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. These families admit a global continuation up to excentricity $e=1$. The same techniques are applicable to the families obtained by continuation from the circular problem ($e=0$). They lead to a refinement of a result obtained by J. Llibre and R. Ortega.
Highlights
The Sitnikov problem is a restricted three-body problem where two primaries of equal mass move in elliptic orbits lying on the plane x, y and the particle of zero mass moves on the z axis
One of the first questions that can be posed about (1) is the study of families of periodic solutions which depend continuously on the eccentricity. This question becomes simpler if one is restricted to the symmetric case: families of even or odd periodic solutions
The standard transversality conditions are not easy to check and we present a slight variant of the theorem on pitchfork bifurcation adapted to our problem
Summary
The Sitnikov problem is a restricted three-body problem where two primaries of equal mass move in elliptic orbits lying on the plane x, y and the particle of zero mass moves on the z axis. One of the first questions that can be posed about (1) is the study of families of periodic solutions which depend continuously on the eccentricity. The method of global continuation of Leray and Schauder has been applied more recently in [10] and its has been shown that some of these families can be continued to all the values of the eccentricity These families are labelled according to the number of zeros in the same fashion as it occurs in the well known work by Rabinowitz [17] for other non-linearities. As a consequence the critical values e = En,N act as barriers for the families and this allows us to control their behaviour This technique is applicable to the families emerging from the circular problem and we derive a sharpened version of the main result in [10]. At the end of the paper we show how to reduce this problem to a question concerning a linear equation of Hill’s type
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