3RD-ORDER RESONANCE IN A HAMILTONIAN SYSTEM WITH ONE DEGREE-OF-FREEDOM

Non-linear oscillations of a nearly integrable Hamiltonian system with one degree of freedom, which is 2pi-periodic in t, are investigated i n a small finite neighbourhood of equilibrium. The Hamiltonian is assu med to be analytic, the linearized system is stable, and its character istic exponents +/-iv are purely imaginary, where 3nu is an integer. T he equilibrium position of such a system is generally unstable and six trajectories exist that asymptotically approach the equilibrium point as t --> +/-infinity [1,2]. It is shown that for most initial data th e motion is quasi-periodic in the neighbourhood of the equilibrium. Th e existence of stable 6pi-periodic motions near an unstable equilibriu m position is established. It is shown that, irrespective of instabili ty, trajectories beginning sufficiently close to an equilibrium point will always remain at a finite distance from it. An estimate is obtain ed for this distance. The stochastic nature of the motion near traject ories asymptotic to the equilibrium point is discussed.