How to detect Wada basins

Figure 1.  Sketch of the condition 1 of the Nusse-Yorke method. The small disks represent areas of the basins $ B_1 $, $ B_2 $ and $ B_3 $. The unstable manifold of the unstable periodic orbit $ P $ intersects the three basins. The preimages of the disks are stretched exponentially and asymptotically approach the stable manifold, which ultimately is the Wada boundary

Figure 2.  Wada detection with the Nusse-Yorke method (a) Basins of attraction of the forced damped pendulum $ \ddot{x}+0.2\dot{x}+\sin x = 1.66\cos t $, with the unstable manifold of a period three orbit (crosses on the basins). The unstable manifold intersects the three basins. (b) Basins of attraction of the forced damped pendulum $ \ddot{x}+0.2\dot{x}+\sin x = 1.71\cos t $. There are four basins and we have found an accessible periodic orbit whose unstable manifold crosses only two basins. There is also a period-three periodic orbit similar to the case in (a). This basin is partially Wada

Figure 3.  Sketch of the grid method. We set up a grid of boxes $ box_j $ covering the whole disk. The center point of each box defines its color. In the first step, we see that $ box_1 $ belongs to the interior because its surrounding 8 boxes have the same color. On the other hand, $ box_2 $ and $ box_3 $ are in the boundary of two attractors, i.e., they are adjacent to boxes whose color is different. In the next step the algorithm classifies $ box_2 $ still in $ G_2 $ (boundary of two), while $ box_3 $ is now classified in $ G_3 $ (boundary of three). Ideally the process would keep on forever redefining the sets $ G_1, G_2 $ and $ G_3 $ at each step, though in practice we can impose some stopping condition. This plot constitutes an example of partially Wada basins

Figure 4.  Wada detection with the grid method. (a) Basin of attraction of the forced damped pendulum $ \ddot{x}+0.2\dot{x}+\sin x = 1.66\cos t $, (b) All $ 1000 \times 1000 $ boxes are labeled either in the interior (white) or in the boundary of the three basins (black). (c) Histogram showing the number of points $ N $ that take $ q $ steps to be classified as boundary of three basins. (d) After reaching a maximum, there is an exponential decay of the computational effort related to the fractal structure of the basins. The log-plot reflects this tendency

Figure 5.  Forced damped pendulum with eight basins. (a) The following damped forced pendulum $ \ddot{x}+0.2\dot{x}+\sin x = 1.73\cos t $ shows eight basins of attraction mixed intricately. (b) Some boxes are classified to be in the boundary of eight basins (black dots), but not all of them (red dots), which is a clear example of a partially Wada basin. (c) The computational effort presents the usual shape for the Wada boundary, but the points which are not Wada keep refining until the algorithm meet the stop criterion at $ q = 15 $ (the red bar at rightmost represent the number of boxes not classified as Wada at this stage.). The grid method works best in systems with the Wada property. (d) Evolution of the proportion of boxes in the Wada boundary ($ W_8 $ in black) and proportion of boxes in a boundary which is not Wada ($ W_{2 - 7} $) as a function of the $ q $-step. The convergence of $ W_8 $ is used to determine the stopping rule

Figure 6.  Graphical description of the merging of basins. In (a) upper left corner we have the original basin of the forced damped pendulum described by $ \ddot{x}+0.2\dot{x}+\sin x = 1.66\cos t $. The other three panels are the modified basins with two merged basins. The colors above indicate which of the original basins have been merged. In (b) The case of the damped pendulum defined by $ \ddot{x}+0.2\dot{x}+\sin x = 1.71\cos t $ is shown, which possesses four attractors. We have displayed only three of the four possible combinations of merging. However, these examples are enough to show that the boundaries are not identical

Figure 7.  Interpretation of the Hausdorff distance. The figure represents two superimposed slim boundaries computed from two different merged basins. One of the boundaries is plotted with red pixels and the other one with green pixels. While it appears that most of the boundaries overlap, some parts of the red boundary do not coincide with the green boundary. The largest distance between the two boundaries is represented by a red circle of radius $ max_d $ that corresponds to the Hausdorff distance between the two sets of points

Figure 8.  Sketch of the saddle-straddle algorithm. Initially, two points are selected in such a way that each one lies on a different basin. Then, a bisection method is applied to reduce the distance between the two points to a desired accuracy. After that, the resulting points are iterated and the segment expands, so that the process must start over again. As a result, we obtain a set of arbitrarily small segments straddling the saddle

Figure 9.  Computations of saddles with the saddle-straddle algorithm. (a) The picture represents the chaotic saddle embedded in the only boundary of the forced damped pendulum with equation $ \ddot{x}+0.2\dot{x}+\sin x = 1.66\cos t $. (b) We have represented the computation of the saddle associated to the boundary between basins $ B_2 $ and $ M_2 $ of the forced damped pendulum with equation: $ \ddot{x}+0.2\dot{x}+\sin x = 1.71\cos t $. In (c) we have the saddle corresponding to the boundary between basins $ B_1 $ and $ M_1 $. (d) shows the chaotic saddle of the boundary in the Hénon-Heiles Hamiltonian for the energy $ E = 0.25 $