Global bifurcations from the center of mass in the Sitnikov problem

Abstract

The Sitnikov problem is a restricted three body problem where the eccentricity of the primaries acts as a parameter. We find families of symmetric periodic solutions bifurcating from the equilibrium at the center of mass.

Periodic Solutions in the Generalized Sitnikov $(N+1)$-Body Problem

“…We will see that most of the results in [11,13] can be obtained in the (N + 1)-body problem under consideration, but with two major differences in relation to the classical Sitnikov problem. These differences will focus on the second group of families mentioned above.…”

“…These families can be continued for all larger values e ∈ ]e * , 1[. An analytical study of these periodic solutions that bifurcate from the equilibrium is given in [13]. Here we consider the Sitnikov problem with N primaries with equal mass moving around the origin in the plane x, y as generalized solutions of the Lagrange problem.…”

“…Here we apply the same techniques found in [11,13] to our GSP in order to analytically study some properties of two groups of families of periodic solutions; the first group is families that are globally continued from the circular GSP (e = 0) to the elliptic GSP (e = 0), and the second is families that arise as bifurcations from equilibrium solution at certain special values of the eccentricity.…”

“…The function r(·, e) has a minimal period 2π and is given implicitly by the equation with t 0 the time of pericenter passage. The Sitnikov problem has been studied by many authors (we refer the reader to [1,4,5,6,7,8,9,10,11,13,20,22]) and became important when Sitnikov [22] showed for the first time the existence of oscillatory motions in this restricted 3-body problem. The study of families of symmetric periodic orbits for (1) has been studied in [6,7] using arguments of continuation from periodic solutions of the circular Sitnikov problem (e = 0) to the elliptic Sitnikov problem (e = 0) for small values of the eccentricity.…”

Quantifying Poincare’s Continuation Method for Nonlinear Oscillators

“…This approach has been successfully applied in the study of periodic solutions on a restricted threebody problem (see [10,11]). In order to apply this tool it is necessary to compute a priori bounds over the zeros of (see Theorem 1 in Section 2 for more details) but the conclusion of the Leray-Schauder Theorem says nothing about the linear stability of the associated periodic solutions.…”