A note on the relation between period and energy of periodic orbits near equilibrium points

We investigate the growth of the periods of periodic solutions of ordinary differential equations which are arbitrarily close to an isolated equilibrium point. More precisely, we estimate the period of periodic orbits near a completely degenerate equilibrium point (the derivative of the vector field at the equilibrium point is identically zero) and show that it tends to infinity as the amplitude of the orbits tends to zero. Based on this, we give estimates on the relation of the period and the energy of periodic orbits near singular points in Hamiltonian systems.