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Visualizing chaos: Lyapunov surfaces and volumes
March/April 1990 (vol. 10 no. 2)
pp. 15-19

The use of a graphic technique to visualize chaos for a dual-parameter one-dimensional map is described. It involves plotting the Lyapunov exponent with both height and color in a three-dimensional map, as a function of the two parameters. Color is determined using a geographic lookup table. A graphics supercomputer can rotate the map in real time. The technique demonstrates graphically interesting behavior in chaotic systems.

1. A. Wolf et al., "Determining the Lyapunov Exponents from a Time Series,"Physica, Vol. 16D, 1985, pp. 285-317.
2. R. May, "Simple Mathematical Models with Very Complicated Dynamics,"Nature, Vol. 261, June, 1976, pp. 459-467.
3. C. Pickover, "The Use of Image Processing Techniques in Rendering Maps with Deterministic Chaos,"The Visual Computer, Vol 4, No. 5, Nov. 1988, pp. 271-276. See also C. Pickover, "Graphics, Bifurcation, Order and Chaos,"Computer Graphics Forum, Vol. 6, No. 1, Jan. 1987, pp. 26-33.
4. F. Moon,Chaotic Vibrations, John Wiley and Sons, New York, 1987, pp. 191-195.
5. J. Crutchfield, J. Farmer, and B. Huberman, "Fluctuations and Simple Chaotic Dynamics,"Physics Reports, Vol. 92, No. 2, 1989, pp. 45-82.
6. M. Markus and B. Hess, "Lyapunov Exponents of the Logistic Map with Periodic Forcing," to be published inComputers and Graphics, Vol. 13, No. 4,1989. See also F. Ward, "Images for the Computer Age,"National Geographic, Vol. 175, No. 6, June 1989, pp. 718-739.
7. M. Feigenbaum, "The Universal Metric Properties of Nonlinear Transformation,"J. Statistical Physics, Vol. 21, 1979, pp. 669-706.
8. M. Feigenbaum, "Universal Behavior in Nonlinear Systems,"Los Alamos Science, Vol. 1, 1981, pp. 4-27.
9. C. Pickover, "A Note on Computer Experiments with Chaotic Shattering of Level Sets,"Computers in Physics, Vol. 3, No. 6, Nov./Dec., 1989, pp. 69-73.

Clifford A. Pickover, "Visualizing chaos: Lyapunov surfaces and volumes," IEEE Computer Graphics and Applications, vol. 10, no. 2, pp. 15-19, March-April 1990, doi:10.1109/38.50669
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