平面不连续微分系统的极限环分支

This project investigates some problems related to the bifurcation of limit cycles for planar discontinuous differential systems. The main concerns are presented as follows. First, we judge that whether the Melnikov function method and the averaging method are equivalent for studying the number of limit cycles, which bifurcate from the period annulus of planar discontinuous piecewise analytic differential systems. Further, we derive the formula of the Melnikov function of higher order. Second, we study the number of limit cycles bifurcated from the period annuli of four classes of cubic isochronous centers with homogenous nonlinearities, when they are perturbed inside piecewise polynomial differential systems of degree n . Third, we analyze the possible number of limit cycles and their distributions that a discontinuous quadratic differential system can have. . All these studies will play a key role in understanding the theory on bifurcation of limit cycles and global dynamics of planar discontinuous differential systems. Moreover, they are helpful for providing the methodological guidance for the study on bifurcation of limit cycles. Accordingly, the project is of great academic value. Furthermore, our project has good prospects for application, since lots of physical phenomena in the real world are described by discontinuous differential systems.

本项目考虑平面不连续微分系统极限环分支相关的若干问题，主要包括：1、判断研究不连续微分系统极限环分支的Melnikov函数法和平均法是否具有等价性，并给出高阶Melnikov函数的表达式；2、研究四类具有齐次非线性项的三次等时中心在分段n次多项式扰动下从中心的周期环域分支出的极限环个数；3、考虑不连续二次系统可能具有的极限环个数及位置分布。 .这些工作将加深对平面不连续微分系统极限环分支相关理论、全局结构等的理解，并为极限环分支的研究提供方法上的帮助。因此，本项目的研究具有重要的学术价值。而且，不连续微分系统更能真实地描述现实世界的很多物理现象，因此本项目的研究还有广阔的应用前景。

本项目研究了平面不连续微分系统极限环分支相关的若干问题，包括利用一阶平均法和一阶Melnikov函数法研究四类具有齐次非线性项的三次等时中心在分段n次多项式扰动下从中心的周期环域分支出的最大极限环个数和不连续二次系统可能具有的极限环个数及几种位置分布。此外，还给出了新的判定极限环个数的切比雪夫准则并对光滑微分系统的若干定性性质与分支问题进行了研究，如多项式系统的临界周期个数新下界，椭圆Sitnikov问题对称周期解的稳定性，两类哈密顿系统的全局相图及扰动的五次BBM方程行波解的存在性等。这些工作不仅加深了对平面不连续微分系统极限环分支研究方法、全局结构等的理解，还对光滑系统的临界周期个数、周期解存在性、稳定性等的研究有了新认识。项目结果丰富了平面微分系统极限环分支及相关问题的研究。