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

Abstract

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.