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January  2021, 26(1): 717-739. doi: 10.3934/dcdsb.2020330

How to detect Wada basins

Alexandre Wagemakers 1,, , Alvar Daza 1,2, and  Miguel A. F. Sanjuán 1,3,

1. 

Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, Spain
2. 

Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
3. 

Department of Applied Informatics, Kaunas University of Technology, Studentu 50-415, Kaunas LT-51368, Lithuania

* Corresponding author: Alexandre Wagemakers

Received  April 2020 Revised  October 2020 Published  January 2021 Early access  November 2020

We present a review of the different techniques available to study a special kind of fractal basins of attraction known as Wada basins, which have the intriguing property of having a single boundary separating three or more basins. We expose several approaches to identify this topological property that rely on different, but not exclusive, definitions of the Wada property.

Keywords: Wada basins, Wada detection, fractals basins, numerical methods, predictability.
Mathematics Subject Classification: 37M05, 37M21, 65P20, 28A80, 37C70.
Citation: Alexandre Wagemakers, Alvar Daza, Miguel A. F. Sanjuán. How to detect Wada basins. Discrete and Continuous Dynamical Systems - B, 2021, 26 (1) : 717-739. doi: 10.3934/dcdsb.2020330
References:
[1]

J. Aguirre and M. A. F. Sanjuán, Unpredictable behavior in the Duffing oscillator: Wada basins, Physica D, 171 (2002), 41-51.  doi: 10.1016/S0167-2789(02)00565-1.

[2]

J. Aguirre, J. C. Vallejo and M. A. F. Sanjuán, Wada basins and chaotic invariant sets in the Hénon-Heiles system, Phys. Rev. E, 64 (2001), 066208.

[3]

P. M. Battelino, C. Grebogi, E. Ott, J. A. Yorke and E. D. Yorke, Multiple coexisting attractors, basin boundaries and basic sets, Physica D, 32 (1988), 296-305.  doi: 10.1016/0167-2789(88)90057-7.

[4]

X. Chen, T. Nishikawa and and A. E. Motter, Slim fractals: The geometry of doubly transient chaos, Phys. Rev. X, 7 (2017), 021040. doi: 10.1103/PhysRevX.7.021040.

[5]

J. C. P. Coninck, S. R. Lopes, R. L. Viana and Ri cardo, Basins of attraction of nonlinear wave–wave interactions, Chaos, Solitons & Fractals, 32 (2007), 711-724.  doi: 10.1016/j.chaos.2005.11.061.

[6]

A. Daza, J. O. Shipley, S. R. Dolan and M. A. F. Sanjuán, Wada structures in a binary black hole system, Phys. Rev. D, 98 (2018), 084050, 13 pp. doi: 10.1103/physrevd.98.084050.

[7]

A. Daza, A. Wagemakers, B. Georgeot, D. Guéry-Odelin and M. A. F. Sanjuán, Basin entropy: A new tool to analyze uncertainty in dynamical systems, Sci. Rep., 6 (2016), 31416. doi: 10.1038/srep31416.

[8]

A. Daza, A. Wagemakers and M. A. F. Sanjuán, Ascertaining when a basin is Wada: The merging method, Sci. Rep., 8 (2018), 9954. doi: 10.1038/s41598-018-28119-0.

[9]

A. Daza, A. Wagemakers and M. A. F. Sanjuán, Wada property in systems with delay, Commun. Nonlinear Sci. Numer. Simul., 43 (2017), 220-226.  doi: 10.1016/j.cnsns.2016.07.008.

[10]

A. Daza, A. Wagemakers, M. A. F. Sanjuán and J. A. Yorke, Testing for basins of Wada, Sci. Rep., 5 (2015), 16579. doi: 10.1038/srep16579.

[11]

G. Edgar, Measure, Topology, and Fractal Geometry, Springer, New York, 2008. doi: 10.1007/978-0-387-74749-1.

[12]

B. I. Epureanu and H. S. Greenside, Fractal basins of attraction associated with a damped Newton's method, SIAM Rev., 40 (1998), 102-109.  doi: 10.1137/S0036144596310033.

