Weak dissipative effects on trajectories from the edge of basins of attraction

The purpose of this work is to present convergence properties of regular and chaotic conservative trajectories under small dissipation. It is known that when subjected to dissipation, stable periodic points become sinks attracting the surrounding trajectories which belong to rational/irrational tori, while chaotic trajectories converge to a chaotic attractor, if it exists. Using the standard map and a mixed plot we show that this simple scenario can be rather complicated and strongly depends on the dissipation intensity. For small dissipations the huge amount of attractors of the phase-space generates a complicated and intricate dynamics where trajectories are steered to their attractors based on the local (non)hyperbolicity, measured by the Lyapunov vectors. Dissipation creates holes (or attracting channels) on the torus from the conservative limit and allows trajectories to penetrate it. These holes are regions of large local hyperbolicity and are related to sticky channels reported recently. For stronger dissipation sinks tend to attract all trajectories, prevailing over the chaotic attractor.