  We are interested in the periodic motion and bifurcations near the surface of
an asteroid. The gravity field of an irregular asteroid and the equation of
motion of a particle near the surface of an asteroid are studied. The periodic
motions around the major body of triple asteroid 216 Kleopatra and the OSIRIS
REx mission target asteroid 101955 Bennu are discussed. We find that motion
near the surface of an irregular asteroid is quite different from the motion
near the surface of a homoplastically spheroidal celestial body. The periodic
motions around the asteroid 101955 Bennu and 216 Kleopatra indicate that the
geometrical shapes of the orbits are probably very sophisticated. There exist
both stable periodic motions and unstable periodic motions near the surface of
the same irregular asteroid. This periodic motion which is unstable can be
resonant or non resonant. The period doubling bifurcation and pseudo period
doubling bifurcation of periodic orbits coexist in the same gravity field of
the primary of the triple asteroid 216 Kleopatra. It is found that both of the
period doubling bifurcations of periodic orbits and pseudo period-doubling
bifurcation of periodic orbits have four different paths. The pseudo period
doubling bifurcation found in the potential field of primary of triple asteroid
216 Kleopatra shows that there exist stable periodic orbits near the primary s
equatorial plane, which gives an explanation for the motion stability of the
triple asteroid 216 Kleopatra s two moonlets, Alexhelios and Cleoselene.
