几类近可积多项式系统和分段光滑系统的极限环分支

项目名称： 几类近可积多项式系统和分段光滑系统的极限环分支

项目编号： No.11501055

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 王言芹

作者单位： 常州大学

项目金额： 18万元

中文摘要： 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。

中文关键词： 极限环；分支；同宿轨；异宿轨；多项式系统

英文摘要： 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.

英文关键词： Limit cycle;Bifurcation;Homoclinic orbit;Heteroclinic orbit;Polynomial system