Periodic orbits and bifurcations in the Sitnikov four-body problem |
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Authors: | P S Soulis K E Papadakis T Bountis |
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Institution: | (1) Department of Mathematics and Center for Research and Applications of Nonlinear Systems, University of Patras, 26504 Patras, Greece;(2) Department of Engineering Sciences, Division of Applied Mathematics and Mechanics, University of Patras, 26504 Patras, Greece |
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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” 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 along 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. |
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Keywords: | Four-body problem Sitnikov motions Stability Critical periodic orbits 3-Dimensional periodic orbits Ordered motion Chaos Sticky orbits Escape regions Poincaré map |
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