# Application of the Poincaré-Bendixson theorem

I am trying to solve the following exercise (19) from this magnificent notes but I am encountering some problems:

Prove that the system

$$\begin{cases} \dot{x} = 2x-x^5-xy^4 \\ \dot{y} = y-y^3-yx^2 \end{cases}$$

has no periodic orbits. I understand this comes from the following corollary of the Poincaré-Bendixson theorem, also stated in the notes:

If the flow has a periodic orbit such that the region bounded by it is contained in Ω then that region contains an equilibrium

I have been working with the hint given (i.e. restricting the flow to the axes where all the equilibrium points are) but I do not know how to prove that the system has no periodic orbits. I would appreciate your help. Thank you very much in advance!

The hint says that all equilibria are on the axes. After some computations you will find that these are $$(0,-1), \quad (0,0),\quad (0,1),\quad (-2^{1/4}, 0),\quad (2^{1/4}, 0).$$ As you wrote, if the flow has a periodic orbit, then the region that it bounds contains an equilibrium. So necessarily the orbit must cross at least one of the axes since this is the only way that one of those $$5$$ points is "inside" the region bounded by the periodic orbit.
But both axes are invariant and so no orbit can cross them. Indeed, for the horizontal axis, taking $$y=0$$ the second equation is automatically verified and so the flow is obtained solving $$\dot x=2x-x^5$$ (which is possible because it does not depend on $$y$$). A similar observation applies to the vertical axis.