Stability and motion around equilibrium points in the rotating plane-symmetric potential field

Abstract

This study presents a study of equilibrium points, periodic orbits, stabilities, and manifolds in a rotating plane-symmetric potential field. It has been found that the dynamical behaviour near equilibrium points is completely determined by the structure of the submanifolds and subspaces. The non-degenerate equilibrium points are classified into twelve cases. The necessary and sufficient conditions for linearly stable, non-resonant unstable and resonant equilibrium points are established. Furthermore, the results show that a resonant equilibrium point is a Hopf bifurcation point. In addition, if the rotating speed changes, two non-degenerate equilibria may collide and annihilate each other. The theory developed here is lastly applied to two particular cases, motions around a rotating, homogeneous cube and the asteroid 1620 Geographos. We found that the mutual annihilation of equilibrium points occurs as the rotating speed increases, and then the first surface shedding begins near the intersection point of the -x axis and the surface. The results can be applied to planetary science, including the birth and evolution of the minor bodies in the Solar system, the rotational breakup and surface mass shedding of asteroids, etc.

The bifurcation of periodic orbits and equilibrium points in the linked restricted three-body problem with parameter ω

“…17 Perdios and Ragos studied the nonsymmetric asymptotic motion to the collinear equilibrium points (EPs) and discussed their relation to families of symmetric periodic solutions. 18 Recently, many researchers are devoted to the dynamics in the asteroid's model with varying spinning velocities, such as Wang et al, 19 and Jiang et al 20,21 In particular, Ref. 16 gave a global distribution of (µ, ω) pair related to the linear stability of triangular libration points and discussed their nonlinear stability in resonance cases by the Kolmogorov-Arnold-Moser theorem.…”