Bifurcations of relative equilibrium points during homotopy deformation of asteroids

Abstract

The bifurcations of relative equilibrium points in the gravitational field of asteroids during homotopy deformation are investigated in this paper. A theorem concerning the continuity of non-degenerate equilibrium points under small variations of parameters and degeneration of equilibrium points after collisions is presented. This can contribute to understanding the bifurcations of the number of equilibrium points. We concentrate on four types of homotopy deformation: from a rotating ellipsoid to a rotating sphere; from 216 Kleopatra to a rotating ellipsoid, a cube and a rectangular parallelepiped. The results show that during the process of deforming a rotating ellipsoid to a sphere, the number of relative equilibria may vary from 1 to 3 to 5. Infinitely many relative equilibria may occur in some critical cases. When deforming 216 Kleopatra to a rotating ellipsoid or a rectangular parallelepiped, the number of relative equilibria changes from 7 to 5 and collisional annihilation of two relative equilibria inside the body occurs, corresponding to a saddle-node bifurcation. The number of relative equilibria changes from 7 to 5 to 9 when 216 Kleopatra is deformed into a rotating cube. Both the saddle-node bifurcation and Hopf bifurcation occur during deformation. Moreover, the positions, eigenvalues, topological types and stability of equilibrium points are studied here.