An Example of the Quasiperiodic Solution of the Restricted Three Body   Problem at 2:1 Resonance
  The 2:1 mean motion resonance orbit was integrated at the restricted planar 3-body problem in absolute frame. Orbit of Jupiter was assumed circular. Initial Jupiter longitude was assumed zero. The Runge-Kutta method was used. The start of first series of integration was from conjunction point at zero inclination and fixed eccentricity e=0.4 and different pericenter longitudes.   The orbit with encounter at apocenter shows fast clockwise rotation. The orbit with encounter at pericenter rotated counterclockwise. It means, that periodic orbit exist between two investigated ones. It was found, that two orbits with e=0.4, initial perihelion longitude close to 100o has not apsidal line rotation; however it has significant semimajor axis variations. The orbital elements show a very regular behavior on time interval about 3000 years. Due to Laplace theorem, at low perturbation, semimajor axis has only small short periodic oscillations. It means, that motion may be exactly periodic. The cases of another initial eccentricity are considered at range from circular orbit to intersecting orbit. The dependence pericenter longitude of quasi-periodic orbits on eccentricity was found. The orbits with e large 0.5 have catastrophic close encounters with Jupiter and may be periodic only at special value of eccentricity. The additional series of integrations, at small shift from exact mean motion commensurability, was done. 
A. E. Rosaev