等时性及其相关问题研究

The problems of center and limit cycle are important for investigation of qualitative properties for planar differential systems. We will study the isochronicity and related problems for planar polynomial systems. We mainly deal with isochronicity problems of cubic, quartic reversible systems, the monotonicity of period function of the quadratic systems, critical period and limit cycles which bifurcate from .period annulus of the isochronous systems. We expect that one to three papers will been published in the major domestic and international mathematics academic journals.

中心及极限环是平面微分方程定性理论的主要研究内容。本项目研究平面多项式系统等时中心的判定及相关问题。主要研究三次、四次可逆多项式系统的等时性问题，二次可逆系统的周期函数的单调性问题，等时系统的临界周期及极限环分支问题。 争取在国内外重要数学学术期刊上发表一至三篇学术论文。

极限环个数的判定是平面微分方程定性理论的主要研究内容. 2009年, Iliev等人把具有亏格1中心的平面二次系统分成18类, 目前该18类系统中绝大部分均已经获得完整的结果. 本课题研究一类具有非亏格一中心的二次可逆Lotka-Volterra在小扰动下产生极限环的个数问题, 并证明其周期环域在二次扰动下之多产生两个极限环.