the method of Bai J. X. and Liu Y. R., the lower bound of Hilbert number is related with the number of small parameters directly. In the fourth chapter, we construct some special system with more small parameter so that the each focus quantity has better relationship of recursive, base on the converting computation of focus quantity we get some weak focus with higher order by using the recursive. Therefore, we obtain the better lower bound H{n)≥n~2-1 for even n. It not only improve the other's results, but is the result which was proved by Bautin in 1954, it means that the system has 3 limit cycles when n = 2, and is highest lower bound for single point.