$$ x'=y \mbox{ and } y'=ax-by-x^2y-x^3 $$ I need non-existence of periodic orbits. Which conditions $a$ and $b$ in $\mathbb{R}$ must satisfy?
First, one can see that if $a\leq 0$, then the system has just one equilibrium point, (0,0).
If $a>0$, then there are 3 equilibrium points (0,0), $(\pm a^{\frac{1}{2}}, 0)$.
I tried to use a Lyapunov function of the form $V(x,y)=\pm ax^2 +y^2$ (depending on $a$) to limit (or not) an orbit, but it is really boring because there was a lot of possibilities to analyze.
Is there any simpler way to look at this problem?
from bendxson negative test i think it is enough to show that fx +gy change sign fx=0 gy=-b-x^2 fx+gy=0-b-x^2=-(b+x^2) there is no sign change it means always it is negative so we conclude that there is no periodic solutions