具有中心奇点的平面多项式可积系统的极限环分岔和临界周期

The theory of planar polynomials differential equations have an extensive applications in the human society and natural science. This project is going to investigate two issues related to the Hilbert's 16th problem in the planar polynomials dynamic system: Bifurcation of limit cycles and monotonicity of period function. Both of these two issue have close relationship with the theory of the Abel integration. Firstly, the planar quadratic system with a center of genus one are classed by Iliya D. Iliev and his cooperators. A conjecture about the cyclicity of such systems are also proposed by them. In our project, we intend to study the cyclicity of at least one class of unresolved system as well as the distribution of limit cycles bifurcating from the period annuli. In particular, we will study the corresponding issue about the codimension-four system and will try to establish an newauxiliary differential equation by which we will improve the results of the cyclicity of codimension-four system in the literature. Moreover, as a research on the Hilbert's 16th problem for n>2, we are going to investigate the cyclicity of some special Hamiltonian systems under perturbations. Secondly, we are going to study the Chicone's conjecture and it's weak version: the monotonicity of period function of the quadratic system. More accurately, we intend to investigate the monotonicity of period function of a class of Lotka-Volterra system or a class of quadratic reversible system.

平面动力系统理论在人类社会和自然科学中有着十分广泛的应用。本项目拟研究平面多项式微分动力系统中两个与希尔伯特第十六问题相关的课题：极限环分支和周期函数的单调性。首先，Iliev等人把具有亏格1中心的平面二次系统进行了分类，并对它们的环性进行猜测。我们拟研究其中若干类目前尚未解决的系统的环性，同时研究分支出来的极限环的分布；特别是对于其中的余维4系统，将尝试建立不同于Iliev文中的辅助微分方程，并借助它来改进目前已有文献关于余维4的环性的结果；另外作为对n>2时对希尔伯特第十六问题的探讨，还将研究一些特殊的哈密顿多项式系统在某类扰动下的环性。其次，我们拟研究Chicone猜想及其弱问题：具有中心的二次系统的周期单调性问题。包括：研究具有一定广泛性的Lotka-Volterra系统和至少一类二次可逆系统的周期函数的单调性。

本项目研究平面多项式微分动力系统的极限环分支和周期单调性。这两个问题均与希尔伯特第十六问题密切相关且在生物数学等领域有广泛应用。我们着重研究了具有亏格1中心的平面二次可逆系统在二次扰动下分支出来的极限环的个数（称为环性）。以PICARD-FUCHS方程、质心曲线为工具，获得了两类含有参数的可逆系统的环性，以切比雪夫判别法为工具，分别获得一类二次、一类三次可逆Lotka-Volterra系统的环性。我们还研究了四次平面拟齐次动力系统的全局相图，证明了它们共有26种不同的拓扑结构。此外，创建了一种计算李雅普诺夫常数线性部分的并行算法，并利用它得到三个希尔伯特数H(6)、H(8)、H(10)至今最高的下界。. 对上述问题的研究及所获成果，有助于我们更好理解平面动力微分系统的定性性质，有一定的学术意义和应用价值。