Topological Classification and Stability of Large-Scale Periodic Orbits

Abstract

Periodic orbit is one of the most important issues among all kinds of large-scale motion types, and has been regarded as the breakthrough of three-body problem by Poincare. Specifically, to orbital dynamics around small bodies, there are at least two reasons for us to focus on the periodic motion, which is related to orbital evolution of real small body system, and is crucial for mission design of approaching the target body. Differing from equilibrium points, to search periodic orbits in the phase space is still an art to date, especially for the highly dimensional systems like Eq. 2.23. The orbital motion equation near a small body Eq. 2.22 has similar formulation with the equation of CRTBP, which suggests that it might include periodic motion of the same level of abundance as the latter. This chapter starts with a specific asteroid. Section 4.2 proposes an algorithm to search large-scale periodic orbits around irregular bodies, which is then applied to find out the periodic orbital families of the target small body. Section 4.3 surveys the stabilities of these orbits, and Sect. 4.4 further describes the topologies of different orbital families, based on which a classification method is proposed to track the topological evolution. Section 4.5 discusses the general motion forms about the periodic orbits, which discriminates different orbital patterns according to the linearized map on the section.