I have the Hamiltonian system given by $$H=\frac{1}{2}x^2-\frac{1}{3}x^3+xy^2+\frac{1}{2}y^2$$ Using computer software I managed to plot the dynamical system in the phase plane. I am aware that the equilibrium point given by $(0,0)$ is a centre of the system and graphing shows that there are certainly periodic solutions close to this equilibrium point but I'm not sure exactly how to give a description of the region in which all phase paths are periodic.
I am thinking to use Poincare-Benedixson to try and find a invariant compact set in the plane with no equilibrium points to show that each path must be periodic and it is clear that around $(0,0)$ this is true but I'm not sure how "close" we need to be and how to actually write down a set of the form $\{(x,y)\in \Bbb{R^2}:\text{Some restirction on x,y}\}$ such that the phase path through $(x,y)$ is periodic.
Any help?
Apologies for the short answer, but I believe the answer to this question contains a lot of useful information, as the question is about the same Hamiltonian system.
Hint: The trapping region seems to be an equilateral triangle. An equilateral triangle has a symmetry group $C_3$ (see here), which just means that you can rotate the triangle over an angle $\frac{2 \pi}{3}$ to obtain the same triangle. It is very worthwhile to test if the dynamical system associated with your Hamiltonian has the same symmetry. Suppose that the system is indeed invariant under such rotations, how could you use this to show the existence of closed orbits?