【摘要】 讨论了一类可积非Hamilton二次系统经二次扰动的Poincare分支.系统x=-y+ax2+by2,y=x(1-2y),经二次扰动的Poincare分支问题当a=-3时已经解决,a=±4时只有一些初步结果.当a=-2,b≠0时,其Able积分的被积函数是对数函数的平方根,Able积分所对应的Picard-Fuchs方程是无穷维的,对应的Poincare分支问题目前尚未得到解决.对于a=-2,b=0时的情形,通过方程变换和计算Able积分等方法,证明了此时上述系统的Poincare分支至多能分支出2个极限环.更多还原

【Abstract】 This paper aims at discussing the Poincare bifurcations of a class of integrable non-Hamilton quadratic system disturbed by quadratic polynomial.It has been solved that Poincare bifurcations of the system x=-y+ax2+by2,y=x(1-2y)disturbed by quadratic polynomial when a=-3,Some preliminary results have been obtained when a=±4.As a=-2,b≠0,the integrand of Abel integral is the square root of logarithmic function.The Picard-Fuchs equation corresponding to the Abel integral is infinite dimensional.The corresponding Poincare bifurcation problem is not yet resolved at present.The case a=-2,b=0is mainly discussed in this paper by means of equation transformation and calculating integration etc.It is proved that the system in this case can be bifurcated out at most two limit cycles after a quadratic polynomial disturbance.更多还原