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The order and chaos of the motion near equilibrium points in the potential of a rotating highly irregular-shaped celestial body are investigated from point of view of the dynamical system theory. The positions of the non-degenerate equilibrium points vary continuously when the parameter changes. The topological structures in the vicinity of equilibrium points are classified into several different cases. Bifurcations at equilibrium points and the topological transfers between different cases for equilibrium points are also discussed. The conclusions can be applied to all kinds of rotating celestial bodies, simple-shaped or highly irregular-shaped, including asteroids, comets, planets, and satellites of planets to help one to understand the dynamical behaviors around them. Applications to asteroids 216 Kleopatra, 2063 Bacchus, and 25143 Itokawa are significant and interesting: Eigenvalues affiliated to the equilibrium points for the asteroid 216 Kleopatra move and always belong to the same topological cases, while eigenvalues affiliated to two different equilibrium points for the asteroid 2063 Bacchus and 25143 Itokawa move through the resonant cases of equilibrium points, and the collision of eigenvalues in the complex plane occurs. Poincaré sections in the potential of the asteroid 216 Kleopatra show the chaos behaviors of the orbits in large scale.

Order and chaos near equilibrium points in the potential of rotating highly irregular-shaped celestial bodies