Box-Counting Algorithm of the Hénon Map

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
where is the number of boxes with size covering the attractor [5–7]. Here is the box-counting step and is the size of the box at the initial step .
• Drag the blue locator to change the position of the center .
• Drag the red locator to change the initial condition .
The sliders let you control the following:
• opacity, the opacity of the plotted points.
• point size, the size of the plotted points.
, the sizes of the horizontal and vertical plot range; .
, the number of iterations.
, the number of initial iterations to be dropped.
, the main control parameter value.
, the dissipation parameter value.
, the box-counting step; the size of the boxes at the box-counting step is given by .
, the maximum box-counting step.
The dropdown menu lets you control:
, the number of initial box-counting steps to be dropped.
And, finally, the resets let you reset the position of the locators.


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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].
[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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