# Box-Counting Algorithm of the Hénon Map

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This Demonstration shows the orbit diagram (OD), the box counting diagram (BC), and the dimension estimation plot (DE) of the Hénon map [1–4]. It is known that the chaotic attractors in the Hénon map are neither area filling (dimension 2) nor a simple curve (of dimension 1) [4]. Therefore the dimensions of these complicated geometries must be non-integer values between 1 and 2, and the chaotic attractors are then called fractals or strange attractors [4, 5]. The capacity or box-counting dimension is the simplest possible way to measure such pathologies. It can be defined by

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Contributed by: Ki-Jung Moon (January 2014)

Open content licensed under CC BY-NC-SA

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## Details

The original Hénon map, which is given by , is a prototypical example of a two-dimensional dissipative map with chaotic behavior. It was proposed by M. Hénon in 1976 as a simplified model of the Poincaré map (or Poincaré section) for the Lorenz model [1,8]. Hénon showed that the chaotic attractor of the Hénon map with and (also called the Hénon attractor) exhibits strange behavior (self-similarity), the typical behavior of fractal sets, via successive enlargement of the local region of the orbit diagram. Therefore the Hénon attractor is not only *chaotic* but also *strange*. These "Hénon-like" attractors are widely distributed within the parameter range and . They can also be found in other parameter regions. The maximally chaotic attractor for the Hénon map is known to occur at and [9]. This Demonstration uses an alternative definition of the Hénon map with the same dynamics [9].

References

[1] M. Hénon, "A Two-Dimensional Mapping with a Strange Attractor," *Communications in Mathematical Physics*, 50(1), 1976 pp. 69–77.

[2] S. H. Strogatz, *Nonlinear Dynamics and Chaos*, New York: Perseus Books Publishing, 1994.

[3] K. T. Alligood, T. D. Sauer, and J. A. Yorke, *Chaos: An Introduction to Dynamical Systems*, New York: Springer, 1996.

[4] H.-O. Peitgen, H. Jurgens, and D. Saupe, *Chaos and Fractals: New Frontiers of Science*, 2nd ed., New York: Springer, 2004.

[5] B. Mandelbrot, *The Fractal Geometry of Nature*, San Francisco: W. H. Freeman, 1982.

[6] D. A. Russel, J. D. Hanson, and E. Ott, "Dimension of Strange Attractor," *Physical Review Letters*, 45(14), 1980 p. 1175.

[7] P. Grassberger and I. Procaccia, "Measuring the Strangeness of Strange Attractors," *Physica*, 9D(1–2), 1983 pp. 189–208.

[8] E. N. Lorenz, "Deterministic Nonperiodic Flow," *Journal of the Atmospheric Sciences*, 20(2), 1963 pp. 130-141.

[9] J. C. Sprott, "Maximally Complex Simple Attractors," *Chaos: An Interdisciplinary Journal of Nonlinear Science*, 17, 2007 p. 033124.

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