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.
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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