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How to detect Wada basins
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 |
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.
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. |









Dynamical system | Wada? | |||
Forced pendulum |
0.0365 | 0.0219 | 0.667 | YES |
Forced pendulum |
0.368 | 0.0439 | 7.3826 | NO |
Forced pendulum |
0.3976 | 0.0655 | 5.0702 | NO |
Hénon-Heiles Hamiltonian |
0.0206 | 0.0168 | 0.2262 | YES |
Hénon-Heiles Hamiltonian |
0.0240 | 0.0236 | 0.0169 | YES |
Newton method |
0.0300 | 0.0240 | 0.2499 | YES |
Newton method |
0.0402 | 0.0350 | 0.1485 | YES |
Newton method |
0.0902 | 0.0420 | 1.1476 | YES |
Newton method |
0.0780 | 0.0566 | 0.3780 | YES |
Dynamical system | Wada? | |||
Forced pendulum |
0.0365 | 0.0219 | 0.667 | YES |
Forced pendulum |
0.368 | 0.0439 | 7.3826 | NO |
Forced pendulum |
0.3976 | 0.0655 | 5.0702 | NO |
Hénon-Heiles Hamiltonian |
0.0206 | 0.0168 | 0.2262 | YES |
Hénon-Heiles Hamiltonian |
0.0240 | 0.0236 | 0.0169 | YES |
Newton method |
0.0300 | 0.0240 | 0.2499 | YES |
Newton method |
0.0402 | 0.0350 | 0.1485 | YES |
Newton method |
0.0902 | 0.0420 | 1.1476 | YES |
Newton method |
0.0780 | 0.0566 | 0.3780 | YES |
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. |
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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