[13]

J. H. Friedman, J. L. Bentley and R. A. Finkel, An algorithm for finding best matches in logarithmic expected time, ACM Trans. Math. Softw., 3 (1977), 209-226.  doi: 10.1145/355744.355745.

[14]

M. Hansen, D. R. da Costa, I. L. Caldas and E. D. Leonel, Statistical properties for an open oval billiard: An investigation of the escaping basins, Chaos Solitons Fractals, 106 (2018), 355-362.  doi: 10.1016/j.chaos.2017.11.036.

[15]

J. G. Hocking and G. S. Young, Topology, Dover, New York, 1988.

[16]

J. Kennedy and J. A. Yorke, Basins of Wada, Physica D, 51 (1991), 213-225.  doi: 10.1016/0167-2789(91)90234-Z.

[17]

C. Kuratowski, Sur les coupures irréductibles du plan, Fundamenta Mathematicae, 6 (1924), 130-145.  doi: 10.4064/fm-6-1-130-145.

[18]

G. Lu, M. Landauskas and M. Ragulskis, Control of divergence in an extended invertible logistic map, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 1850129, 15 pp. doi: 10.1142/S0218127418501298.

[19]

A. C. Mathias, R. L. Viana, T. Kroetz and I. L. Caldas, Fractal structures in the chaotic motion of charged particles in a magnetized plasma under the influence of drift waves, Physica A, 469 (2017), 681-694.  doi: 10.1016/j.physa.2016.11.049.

[20]

H. E. Nusse, E. Ott and J. A. Yorke, Saddle-Node Bifurcations on Fractal Basin Boundaries, Phys. Rev. Lett., 75 (1995), 2482-2485.  doi: 10.1103/PhysRevLett.75.2482.

[21]

H. E. Nusse and J. A. Yorke, Wada basin boundaries and basin cells, Physica D, 90 (1996), 242-261.  doi: 10.1016/0167-2789(95)00249-9.

[22]

H. E. Nusse and J. A. Yorke, Dynamics: Numerical Explorations, Springer, New York, 2012.

[23]

H. E. Nusse and J. A. Yorke, Fractal basin boundaries generated by basin cells and the geometry of mixing chaotic flows, Phys. Rev. Lett., 84 (2000), 626-629.  doi: 10.1103/PhysRevLett.84.626.

[24]

L. Poon, J. Campos, E. Ott and C. Grebogi, Wada basin boundaries in chaotic scattering, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 6 (1996), 251-265.  doi: 10.1142/S0218127496000035.

[25]

M. A. F. Sanjuán, J. Kennedy, E. Ott and J. A. Yorke, Indecomposable continua and the characterization of strange sets in nonlinear dynamics, Phys. Rev. Lett., 78 (1997), 1892.

[26]

L. G. Shapiro and G. C. Stockman, Computer vision, Prentice Hall, Upper Saddle River, NJ, 2001.

[27] T. Tél and M. Gruiz, Chaotic Dynamics: An Introduction based on Classical Mechanics, Cambridge University Press, Cambridge, 2006.  doi: 10.1017/CBO9780511803277.
[28]

Z. Toroczkai, G. Károlyi, A. Péntek, T. Tél, C. Grebogi and J. A. Yorke, Wada dye boundaries in open hydrodynamical flows, Physica A, 239 (1997), 235-243. 

[29]

J. Vandermeer, Wada basins and qualitative unpredictability in ecological models: a graphical interpretation, Ecol. Model., 176 (2004), 65-74. 

[30]

A. Wagemakers, A. Daza and M. A. F. Sanjuán, The saddle-straddle method to test for Wada basins, Commun. Nonlinear Sci. Numer. Simul., 84 (2020), 105167, 8 pp. doi: 10.1016/j.cnsns.2020.105167.

[31]

K. Yoneyama, Theory of continuous sets of points, Tokohu Math. J., 11 (1917), 43-158. 

[32]

Y. Zhang and G. Luo, Unpredictability of the Wada property in the parameter plane, Physics Letters A, 376 (2012), 3060-3066.  doi: 10.1016/j.physleta.2012.08.015.

