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 noninteger values between 1 and 2, and the chaotic attractors are then called fractals or strange attractors [4, 5]. The capacity or boxcounting 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 boxcounting 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 boxcounting step; the size of the boxes at the boxcounting step is given by . • , the maximum boxcounting step. The dropdown menu lets you control: • , the number of initial boxcounting 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 twodimensional 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 (selfsimilarity), 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 "Hénonlike" 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 TwoDimensional 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. 130141. [9] J. C. Sprott, "Maximally Complex Simple Attractors," Chaos: An Interdisciplinary Journal of Nonlinear Science, 17, 2007 p. 033124.
