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On the families of periodic orbits which bifurcate from the circular Sitnikov motions. (English) Zbl 0818.70011

Summary: We deal with the circular Sitnikov problem as a subsystem of the three- dimensional circular restricted three-body problem. The paper has a first analytical part, where by using elliptic functions we give the analytical expressions for the solutions of the circular Sitnikov problem and for the period function of its family of periodic orbits. We also analyze the qualitative and quantitative behavior of the period function. In the second numerical part, we study the linear stability of the family of periodic orbits of the Sitnikov problem, and of the families of periodic orbits of the three-dimensional circular restricted three-body problem which bifurcate from them; we follow these bifurcated families until they end in families of periodic orbits of the planar circular restricted three-body problem. Finally, the characteristic curves of some bifurcated families obtained for the mass parameter close to \(1/2\) are also described.

MSC:

70F07 Three-body problems
37G99 Local and nonlocal bifurcation theory for dynamical systems
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