Wagemakers, Alexandre; Daza, Alvar; Sanjuán, Miguel A. F. The saddle-straddle method to test for Wada basins. (English) Zbl 07261592 Commun. Nonlinear Sci. Numer. Simul. 84, Article ID 105167, 8 p. (2020); corrigendum ibid. 90, Article ID 105334, 1 p. (2020). Summary: First conceived as a topological construction, Wada basins abound in dynamical systems. Basins of attraction showing the Wada property possess the particular feature that any small perturbation of an initial condition lying on the boundary can lead the system to any of its possible outcomes. The saddle-straddle method, described here, is a new method to identify the Wada property in a dynamical system based on the computation of its chaotic saddle in the fractalized phase space. It consists of finding the chaotic saddle embedded in the boundary between the basin of one attractor and the remaining basins of attraction by using the saddle-straddle algorithm. The simple observation that the chaotic saddle is the same for all the combinations of basins is sufficient to prove that the boundary has the Wada property. Cited in 1 Review MSC: 37C75 Stability theory for smooth dynamical systems 37C70 Attractors and repellers of smooth dynamical systems and their topological structure 37C45 Dimension theory of smooth dynamical systems Keywords:Wada basins; fractalized phase space; basins of attraction Software:Dynamics PDF BibTeX XML Cite \textit{A. Wagemakers} et al., Commun. Nonlinear Sci. Numer. Simul. 84, Article ID 105167, 8 p. (2020; Zbl 07261592) Full Text: DOI References: [1] Yoneyama, K., Tokohu, Math J, 11, 43 (1917) [2] G. Hocking, J.; S. Young, G., Topology (1988), Dover, New York · Zbl 0718.55001 [3] Kuratowski, C., Fundamenta, Mathematicae, 6, 130 (1924) [4] Sanjuán, M. A.F.; Kennedy, J.; Ott, E.; YorkeA., J., Phys, Rev Lett, 78, 1892 (1997) [5] Kennedy, J.; Yorke, J. A., Physica, D, 51, 213 (1991) [6] Nusse, H. E.; Yorke, J. A., Physica, D, 90, 242 (1996) [7] Nusse, H. E.; Ott, E.; A. Yorke, J., Phys, Rev Lett, 75, 2482 (1995) [8] Nusse, H. E.; Yorke, J. A., Rev. lett, Phys, 626 (2000) [9] Poon, L.; Campos, J.; Ott, E.; Grebogi, C., J. Bifurcation Chaos, Int, 6, 251 (1996) [10] Aguirre, J.; Vallejo, J. C.; Sanjun, M. A.F., Phys, Rev E, 64, 066208 (2001) [11] Toroczkai, Z.; Krolyi, G.; Pntek, A.; Tl, T.; Grebogi, C.; Yorke, J. A., A, Physica, 239, 235 (1997) [12] Daza, A.; Wagemakers, A.; Sanjun, M. A.F., Nonlinear Sci. Numer. Simul, Commun, 43, 220 (2017) [13] Daza, A.; Wagemakers, A.; Sanjun, M. A.F.; Yorke, J. A., Rep, Sci, 5, 16579 (2015) [14] Daza, A.; Wagemakers, A.; Sanjuán, M. A.F., Rep, Sci, 8, 9954 (2018) [15] Zhang, Y.; Luo, G., Lett. a, Phys, 377, 1274 (2013) [16] Grebogi, C.; Ott, E.; Yorke, J. A.; E. Nusse, H., Ann. N.Y. Acad. Sci, 497, 117 (1987) [17] Grebogi, C.; Nusse, H. E.; Ott, E.; Yorke, J. A., Dynamical Systems. Lecture Notes in Mathematics, 220-250 (1988), Springer: Springer Berlin [18] Battelino, P. M.; Grebogi, C.; Ott, E.; Yorke, J. A.; Yorke, E. D., D, Physica, 32, 296 (1988) [19] Nusse, H. E.; Yorke, J. A., Dynamics: numerical explorations, Vol. 101 (2012), Springer [20] Edgar, G., Measure, Topology, and Fractal Geometry (2007), Springer: Springer New York [21] Friedman, J. H.; Bentley, J. L.; Finkel, R. A., Softw, ACM Trans Math, 3, 209 (1977) [22] Aguirre, J.; Viana, R. L.; Sanjuán, M. A.F., Phys, Rev Mod, 81, 333 (2009) [23] Daza, A.; Shipley, J. O.; Dolan, S. R.; Sanjuán, M. A.F., Rev. D, Phys, 98, 84050 (2018) [24] Kantz, H., Rev. E, Phys, 49, 5091 (1994) [25] Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P., Numerical Recipes 3rd edition: The Art of Scientific Computing (2007), Cambridge University Press · Zbl 1132.65001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.