Ap. Markeyev, 3RD-ORDER RESONANCE IN A HAMILTONIAN SYSTEM WITH ONE DEGREE-OF-FREEDOM, Journal of applied mathematics and mechanics, 58(5), 1994, pp. 793-804
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.