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.