Pseudo bifurcation and variety of periodic ratio for periodic orbit families close to asteroid (22) Kalliope

This paper is focused on the pseudo bifurcations and the variety of periodic ratio in the periodic orbits near the primary of binary irregular asteroid system (22) Kalliope, which would help on trajectory design for asteroid missions and give a practical insight into the generation and dynamic behaviour of binary asteroid systems. In this paper, we find three basic pseudo bifurcations in the periodic orbit families near (22) Kalliope during the numerical continuation with the variation of Jacobian constant. We also discover a nonuple mixed bifurcations which possess the highest multiplicity of bifurcations ever found and consist of three pseudo tangent bifurcations, two period-doubling bifurcations, one pseudo period-doubling bifurcation, one Neimark-Sacker bifurcation, one pseudo Neimark-Sacker bifurcation, and one real saddle bifurcation. Moreover, we find that the periodic ratio in the periodic orbit family may change during the continuation. Based on plenty of numerical evidences, we summarize three astonishing and exciting conclusions about the relationship of the periodic ratio and (pseudo) bifurcations in the periodic motion near (22) Kalliope. Firstly, the pseudo period-doubling bifurcation shows up when the periodic ratio comes near a half-integer (i.e. 1.5:1, 2.5:1, 3.5:1). Furthermore, almost all the cuspidal changes of the periodic ratio are accompanied with tangent bifurcation or pseudo tangent bifurcation. In addition, an integer can be admitted as the asymptotic value of the periodic ratio at the end of continuation, if the Jacobian constant isn't stuck into its local extremum, yet a half-integer can not.