分段光滑微分系统的极限环分支

This project intends to solving some interesting but challenging problems on the bifurcation of limit cycles of non-smooth piecewise differential systems, which can be seen as an xtension of the 16th Hilbert’s problem in some sense. The problems of this project are crucial and the study on them is very important and useful for the development bifurcation theory. .There are 4 problems to be investigated in our project. .The first one is about the number of limit cycles for quadratic differential systems.with isochronous centers under piecewise polynomial perturbations. The second problem is on the number of limit cycles for a class of cubic systems with conic integral factors under piecewise polynomial perturbations. The third problem is on the number of limit cycles for several piecewise smooth Hamiltonian systems with degree 2 or 3 under piecewise polynomial perturbations. The last problem is on the hyperbolic periodic orbit of the regular systems of the left and right piecewise systems on R^3. . These problems are to be solved by the averaging method or calculating the Melnikov functions. Since the functions involved in the averaging functions or the Melnikov functions are from two sub-systems and defined on the half plane, it is very difficult to determine the number of isolated zeros of averaging functions or the Melnikov functions. We will first find some useful transformations and reduce the number of the generators in averaging functions or Melnikov functions, and then estimate the number of zeros by argument principle 、 ECT series、deravative-division and some new methods.

本项目研究分段光滑系统的分支理论中的热点、难点问题,是希尔伯特第16问题的延伸.研究难度大,具有挑战性,是国际前沿的研究课题. 有以下四方面的内容:(1)具有等时中心的光滑二次系统在n次多项式分段扰动下分支出的极限环个数的估计(2)积分因子为二次圆锥曲线的一类三次光滑系统在n次多项式分段扰动下分支出的极限环个数的估计(3)几类左、右子系统为2次、3次哈密尔顿系统的分段光滑系统在n次多项式分段扰动下分支出的极限环个数的估计(4)三维空间中左、右子系统的正则化系统的双曲周期轨的存在性.由于在平均函数或Melnikov函数中同时出现两个子系统对应的生成元，且其定义域仅为半个平面，以及生成元满足不同的Picard-Fuchs方程,使得判断函数的零点个数非常困难.我们在探索有效的变换、减少生成元的个数和发现新方法的基础上，综合运用幅角原理、ECT序列、求导-做除等方法，解决这些问题。

(1)在弱化希尔伯特第16问题方面, 研究了一类有交叉项的四次哈密尔顿函数所对应的向量场在任意多项式扰动下从周期环域分支出极限环的个数问题. 这类向量场具有幂零奇点，含有多个复杂周期环域, 阿贝尔积分的表达式中最多的情形下含有9个生成元，满足3组Picard-Fuchs 方程, 这给问题的研究带来了很大的困难。我们首次成功地提出了“消元法”， 通过反复消除生成元，得到了阿贝尔积分零点个数的上界。(2) 在分段光滑系统的分支理论中，首先研究了亏格为1的二次可逆系统的(r19)(r20)和二次等时系统(S1) (S2)的周期环域在切换直线为$y=0$时的极限环分支问题. 首次成功地把光滑系统中的Picard-Fuchs 方程法和“消元法”运用到分段光滑系统中，得到了从(r19)(r20)分支出极限环个数的上界和从(S1) (S2)在2次多项式扰动下分支出极限环的精确个数。 接着研究了二次系统中BT系统、广义LV系统、退化二次系统中具有多角环S^{(2)} 或者S^{(3)} 的向量场当以$x$轴为切换曲线时在任意多项式扰动下从周期环域分支出极限环的个数问题. 这些系统都是二次系统中的重要类型。进一步我们还研究了BT系统当以$y$轴为切换曲线时、二次等时中心(S1)(S2)具有两条垂直分解曲线时在任意多项式扰动下从周期环域分支出极限环的个数问题。 (3) 三维空间中左、右子系统的正则化系统的双曲周期轨的存在性， 研究了一类n维不连续边界由两个超平面相交而成分段光滑系统的周期轨与其正则化系统的周期轨的关系。主要结论表明如果该分段光滑系统具有一条渐近稳定的周期轨，其正则化系统将在此周期轨附近有一个稳定的极限环，且此极限环会收敛到原分段光滑系统的周期轨。