关于Léinard系统的中心问题

On the center of the Liénard system

研究Liénard系统ddt2+f(x)dxdt+g(x)=0的中心问题.首先通过计算线性近似系统特征值的方法,给出了奇点O为局部中心点的判定条件.其次通过引进辅助系统,利用比较定理和线性近似理论给出局部中心点存在的充分条件,推广了文[1-2]的某些结果,从而扩充了局部中心点的可判定性范围.

The conditions of a local center for a Liénard system d^2x/dt^2＋f（x）dx/dt＋g（x）=0 are discussed. Firstly, by computing the eigenvalues of linear approximation system, a critical condition which ensures odd point O is local center is obtained. Secondly, by introducing auxiliary systems and using comparison theorem and linear approximation theory, some sufficient conditions to guarantee the origin is a local center are given. Therefore, some results of Refs. [ 1 -2 ] are generalized, and thereby the range of critical local center is extended.