(a)x'=y+x^2+xy^3 ,(b)x'=2x-y+x^2 y'=-2x-y^3 y'=y(x+1)

(a)x'=y+x^2+xy^3 ,(b)x'=2x-y+x^2
y'=-2x-y^3 y'=y(x+1)

For (a), I get the critical points are(0,0) and(-2^1/5, 4^1/5). Then I get det(0,0)=2 and det(-2^1/5, 4^1/5)=-10, so index of the system is -8 and the index of periodic orbit is 1, so there's no periodic orbit.

For (b), I get the critical points are(0,0),(-1,-1)(-2,0). The det is 2,-1,2, so the index is 3 not equal to 1,so no orbit.