一个二次可积系统在二次保守扰动下的分支

研究了一个具有同宿轨的二次可积系统在二次保守扰动下的分支现象.此时一阶Melnikov函数恒等于零,必须考虑二阶Melnikov函数,而在一阶Melnikov函数中不起作用的扰动参数在二阶Melnikov函数中却起着非常重要的作用,因此扰动部分极为复杂.在此借助于二阶Melnikov函数的计算公式及理论分析,建立了极限环分支定理,得到了在一阶Melnikov函数恒等于零,而二阶Melnikov函数不恒等于零时,可能分支出的极限环个数最大为2的结论.

Bifurcation of a Quadratic Integrable System under Quadratic Conservative Perturbations

The question on bifurcation of limit cycles in quadratic conservative perturbations of a quadratic integrable system is discussed. As a result of conservative perturbations, the first order Melnikov function for a perturbed system is identically equal to zero, so the second order Melnikov function has to be considered. The perturbation parameters which make no use in the first order Melnikov function may be important in the second order Melnikov function. Hence perturbed systems are more complicated. With the help of the formula for the second Melnikov function and theoretical analysis, the following result is obtained: if the first order Melnikov function for a perturbed system is identically equal to zero, and the second order Melnikov function for the perturbed system is not identically equal to zero, the system has at most two limit cycles.