[33]

Y. Zhang and G. Luo, Wada bifurcations and partially Wada basin boundaries in a two-dimensional cubic map, Phys. Lett. A, 377 (2013), 1274-1281.  doi: 10.1016/j.physleta.2013.03.027.

[34]

Y. Zhang, G. Luo, Q. Cao and M. Lin, Wada basin dynamics of a shallow arch oscillator with more than 20 coexisting low-period periodic attractors, International Journal of Non-Linear Mechanics, 58 (2014), 151-161.  doi: 10.1016/j.ijnonlinmec.2013.09.009.

[35]

P. Ziaukas and M. Ragulskis, Fractal dimension and Wada measure revisited: no straightforward relationships in ndds, Nonlinear Dynamics, 88 (2017), 871-882.  doi: 10.1007/s11071-016-3281-4.

show all references

References:
[1]

J. Aguirre and M. A. F. Sanjuán, Unpredictable behavior in the Duffing oscillator: Wada basins, Physica D, 171 (2002), 41-51.  doi: 10.1016/S0167-2789(02)00565-1.

[2]

J. Aguirre, J. C. Vallejo and M. A. F. Sanjuán, Wada basins and chaotic invariant sets in the Hénon-Heiles system, Phys. Rev. E, 64 (2001), 066208.

[3]

P. M. Battelino, C. Grebogi, E. Ott, J. A. Yorke and E. D. Yorke, Multiple coexisting attractors, basin boundaries and basic sets, Physica D, 32 (1988), 296-305.  doi: 10.1016/0167-2789(88)90057-7.

[4]

X. Chen, T. Nishikawa and and A. E. Motter, Slim fractals: The geometry of doubly transient chaos, Phys. Rev. X, 7 (2017), 021040. doi: 10.1103/PhysRevX.7.021040.

[5]

J. C. P. Coninck, S. R. Lopes, R. L. Viana and Ri cardo, Basins of attraction of nonlinear wave–wave interactions, Chaos, Solitons & Fractals, 32 (2007), 711-724.  doi: 10.1016/j.chaos.2005.11.061.

[6]

A. Daza, J. O. Shipley, S. R. Dolan and M. A. F. Sanjuán, Wada structures in a binary black hole system, Phys. Rev. D, 98 (2018), 084050, 13 pp. doi: 10.1103/physrevd.98.084050.

[7]

A. Daza, A. Wagemakers, B. Georgeot, D. Guéry-Odelin and M. A. F. Sanjuán, Basin entropy: A new tool to analyze uncertainty in dynamical systems, Sci. Rep., 6 (2016), 31416. doi: 10.1038/srep31416.

[8]

A. Daza, A. Wagemakers and M. A. F. Sanjuán, Ascertaining when a basin is Wada: The merging method, Sci. Rep., 8 (2018), 9954. doi: 10.1038/s41598-018-28119-0.

[9]

A. Daza, A. Wagemakers and M. A. F. Sanjuán, Wada property in systems with delay, Commun. Nonlinear Sci. Numer. Simul., 43 (2017), 220-226.  doi: 10.1016/j.cnsns.2016.07.008.

[10]

A. Daza, A. Wagemakers, M. A. F. Sanjuán and J. A. Yorke, Testing for basins of Wada, Sci. Rep., 5 (2015), 16579. doi: 10.1038/srep16579.

[11]

G. Edgar, Measure, Topology, and Fractal Geometry, Springer, New York, 2008. doi: 10.1007/978-0-387-74749-1.

[12]

B. I. Epureanu and H. S. Greenside, Fractal basins of attraction associated with a damped Newton's method, SIAM Rev., 40 (1998), 102-109.  doi: 10.1137/S0036144596310033.

[13]

J. H. Friedman, J. L. Bentley and R. A. Finkel, An algorithm for finding best matches in logarithmic expected time, ACM Trans. Math. Softw., 3 (1977), 209-226.  doi: 10.1145/355744.355745.

