三维多项式微分系统在z轴附近的极限环

考虑三维多项式微分系统x=-y(1+x)+ε(ax+F(x,y,z)),y=x(1+x)+ε(ay+c(x,y,z)),z=ε(cz+R(x,y,z))(F(0,0,z)=0,G(0,0,z)=0),利用一阶平均理论得到上面系统可以从x=-y(1+x),y=x(1+x),z=0的周期轨中分支出n2个极限环,最后用一个例子展示主要结果的简洁性和有效性.

Limit cycles for a class of three-dimensional polynomial differential systems near the z-axis

We consider the following polynomial differential system of degree n in R3 near the z -axis x= -y(1+x) +ε(ax+F(x,y,z)),y=x(1+x) +ε(ay+G(x,y,z)),z =ε(cz+R(x,y,z) )(satisfying F(O,O,z) =0,G(0,0,z) =0). By applying the first-order averaging theory,we ob-tain at least n2 limit cycles bifurcating from the periodic orbits of the system x = - y ( 1 + x), y =x (1 + x), z= 0. An example is given to illustrate the simpleness and effectiveness of our main result.