非亏格1中心的平面二次系统的极限环分支问题

The problem of bifurcation of limit cycles is important for qualitative theory of differential equations, which is related to the Hilbert's 16th problem. It is widely believed that the dynamics phenomenas generated by the quadratic reversible systems under quadratic perturbations are very rich. The project is going to investigate the problem of bifurcation of limit cycles of quadratic reversible systems under quadratic perturbation, which contains the following contents: (1) the quadratic reversible systems with non-algebraic first integral; (2) the quadratic reversible centers with non-genus one; (3) the quadratic reversible Lotka-Volterra systems with two centers; (4) the center of system (r6) with genus one. The study of these problems is very important for theoretical significance and academic value.

极限环分支问题是微分方程定性理论的主要研究内容，该问题涉及到Hilbert第十六问题。人们普遍认为二次可逆系统在二次扰动下产生的动力学现象最为丰富。本项目主要研究二次可逆系统在二次扰动下的极限环分支问题, 主要内容包括：1、具有非代数首次积分的二次可逆系统；2、非亏格1中心的二次可逆系统；3、具有双中心的二次可逆Lotka-Volterra系统；4、亏格1中心的系统(r6)。这些问题的研究有着重要的理论意义和学术价值。

该项目研究了几类多项式系统的极限环分支问题及周期轨道周期的单调性问题。这两个问题涉及弱化的Hilbert第十六问题，是微分方程定性理论研究的热点、难点问题。我们主要研究了这几类几类系统的极限环分支问题：亏格1的二次可逆Lotka-Volterra系统、 (4, 3)型的Liénard系统、三次可逆系统、分段光滑的Hamiltonian系统的、分段光滑的拟齐次系统以及Liénard系统的临界周期个数问题。相关的研究成果改进或补充了已有的结果，具有一定的学术意义和应用价值。