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Ziaukas, Pranas; Ragulskis, Minvydas

Fractal dimension and Wada measure revisited: no straightforward relationships in NDDS. (English) Zbl 1375.28023

Nonlinear Dyn. 88, No. 2, 871-882 (2017).
Summary: An extended Newton’s discrete dynamical system with a complex control parameter is investigated in this paper. A novel computational algorithm is introduced for the evaluation of Wada measure. A nontrivial relationship between the fractal dimension and the Wada measure is revealed in NDDS. It is demonstrated that the reduction of the fractal dimension of basin boundaries of coexisting attractors does not automatically imply a lower Wada measure of these boundaries. Computational experiments are used to illustrate what impact the complexity of the relationship between fractal dimension and Wada measure does have in practical applications.

Cited in 4 Documents

MSC:

28A80 Fractals
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior

Keywords:

chaotic attractor; fractal dimension; Wada property
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References:

[1] Aguirre, J., Sanjuán, M.A.: Unpredictable behavior in the Duffing oscillator: Wada basins. Phys. D: Nonlinear Phenom. 171(1-2), 41-51 (2002). doi:10.1016/S0167-2789(02)00565-1 · Zbl 1008.37011 · doi:10.1016/S0167-2789(02)00565-1
[2] Amrein, M., Wihler, T.P.: An adaptive Newton-method based on a dynamical systems approach. Commun. Nonlinear Sci. Numer. Simul. 19(9), 2958-2973 (2014). doi:10.1016/j.cnsns.2014.02.010 · Zbl 1510.65104 · doi:10.1016/j.cnsns.2014.02.010
[3] Atkinson, K.E.: An Introduction to Numerical Analysis. Wiley, Hoboken (1978) · Zbl 0402.65001
[4] Barnsley, M.F., Rising, H.: Fractals Everywhere. Morgan Kaufmann, Burlington (2000)
[5] Breban, R., Nusse, H.E.: On the creation of Wada basins in interval maps through fixed point tangent bifurcation. Phys. D: Nonlinear Phenom. 207(1-2), 52-63 (2005). doi:10.1016/j.physd.2005.05.012 · Zbl 1093.37012 · doi:10.1016/j.physd.2005.05.012
[6] Cartwright, J.H.: Newton maps: fractals from Newton’s method for the circle map. Comput. Gr. 23(4), 607-612 (1999). doi:10.1016/S0097-8493(99)00078-3 · doi:10.1016/S0097-8493(99)00078-3
[7] Cayley, A.: Application of the Newton-Fourier method to an imaginary root of an equation. Q. J. Pure Appl. Math 16, 179-185 (1879) · JFM 11.0067.02
[8] Chandra Sekhar, D., Ganguli, R.: Fractal boundaries of basin of attraction of Newton-Raphson method in helicopter trim. Comput. Math. Appl. 60(10), 2834-2858 (2010). doi:10.1016/j.camwa.2010.09.040 · Zbl 1207.65059 · doi:10.1016/j.camwa.2010.09.040
[9] Daza, A., Wagemakers, A., Sanjuán, M.A.F., Yorke, J.A.: Testing for Basins of Wada. Sci. Rep. 5, 16,579 (2015). doi:10.1038/srep16579 · doi:10.1038/srep16579
[10] Drexler, M., Sobey, I., Bracher, C.: On the fractal characteristics of a stabilised Newton method. Oxford University Computing Laboratory, Oxford (1995)
[11] Drexler, M., Sobey, I., Bracher, C.: Fractal Characteristics of Newton’s Method on Polynomials. Oxford University Computer Laboratory, Oxford (1996)
[12] Epureanu, B.I., Greenside, H.S.: Fractal basins of attraction associated with a damped Newton’s method. SIAM Rev. 40(1), 102-109 (1998). doi:10.1137/S0036144596310033 · Zbl 0912.65039 · doi:10.1137/S0036144596310033
[13] Fedaravičius, A.P., Cao, M., Ragulskis, M.: Control of a dendritic neuron driven by a phase-independent stimulation. Chaos Solitons Fractals 85, 77-83 (2016) · Zbl 1360.92025 · doi:10.1016/j.chaos.2016.01.017
[14] Frame, M., Neger, N.: Newton’s Method and the Wada property: a Graphical Approach. Coll. Math. J. 38(3), 192-204 (2007) · Zbl 1144.37038
[15] Gilbert, W.J.: Generalizations of Newton’s method. Fractals 09(03), 251-262 (2001). doi:10.1142/S0218348X01000737 · Zbl 1046.37027 · doi:10.1142/S0218348X01000737
