几类近可积多项式系统和分段光滑系统的极限环分支

Bifurcation theory of limit cycles is the most important component part not only in applied analysis, but also in dynamical system theory, which has significant theoretical value and wide application prospects. The study of limit cycles of polynomial differential autonomous systems has strong practical background. The problem of bifurcations for this kind of differential systems is hard to be investigated and is quite challenging, and has become one of the most important and hottest problems in the research field of nonlinear dynamic systems. . In this project, we will mainly use the research methods of the Melnikov function and others to study the bifurcation problems for several types of polycyclic and near-integrable polynomial systems perturbed by multiple parameters and piecewise smooth systems. We shall establish the expansion of the Melnikov function for the perturbed systems near singular points or singular closed orbits. By discussing the expansion coefficients，we shall determine the upper and lower bounds of the numbers of limit cycles, aiming to get more numbers of limit cycles. . The results of our project will be very valuable for the development of bifurcation theory of limit cycles and hope to provide important scientific basis for the qualitative research of dynamics and other fields.

极限环的分支理论不仅是应用分析，也是动力系统理论的重要组成部分，有重要的理论价值和广阔的应用前景。多项式微分自治系统的极限环研究有强烈的实际背景，这类微分系统的分支问题研究难度大、很具有挑战性，已成为非线性动力系统研究的重点和热点之一。. 本项目将主要利用Melnikov 函数等研究方法，重点研究几类被多参数扰动的具有多环的近可积多项式系统和分段光滑系统的极限环的分支问题，建立扰动系统在奇点或奇闭轨附近的Melnikov 函数展开式，通过分析展开式的系数，来确定极限环的个数的上界和下界，力求获得更多个数的极限环。. 项目的研究成果对于极限环分支理论的发展具有重要意义，可以为动力学等其他领域的定性研究提供重要的科学依据。

本项目首先研究了具有多条开关直线的扰动的分段光滑多项式系统的极限环分支，在未扰动系统具有一个复合中心、复合同宿环、复合两点环、复合三点环、复合四点环的五种情形下，我们分别推出了扰动系统的首阶 Melnikov 函数的表达式，进一步获得了扰动系统在这五种情形下的极限环最大个数的下界和后面四种情形下的上界。. 其次，应用极限环的 Hopf 分支理论分别研究一类哺乳动物的生物钟系统和具有反应扩散的时滞细胞周期模型，相应的分别得到建立了 Hopf 分支的存在性和空间齐次与非齐次分支周期解的存在性，进一步结合偏微分方程的规范型和中心流形理论，可以确定后一种情形下的分支周期解得稳定性和方向。. 最后，运用拓扑度等理论研究了几类微分系统的周期解的存在性与稳定性问题。. 总之，本项目的研究成果对于极限环分支理论的发展具有重要意义，可以为动力学等其他领域的定性研究提供重要的科学依据。