I am trying to show that no periodic orbits exist for the system:
$$ x_1'=y+x^2+xy^3$$ $$y'=-2x-y^3$$
I have tried using Dulac's criterion to find a function $g(x,y)$ such that $\Phi(x,y)$ given by :
$$\Phi(x,y)=\frac{\partial(gx')}{\partial x}+\frac{\partial(gy')}{\partial y}$$
Is always $>0$ or $<0$ . But I have had no luck guessing such $g(x,y)$ and leaving $g(x,y)$ general results in a complicated first order PDE. Plotting the system on the phase space seems to imply there is no periodic orbits and that the origin is a focus (although a very slowly converging one) . There is one other critical point at $(x,y)=(-2^{1/5} , 2^{2/5})$ .
Any ideas how I can show that no periodic orbits exist?
I think I've figured it out. Apparently $\Phi(x,y)$ needs only be non-zer0 almost everywhere . Where almost everywhere means except on a set of measure zero. In $\mathbb R^2$ this is a straight line or point.
Therefore using dulac function, $g(x,y)=e^y$ , gives:
$$\Phi(x,y)=-3y^2e^y<0, \forall (x,y):y\not=0$$
$\Phi(x,y)$ is non-zero almost everywhere, therefore no periodic orbits exist in $\mathbb R^2$.