[16] Holt, R., Schwartz, I.: Newton’s method as a dynamical system: global convergence and predictability. Phys. Lett. A 105(7), 327-333 (1984). doi:10.1016/0375-9601(84)90273-1 · doi:10.1016/0375-9601(84)90273-1
[17] Huber, P.J.: Robust Statistics. Springer, Berlin (2011) · doi:10.1007/978-3-642-04898-2_594
[18] Straffin Jr., P.: Newton’s method and fractal patterns. Applications of Calculus 3, 68-84 (1991)
[19] Kennedy, J., Yorke, J.A.: Basins of Wada. Phys. D 51(1-3), 213-225 (1991). doi:10.1016/0167-2789(91)90234-Z · Zbl 0746.58054 · doi:10.1016/0167-2789(91)90234-Z
[20] Krause, E.F.: Taxicab Geometry: An Adventure in Non-Euclidean Geometry. Courier Corporation, Mineola (1975)
[21] Landauskas, M., Ragulskis, M.: Clocking convergence to a stable limit cycle of a periodically driven nonlinear pendulum. Chaos: Interdisc. J. Nonlinear Sci. 22(3), 033,138 (2012) · doi:10.1063/1.4748856
[22] Nusse, H.E., Yorke, J.A.: Wada basin boundaries and basin cells. Phys. D 90(3), 242-261 (1996). doi:10.1016/0167-2789(95)00249-9 · Zbl 0886.58072 · doi:10.1016/0167-2789(95)00249-9
[23] Ott, E.: Chaos in Dynamical Systems. Cambridge University Press, Cambridge (2002) · Zbl 1006.37001 · doi:10.1017/CBO9780511803260
[24] Poon, L., Campos, J., Edward, O., Grebogi, C.: Wada basin boundaries in chaotic scattering. Int. J. Bifurc. Chaos 06(02), 251-265 (1996). doi:10.1142/S0218127496000035 · Zbl 0870.58069 · doi:10.1142/S0218127496000035
[25] Portela, S., Iber, E.: Fractal and Wada exit basin boundaries in tokamaks. Int. J. Bifurc. Chaos 17(11), 4067-4079 (2007). doi:10.1142/S021812740701986X · Zbl 1193.78017 · doi:10.1142/S021812740701986X
[26] Sarkar, N., Chaudhuri, B.: An efficient differential box-counting approach to compute fractal dimension of image. IEEE Trans. Syst. Man Cybern. 24(1), 115-120 (1994) · doi:10.1109/21.259692
[27] Schröder, E.: Ueber unendlich viele Algorithmen zur Auflösung der Gleichungen. Math. Ann. 2(2), 317-365 (1870). doi:10.1007/BF01444024 · JFM 02.0042.02 · doi:10.1007/BF01444024
[28] Sobey, I.J.: Characteristics of Newton’s method on polynomials. Oxford Computer Lab, Oxford (1996)
[29] Susanto, H., Karjanto, N.: Newton’s method’s basins of attraction revisited. Appl. Math. Comput. 215(3), 1084-1090 (2009). doi:10.1016/j.amc.2009.06.041 · Zbl 1175.65055 · doi:10.1016/j.amc.2009.06.041
[30] Sweet, D., Ott, E., Yorke, J.A.: Topology in chaotic scattering. Nature 399(6734), 315-316 (1999). doi:10.1038/20573 · Zbl 1369.37049 · doi:10.1038/20573
[31] Vandermeer, J.: Wada basins and qualitative unpredictability in ecological models: a graphical interpretation. Ecol. Model. 176(1-2), 65-74 (2004). doi:10.1016/j.ecolmodel.2003.10.028 · doi:10.1016/j.ecolmodel.2003.10.028
[32] Vandermeer, J., Stone, L., Blasius, B.: Categories of chaos and fractal basin boundaries in forced predator-prey models. Chaos Solitons Fractals 12(2), 265-276 (2001). doi:10.1016/S0960-0779(00)00111-9 · Zbl 0976.92033
[33] Walsh, J.: The dynamics of Newton’s method for cubic polynomials. Coll. Math. J. 26, 22-28 (1995) · Zbl 1291.65148 · doi:10.2307/2687287
[34] Wang, X., Yu, X.: Julia sets for the standard Newton’s method, Halley’s method, and Schröder’s method. Appl. Math. Comput. 189(2), 1186-1195 (2007). doi:10.1016/j.amc.2006.12.002 · Zbl 1122.65348 · doi:10.1016/j.amc.2006.12.002
[35] Zhang, Y., Luo, G.: Unpredictability of the Wada property in the parameter plane. Phys. Lett. A 376(45), 3060-3066 (2012). doi:10.1016/j.physleta.2012.08.015 · Zbl 1369.78641 · doi:10.1016/j.physleta.2012.08.015
[36] Zhang, Y., Luo, G.: Wada bifurcations and partially Wada basin boundaries in a two-dimensional cubic map. Phys. Lett. A 377(18), 1274-1281 (2013). doi:10.1016/j.physleta.2013.03.027 · Zbl 1290.37027 · doi:10.1016/j.physleta.2013.03.027
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