Lyapunov functions for nonautonomous systems close to Hamiltonian

Non-autonomous perturbations of autonomous hamiltonian system with one degree of freedom are considered in the paper. The problem of stability of equilibrium is solved by the second Lyapunov method. Hamiltonian of autonomous system is taken as the leading term of Lyapunov function. The conditions of stability of equilibrium are determined by the coefficients of the perturbations. The stability and unstability theorems are proved. The obtained results are applied to research the stability of solutions of the Painleve equations.