在平面系统的研究中，极限环的数目及其相对问题起到了非常重要的作用，本项目计划在几个方面对平面连续不连续系统的这方面的相关问题做探讨，部分内容是申请人以前工作的深化与发展，另一部分则是新的。具体内容包括：Hopf分岔中焦点量阶数问题的复杂性分析，给出首个非零焦点量的阶数的一个下界估计；通过正规型理论，建立平面分段光滑系统的中心与可积、可逆系统的联系；给出Abelian积分零点个数的估计的新方法，在应用中积分很可能是非代数的；给出周期函数的临界点个数的新估计并将用这种方法解决一些来自于其他方面的问题。

In the research of qualitative theory of planar systems, the number and the relative position of limit cycles play an important role. In this project, we plan to study planar smooth or non-smooth systems about this topic, some of which is the generalization of previous works and some of which is new. Concretely, Study on the following problems: On the complexity of the order of focus value in Hopf bifurcation and give a lower bound on the order of the first non-zero focus value; Use normal form thery to establish a relation of a piecewise smooth system with a center and intrgrable or reversible system; Give a new method to estimate some Abelian integrals, in general, these integrals are not algebraic; Give new estimate of the number of critical points of period function and use this method to solve some problems from other fields.