Periodic orbits and bifurcations in the Sitnikov four-body problem

P. S. Soulis, K. E. Papadakis, T. Bountis

Research output: Contribution to journalArticlepeer-review

39 Citations (Scopus)


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" ż0 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 languageEnglish
Pages (from-to)251-266
Number of pages16
JournalCelestial Mechanics and Dynamical Astronomy
Issue number4
Publication statusPublished - Apr 1 2008


  • 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

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