use the manifolds of periodic orbits to describe the dynamics of spacecraft manoeuvres and station keeping. The first method of constructing trajectories from the Earth to the Moon with lunar ballistic capture using low thrust spiral orbit [*was published by Belbruno (1987).*]{} Determining the weak stability boundary around the Moon was a great leap forward in the study of the low thrust orbits (Kevin, Belbruno & Topputo 2012). Although they used methods similar to ours, but without the conception of transient chaos (survival probability, escape rate, chaotic saddle, etc.) they did not give the explication to the phenomena they experienced (e.g. the magnitude of the thrusting, the stabilization of the motion).

The new dynamical system approach technique seemed to be an extremely useful application in space research, especially for precise orbit determination. Many recent space missions (Genesis (Howell et al. 1998), MAP (Bennett 1996) and Herschel-Planck (Rosa et al. 2005)) were designed using this so-called space manifold dynamics method (SMD) (Belló, Gómez & Masdemont 2010; Belbruno 2010).

The basic purpose of this technique is to find an orbit from one place to other, or keep a spacecraft at a given position around an unstable periodic orbit as cheaply as possible, 