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Strange Attractor

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An attracting set that has zero measure in the embedding phase space and has fractal dimension. Trajectories within a strange attractor appear to skip around randomly.

StrangeAttractors1
StrangeAttractors2

A selection of strange attractors for a general quadratic map

x_(n+1)=a_1+a_2x_n+a_3x_n^2+a_4x_ny_n+a_5y_n+a_6y_n^2
(1)
y_(n+1)=a_7+a_8x_n+a_9x_n^2+a_(10)x_ny_n+a_(11)y_n+a_(12)y_n^2
(2)

are illustrated above, where the letters A to Y stand for coefficients of the quadratic from -1.2 to 1.2 in steps of 0.1 (Sprott 1993c). These represent a small selection of the approximately 1.6% of all possible 25^(12) approx 6×10^(16) such maps that are chaotic (Sprott 1993bc).

See also

Basin of Attraction, Correlation Exponent, Fractal, Gingerbreadman Map, Hénon-Heiles Equation, Hénon Map, Lorenz Attractor, Lozi Map, Phase Space, Rössler Attractor, Standard Map, Wada Basin

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References

Benmizrachi, A.; Procaccia, I.; and Grassberger, P. "Characterization of Experimental (Noisy) Strange Attractors." Phys. Rev. A 29, 975-977, 1984.Dewdney, A. K. "Probing the Strange Attractors of Chaos." Sci. Amer. 235, 90-93, 1976.Farmer, J. D.; Ott, E.; and Yorke, J. A. "The Dimension of Chaotic Attractors." Physica 7D, 153, 1983.Gleick, J. "Strange Attractors." Chaos: Making a New Science. New York: Penguin Books, pp. 119-153, 1988.Grassberger, P. "On the Hausdorff Dimension of Fractal Attractors." J. Stat. Phys. 26, 173-179, 1981.Grassberger, P. and Procaccia, I. "Measuring the Strangeness of Strange Attractors." Physica D 9, 189-208, 1983a.Grassberger, P. and Procaccia, I. "Characterization of Strange Attractors." Phys. Rev. Let. 50, 346-349, 1983b.Lauwerier, H. Fractals: Endlessly Repeated Geometric Figures. Princeton, NJ: Princeton University Press, pp. 137-138, 1991.Peitgen, H.-O. and Richter, D. H. The Beauty of Fractals: Images of Complex Dynamical Systems. New York: Springer-Verlag, 1986.Pickover, C. "A Note on Rendering 3-D Strange-Attractors." Comput. & Graphics 12, 263, 1988.Sprott, J. C. Strange Attractors: Creating Patterns in Chaos. New York: Henry Holt, 1993a.Sprott, J. C. "How Common Is Chaos?" Phys. Lett. A 173, 21, 1993b.Sprott, J. C. "Automatic Generation of Strange Attractors." Comput. & Graphics 17, 325-332, 1993c. Reprinted in Chaos and Fractals, A Computer Graphical Journey: Ten Year Compilation of Advanced Research (Ed. C. A. Pickover). Amsterdam, Netherlands: Elsevier, pp. 53-60, 1998.Viana, M. "What's New on Lorenz Strange Attractors." Math. Intell. 22, 6-19.

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Strange Attractor

Cite this as:

Weisstein, Eric W. "Strange Attractor." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/StrangeAttractor.html

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