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

This paper is devoted to the bifurcation of periodic orbits and libration points in the linked restricted three-body problem (LR3BP). Inherited from the classic circular restricted three-body problem (CR3BP), it retains most of the dynamical structure of CR3BP, while its dynamical flow is dominated by angular velocity ω and Jacobi energy C. Thus, for the first time, the influence of the angular velocity in the three-body problem is discussed in this paper based on ω-motivated and C-motivated bifurcation. The existence and collision of equilibrium points in the LR3BP are investigated analytically. The dynamic bifurcation of the LR3BP under angular velocity variation is obtained based on three typical kinds of periodic orbits, i.e., planar and vertical Lyapunov orbits and Halo orbits. More bifurcation points are supplemented to Doedel's results in the CR3BP for a global sketch of bifurcation families. For the first time, a new bifurcation phenomenon is discovered that as ω approaches to 1.4, two period-doubling bifurcation points along the Halo family merge together. It suggests that the number and the topological type of bifurcation points themselves can be altered when the system parameter varies in LR3BP. Thus, it is named as “bifurcation of bifurcation” or “secondary bifurcation” in this paper. At selected values of ω, the phase space structures of equilibrium points L2 and L3 are revealed by Lie series method numerically, presenting the center manifolds on the Poincaré section and detecting three patterns of evolution for center manifolds in LR3BP. Holding the key to the origin of the universe, small bodies, e.g., asteroids are attracting more and more interest from academic and industrial fields. Current simulation on asteroid is implemented based on the regular spinning rate of an asteroid body. However, recently, the observation results on some asteroids show that their spinning velocity varies due to the solar radiation pressure, such as 2000 PH5, whose spinning velocity increases by (2.0 ± 0.2) × 10−4°/day2. The effect of the variable spinning velocity has not been fully understood. To cope with the orbital dynamics near a celestial object with varying angular velocity, a linked restricted three-body problem (LR3BP) is proposed in this paper given that the primary and the secondary are connected by a massless link. The bifurcations motivated by both angular velocity and Jacobi energy are detected to present the influence of the angular velocity. The expected results will provide new insights into orbital dynamics near asteroids, serving for future asteroid exploration mission. The LR3BP and the discovered bifurcation phenomena are important theoretical supplementation to the classic three-body problem theory.