TABLE OF CONTENTS Let us now examine the manifolds of the hilltop saddle of the system. To this end, we apply the method of Poincar map, the method that transforms a T-periodic solution into a single point in the phase plane (see Section 3.2). In our case, the hilltop saddle represents the unstable T-periodic solution, which appears close to the top of the potential energy. For the sake of simplicity, we will concentrate our attention on the region of control parameters F, ? where only one resonant attractor S r exists in each potential well. In Figure 4.5, the defined region is situated to the right from the bifurcation line snA.
Figure 4.6(a) displays, in the Poincar plane 
, the resonant attractors in both left and right potential wells (represented by points 
and 
, respectively), as well as the hilltop saddle (denoted D H), and its stable and unstable manifolds (depicted W s and W u, respectively). At this point, we recall that the stable manifolds separate the basins of attraction of different coexisting attractors, whereas the unstable manifolds tend directly to the attractors (in this case, to the points , ). It follows that all trajectories starting in the white region tend to the left attractor (i.e. periodic oscillations of a ball in the left well), whereas all trajectories starting in the blue region tend to the right attractor . The boundary of both basins of attraction is a smooth,...
