Given a Hamiltonian $H$ on $\mathbb{R}^{2n}$ and a periodic orbit $\gamma$, what in general can one say about the existence of periodic orbits near $\gamma$?
I'm almost embarrassed to ask this question because about a decade ago I had a class from Hofer, and I have the Hofer-Zehnder book, but on the other hand Hofer's approach made it difficult for me to see the forest for the trees even at the time.
Here is an article on Hamilton systems near strongly resonant periodic orbits. From the first page:
In a Hamiltonian system periodic orbits are not usually isolated but form one-parametric families. Naturally the value of the Hamiltonian function H plays the role of the parameter. Thus even in the case when the original Hamiltonian does not explicitly contain any parameter, it is possible to observe bifurcations of periodic orbits. A bifurcation corresponds to a resonance between the frequency of a periodic orbit and the frequency of small oscillations around it. In a generic situation there is a family of hyperbolic periodic orbits of a multiple period, which shrinks to the resonant periodic orbit at an exact resonance. Separatrices of the hyperbolic periodic orbit have to intersect due to Hamiltonian nature of the problem. Segments of separatrices of the corresponding resonant normal form make up a closed loop around the periodic trajectory. In section E, appendix 7 of ref. 1, Arnold pointed out that there should be an important qualitative difference between the original Hamiltonian system and its normal form due to the splitting of separatrices.
I would refer you to the version for hamiltonian systems of a result known as Poincaré–Lyapunov theorem that describes the periodic orbits around a known one when a certain condition is satisfied.
Let $(M,\omega)$ be a $2n$-dimensional symplectic manifold, an $H$ a smooth regular function on $M$.
Let $\Lambda$ be a $1$-dimensional compact connected submanifold of $M$ which is invariant under the flow of $X_H$, i.e. $\Lambda$ is the image of a periodic integral curve of $X_H$.
If $1$ is not an eigenvalue for the derivative of the first recurrence map for $X_H$ in a point of $\Lambda$ then there exists a $2$-dimensional symplectic submanifold $N$ of $(M,\omega)$ containing $\Lambda$ such that $H|_N$ is a summersion whose fibers are compact connected and invariant under the flow of $X_H$.
So under the stated non-degeneracy condition a periodic trajectory of $X_H$ is included in a family of periodic orbits forming a symplectic submanifold and parametrized by $H$.
For a reference and a generalization which joins together the Poincaré–Lyapunov theorem with the Liouville–Arnol'd theorem, I would suggest N.N. Nekhoroshev: The Poincaré–Lyapunov–Liouville–Arnol'd theorem. Funct. Anal. Appl. 28 (1994), no. 2, 128–129. Zbl 0847.58035, MR1283258.
Perhaps an partially answer: It is a well known fact, that if $\gamma$ is a non-degenerate periodic orbit, then it is isolated. (see for example Arnold, Dynamical Systems III, Encyclopedia of Mathematical Sciences, Vol 3.)
Are you just interested in the existence or in further properties like stability? If you are interested in stability, perhaps the scholarpedia site will we helpful: Periodic orbit
Perhaps useful though not quite what you are asking for. I remember a counting argument in one of Pierre Lochak's papers using a diophantine type of inequality to show there exist nearly periodic orbits with period less than $T$ near any point for any $T$. As $T$ gets larger there exists a nearer nearly periodic orbit. The argument was used to prove the Nekhoroshev theorem via simultaneous approximation. This counting argument didn't find periodic orbits, as you are asking, but did count nearby nearly periodic orbits.
