Yesterday I asked a question on here. Unfortunately I closed off the page without fully signing up for my account so I could not comment on the answer I received, whilst the answer was very good there were a few things I did not quite get and wanted some extra clarification. I could not get access to my old account so I will have to post here sorry for any inconvenience this causes.
Here is the question for reference: How to determine which initial conditions will make the solution of a Hamiltonian system periodic?
And the answer provided by @Jonas.
Hint: Notice that the level set $H=0$ is the union of the line $y=0$ and a parabola. Simply drawing these two lines, you will find that they determine a compact invariant set containing your fixed point $(1/2,-1/4)$. So the initial conditions giving a periodic orbit are precisely those that are inside this compact invariant set, other than the fixed point.
Indeed, periodic orbits must have fixed points inside and the other other two fixed points can be ruled out because they are on the line $y=0$, which then would have to be crossed by the periodic orbit.
A few comments I want clearing up in regards to this.
Why do we look straight away for $H=0$ why is this necessary to determine the periodic orbits?
Why do periodic orbits have to contain fixed points inside?
Finally why does the fact that that the two other fixed points are on $y=0$ mean that they would be crossed by the periodic orbit? And how can we then rule them out?
In summary I don't understand the last paragraph.
Thanks for reading.