## Abstract

We study the existence, linear stability and bifurcations of what we call the Sitnikov family of straight line periodic orbits in the case of the restricted four-body problem, where the three equal mass primary bodies are rotating on a circle and the fourth (small body) is moving in the direction vertical to the center mass of the other three. In contrast to the restricted three-body Sitnikov problem, where the Sitnikov family has infinitely many stability intervals (hence infinitely many Sitnikov critical orbits), as the "family parameter" ż_{0} varies within a finite interval (while z _{0} tends to infinity), in the four-body problem this family has only one stability interval and only twelve 3-dimensional (3D) families of symmetric periodic orbits exist which bifurcate from twelve corresponding critical Sitnikov periodic orbits. We also calculate the evolution of the characteristic curves of these 3D branch-families and determine their stability. More importantly, we study the phase space dynamics in the vicinity of these orbits in two ways: First, we use the SALI index to investigate the extent of bounded motion of the small particle off the z-axis alo ng its interval of stable Sitnikov orbits, and secondly, through suitably chosen Poincaré maps, we chart the motion near one of the 3D families of plane-symmetric periodic orbits. Our study reveals in both cases a fascinating structure of ordered motion surrounded by "sticky" and chaotic orbits as well as orbits which rapidly escape to infinity.

Original language | English |
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Pages (from-to) | 251-266 |

Number of pages | 16 |

Journal | Celestial Mechanics and Dynamical Astronomy |

Volume | 100 |

Issue number | 4 |

DOIs | |

Publication status | Published - Apr 1 2008 |

## Keywords

- 3-Dimensional periodic orbits
- Chaos
- Critical periodic orbits
- Escape regions
- Four-body problem
- Ordered motion
- Poincaré map
- Sitnikov motions
- Stability
- Sticky orbits

## ASJC Scopus subject areas

- Modelling and Simulation
- Mathematical Physics
- Astronomy and Astrophysics
- Space and Planetary Science
- Computational Mathematics
- Applied Mathematics