[14]

M. Hansen, D. R. da Costa, I. L. Caldas and E. D. Leonel, Statistical properties for an open oval billiard: An investigation of the escaping basins, Chaos Solitons Fractals, 106 (2018), 355-362.  doi: 10.1016/j.chaos.2017.11.036.

[15]

J. G. Hocking and G. S. Young, Topology, Dover, New York, 1988.

[16]

J. Kennedy and J. A. Yorke, Basins of Wada, Physica D, 51 (1991), 213-225.  doi: 10.1016/0167-2789(91)90234-Z.

[17]

C. Kuratowski, Sur les coupures irréductibles du plan, Fundamenta Mathematicae, 6 (1924), 130-145.  doi: 10.4064/fm-6-1-130-145.

[18]

G. Lu, M. Landauskas and M. Ragulskis, Control of divergence in an extended invertible logistic map, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 1850129, 15 pp. doi: 10.1142/S0218127418501298.

[19]

A. C. Mathias, R. L. Viana, T. Kroetz and I. L. Caldas, Fractal structures in the chaotic motion of charged particles in a magnetized plasma under the influence of drift waves, Physica A, 469 (2017), 681-694.  doi: 10.1016/j.physa.2016.11.049.

[20]

H. E. Nusse, E. Ott and J. A. Yorke, Saddle-Node Bifurcations on Fractal Basin Boundaries, Phys. Rev. Lett., 75 (1995), 2482-2485.  doi: 10.1103/PhysRevLett.75.2482.

[21]

H. E. Nusse and J. A. Yorke, Wada basin boundaries and basin cells, Physica D, 90 (1996), 242-261.  doi: 10.1016/0167-2789(95)00249-9.

[22]

H. E. Nusse and J. A. Yorke, Dynamics: Numerical Explorations, Springer, New York, 2012.

[23]

H. E. Nusse and J. A. Yorke, Fractal basin boundaries generated by basin cells and the geometry of mixing chaotic flows, Phys. Rev. Lett., 84 (2000), 626-629.  doi: 10.1103/PhysRevLett.84.626.

[24]

L. Poon, J. Campos, E. Ott and C. Grebogi, Wada basin boundaries in chaotic scattering, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 6 (1996), 251-265.  doi: 10.1142/S0218127496000035.

[25]

M. A. F. Sanjuán, J. Kennedy, E. Ott and J. A. Yorke, Indecomposable continua and the characterization of strange sets in nonlinear dynamics, Phys. Rev. Lett., 78 (1997), 1892.

[26]

L. G. Shapiro and G. C. Stockman, Computer vision, Prentice Hall, Upper Saddle River, NJ, 2001.

[27] T. Tél and M. Gruiz, Chaotic Dynamics: An Introduction based on Classical Mechanics, Cambridge University Press, Cambridge, 2006.  doi: 10.1017/CBO9780511803277.
[28]

Z. Toroczkai, G. Károlyi, A. Péntek, T. Tél, C. Grebogi and J. A. Yorke, Wada dye boundaries in open hydrodynamical flows, Physica A, 239 (1997), 235-243. 

[29]

J. Vandermeer, Wada basins and qualitative unpredictability in ecological models: a graphical interpretation, Ecol. Model., 176 (2004), 65-74. 

[30]

A. Wagemakers, A. Daza and M. A. F. Sanjuán, The saddle-straddle method to test for Wada basins, Commun. Nonlinear Sci. Numer. Simul., 84 (2020), 105167, 8 pp. doi: 10.1016/j.cnsns.2020.105167.

[31]

K. Yoneyama, Theory of continuous sets of points, Tokohu Math. J., 11 (1917), 43-158. 

[32]

Y. Zhang and G. Luo, Unpredictability of the Wada property in the parameter plane, Physics Letters A, 376 (2012), 3060-3066.  doi: 10.1016/j.physleta.2012.08.015.

[33]

Y. Zhang and G. Luo, Wada bifurcations and partially Wada basin boundaries in a two-dimensional cubic map, Phys. Lett. A, 377 (2013), 1274-1281.  doi: 10.1016/j.physleta.2013.03.027.

