On the stability of equilibria in degenerate periodic Hamiltonian systems

This paper addresses the problem of Lyapunov stability for equilibrium solutions in periodic Hamiltonian systems under degenerate conditions. The degeneracy occurs in resonant cases when the leading coefficient function in the normal form possesses only zeros of high multiplicity, rendering standard criteria inapplicable. By employing Lie normal form theory, we establish two general stability criteria that resolve the stability problem for nearly all such degenerate scenarios. Our results explicitly demonstrate the decisive role of higher-order terms and provide easily verifiable conditions that generalize several known results in the literature.