On the stability of periodic Hamiltonian systems with one degree of freedom in the case of degeneracy (Q283348)

It is considered a Hamiltonian system defined by the pair of canonical differential equations \[ \frac{dx}{dt}=\frac{\partial H}{\partial y},\quad \frac{dy}{dt}=-\frac{\partial H}{\partial x} \] Here the Hamiltonian \(H(x,y,t)\) is an analytic function of \(x\), \(y\), \(t\) in a small neighborhood of the origin and \(2\pi\)-periodic in \(t\), the characteristic exponents of the linearized system have zero real part, i.e., a nonlinear analysis is necessary to study stability in the sense of Lyapunov. The authors studied the degenerate cases and establish general conditions for stability. These results are applied to obtain new stability criteria for the cases of degeneracy, which appear in the presence of first, second, third and fourth order resonance.