Stabilization of Chaotic Behavior in the Restricted Three-Body Problem

Henri Poincaré in his work on Celestial Mechanics underlined a possibility of chaotic behavior in threebody problem by the destruction of homoclinic contours. Later, an existence of transverse homoclinic points in the three-body problem was analytically verified. A well-known modification of the restricted three-body problem is that of Sitnikov [1960]. The Sitnikov problem takes place when two equal masses M orbit around their barycentre. A third, massless or small but finite mass body (particle) μ moves in their gravitational field perpendicular to the motion surface of the primaries (see Fig. 1). It can be shown, that the oscillation of the third body is chaotic (under certain additional conditions). In this work we consider the problem of stabilization of this chaotic behavior. In general, this problem is related to stabilization and control of unstable and chaotic behavior of dynamical systems by external forces. A comprehensive study of chaotic systems with external controls may thus provide a key to the understanding many nonlinear processes in both localized and distributed systems. This could be of interest, for example, in the study of planetary systems of binary stars. μ