nature.

The large number of studies dealing with escape dynamics shows the importance of this issue in dynamical astronomy. From the very simple systems such as the Hill approximation of the restricted three body problem(Simó & Stuchi 2000) or the Sitnikov problem (Kovács & Érdi 2009) to more complex dynamics, e.g. 3D model of barred galaxies (Contopoulos & Harsoula 2013; Jung & Zotos 2015; Maffione et al. 2015) or open star clusters (Ernst et al. 2015), one can find thorough investigations of finite time chaotic motion. Numerical simulations and also analytic computations have pointed out that the stable and unstable manifolds of the unstable periodic orbits play a crucial role in escape and capture (Ashtakov et al. 2004; Branicki & Wiggins 2010; Harsoula, Contopoulos & Efthymiopoulos 2015; Efthymiopoulos 2012). In computing the unstable periodic orbits and their manifolds, two suitable analytical methods are commonly used, one of these two is called the hyperbolic normal form (Moser 1958), and the other method is the normally hyperbolic invariant manifolds, [*introduced by Fenichel (1979)*]{}. The chaotic saddle can also be constructed numerically by using the sprinkler method (Tél & Gruiz 2006).

At the end of the last century, researchers and engineers started to 