平面微分系统高余维分支及其应用

By using the qualitative and bifurcation theory in ordinary differential equation and dynamical system, and combining the theoretical analysis with symbolic and numeric computation, we will study the following three problems: Firstly, the maximum number of limit cycles bifurcating from codimension 3 Bogdanov-Takens bifurcations of saddle and elliptic cases; Secondly, the bifurcations of high codimension in some epidemic systems with general nonmonotonic incidence rate; Thirdly, the bifurcations of high codimension in some systems that modeling the control of invasive hosts by generalist parasitoids.. Bifurcation with high codimension is a challenging and difficult problem in ordinary differential equation and dynamical system; As an application of bifurcation theory, the study of bifurcation phenomena in epidemic and biological models has important direction for epidemic control and scientific species management.

本项目将应用常微分方程与动力系统中的定性理论和分支理论，将理论分析和符号数值计算相结合，研究以下三个问题：平面微分系统鞍点型和椭圆型余维3 Bogdanov-Takens 分支所能分支出的最大极限环个数；一类带有一般非单调发生率的传染病模型的高余维分支；一类多寄主型寄生性天敌昆虫对寄主的生物控制模型的高余维分支。. 高余维分支是常微分方程与动力系统中具有挑战性和困难的问题；作为分支理论的应用，传染病与生物模型中分支现象的研究对于传染病的控制与生物物种的科学管理具有重要的指导意义。

课题组在国家自然科学基金面上项目 “平面微分系统高余维分支及其应用”（11871235）的资助下，对所要研究的问题基本上都得到了较好的理论结果，在 SIAM Journal on Applied Dynamical Systems、J. Differential Equations、J. Dynamics and Differential Equations、Journal of Mathematical Biology、Stud. Appl. Math. 等国际重要期刊发表27篇SCI文章。在如下方面取得重要进展和较好结果：平面自治常微分系统的定性与高余维分支分析及其应用；平面非自治常微分系统的定性与高余维分支分析及其应用；平面离散差分系统的定性与分支分析及其应用；反应扩散系统定性与分支分析及其应用；非光滑常微分系统定性与高余维分支及其应用。