[34]

Y. Zhang, G. Luo, Q. Cao and M. Lin, Wada basin dynamics of a shallow arch oscillator with more than 20 coexisting low-period periodic attractors, International Journal of Non-Linear Mechanics, 58 (2014), 151-161.  doi: 10.1016/j.ijnonlinmec.2013.09.009.

[35]

P. Ziaukas and M. Ragulskis, Fractal dimension and Wada measure revisited: no straightforward relationships in ndds, Nonlinear Dynamics, 88 (2017), 871-882.  doi: 10.1007/s11071-016-3281-4.

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
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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
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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
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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
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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
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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
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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
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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
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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 $
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Table 1.  Results of the computation of the Wada merging method for different systems with fractal basin boudaries. Some of these examples show a fractal basin according to the merging method. All the basins have been computed with a finite resolution of $ 1000 \times 1000 $
Dynamical system $ max_d $ $ min_d $ $ (max_d - min_d)/ min_d $ Wada?
Forced pendulum $ N_A =3 $ 0.0365 0.0219 0.667 YES
Forced pendulum $ N_A =4 $ 0.368 0.0439 7.3826 NO
Forced pendulum $ N_A =8 $ 0.3976 0.0655 5.0702 NO
Hénon-Heiles Hamiltonian $ E_0 = 0.2 $ 0.0206 0.0168 0.2262 YES
Hénon-Heiles Hamiltonian $ E_0 = 0.3 $ 0.0240 0.0236 0.0169 YES
Newton method $ N_A=3 $ 0.0300 0.0240 0.2499 YES
Newton method $ N_A=4 $ 0.0402 0.0350 0.1485 YES
Newton method $ N_A=5 $ 0.0902 0.0420 1.1476 YES
Newton method $ N_A=6 $ 0.0780 0.0566 0.3780 YES
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Dynamical system $ max_d $ $ min_d $ $ (max_d - min_d)/ min_d $ Wada?
Forced pendulum $ N_A =3 $ 0.0365 0.0219 0.667 YES
Forced pendulum $ N_A =4 $ 0.368 0.0439 7.3826 NO
Forced pendulum $ N_A =8 $ 0.3976 0.0655 5.0702 NO
Hénon-Heiles Hamiltonian $ E_0 = 0.2 $ 0.0206 0.0168 0.2262 YES
Hénon-Heiles Hamiltonian $ E_0 = 0.3 $ 0.0240 0.0236 0.0169 YES
Newton method $ N_A=3 $ 0.0300 0.0240 0.2499 YES
Newton method $ N_A=4 $ 0.0402 0.0350 0.1485 YES
Newton method $ N_A=5 $ 0.0902 0.0420 1.1476 YES
Newton method $ N_A=6 $ 0.0780 0.0566 0.3780 YES
Table 2.  Comparison of the principal procedures to test if a basin of attraction has the Wada property. The time noted with $ ^* $ refers only to the computation time and does not take into account the previous study of the system
Name Type of system Dim. Computation What we need
time
Nusse-Yorke method [21] ODEs Hamiltonians Maps 2D 1$ ^* $ It requires a detailed knowledge of the basin and the boundaries (accessible unstable periodic orbit embedded in the basin boundary).
Grid method [10] Any dynamical system n-D 100 It requires the basins and the dynamical system to compute parts of the basin at a higher resolution.
Merging method [8] Any dynamical system n-D 0.01 It needs to know the basins, but not the dynamical system.
Saddle-straddle method [30] ODEs Hamiltonians Maps 2D 1 It needs to know the dynamical system, but not the basins.
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Name Type of system Dim. Computation What we need
time
Nusse-Yorke method [21] ODEs Hamiltonians Maps 2D 1$ ^* $ It requires a detailed knowledge of the basin and the boundaries (accessible unstable periodic orbit embedded in the basin boundary).
Grid method [10] Any dynamical system n-D 100 It requires the basins and the dynamical system to compute parts of the basin at a higher resolution.
Merging method [8] Any dynamical system n-D 0.01 It needs to know the basins, but not the dynamical system.
Saddle-straddle method [30] ODEs Hamiltonians Maps 2D 1 It needs to know the dynamical system, but not the basins.
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