I was learning qualitative theory of ODEs and was confused about the relation between the existence of singularity and periodic orbits.(We care about flows generated by $C^1$ vector fields from $\mathbb{R}^n$ to itself)
Namely, is it true that if a system have periodic orbit, then there is a singular point?
Or in general how to determine whether a system have periodic orbits or not, or the behavior (i.e. Lyapunov Stability, etc.) of singularities, or the relationship between them?
I know the Poincare-Bendixson theorem states that if the orbit is bounded and the limit set is without singularity then it is a periodic orbit, but what about in other conditions, is there some regular method to analyze it?
If you have a periodic orbit then it must enclose a critical point. This is because the Poincare Index of a periodic orbit must be equal to 1.
In general, proving the existence of a periodic orbit is highly non-trivial. There are some techniques that may assist you in finding them (conserved quantities, integral curves, Poincare-Bendixon) but this is in general difficult to do.
If you're looking at Poincare-Bendixon, be careful in $\mathbb{R}^n$ as this does not hold in dimensions greater than $2$.
Hopefully this somewhat answers your question!