Periodic solutions of even Hamiltonian systems on the torus (Q1358195)

We consider the Hamiltonian system (HS) \(-J\dot z=H_z(t,z)\) where \(H\in {\mathcal C}^2 (\mathbb{R} \times \mathbb{R}^{2N}, \mathbb{R})\) is \(2\pi\)-periodic in all variables, so (HS) induces a Hamiltonian system on the torus \(T^{2N}\). In addition we assume that \(H\) is even in the \(z\)-variable. This implies the existence of \(2^{2N}\) trivial stationary solutions of (HS). We are interested in the existence of nontrivial periodic solutions. Observe that the Arnold conjecture does not guarantee any more periodic solutions than those already given by the stationary solutions. Let \(w_0\) be such a trivial stationary solution (considered as a \(2\pi\)-periodic solution) and let \(i(w_0)\) be its Maslov index. Under certain nondegeneracy assumptions we prove the existence of at least \(|i(w_0) |-N\) pairs \(\pm z(t)\) of nontrivial \(2\pi\)-periodic solutions.